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Escape of mass on Hilbert modular varieties Zaman, Asif Ali 2012

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Escape of Mass on Hilbert Modular Varieties by Asif Ali Zaman B.Sc., Simon Fraser University, 2010 a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the faculty of graduate studies (Mathematics) The University Of British Columbia (Vancouver) August 2012 c Asif Ali Zaman, 2012Abstract Let F be a number  eld, G = PGL(2; F1), and K be a maximal compact subgroup of G. We eliminate the possibility of escape of mass for measures associated to Hecke-Maa cusp forms on Hilbert modular varieties, and more generally on congruence locally symmetric spaces covered by G=K, hence enabling its application to the non-compact case of the Arithmetic Quantum Unique Ergodicity Conjecture. This thesis generalizes work by Soundararajan in 2010 eliminating escape of mass for congruence surfaces, including the classical modular surface SL(2;Z)nH2, and follows his approach closely. First, we de ne M , a congruence locally symmetric space covered by G=K, and articulate the details of its structure. Then we de ne Hecke-Maass cusp forms and provide their Whittaker expansion along with identities regarding the Whittaker coe cients. Utilizing these identities, we introduce mock P-Hecke multiplicative functions and bound a key related growth measure following Soundararajan’s paper. Finally, amassing our results, we eliminate the possibility of escape of mass for Hecke-Maa cusp forms on M . iiTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 Hyperbolic 2- and 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1 Hyperbolic 2-space and PGL(2;R) . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Hyperbolic 3-space and PGL(2;C) . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Congruence Locally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . 11 2.1 Symmetric Space of G = PGL(2; F1) . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Congruence Subgroups and their Cusps . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Cusp Stabilizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Distance to Cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 Fundamental Domain of  nX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Hecke-Maa Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Maa Forms and their Fourier Expansion . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Maa Cusp Forms and their Whittaker Expansion . . . . . . . . . . . . . . . . . 33 3.3 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Hecke-Maa Cusp Forms and their Whittaker Coe cients . . . . . . . . . . . . . 43 iii4 Mock P-Hecke Multiplicative Functions . . . . . . . . . . . . . . . . . . . . . 48 4.1 Statement of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 A Large Set of Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5 Elimination of Escape of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.1 Decay High in the Cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 No Escape of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 ivList of Figures Figure 2.1 A fundamental domain for SL(2;Z)nH2 . . . . . . . . . . . . . . . . . . . . . 12 Figure 2.2 Depiction of  nX with 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . 29 vList of Symbols a integral ideal associated to  2 P1(F ), except in Chapter 4 . . . . . . . . . . . . . . . . . . . . 16 Av area measure of Fv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 AQUE Arithmetic Quantum Unique Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 B v (sv; yv) special function composed of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B (s; y) Q vj1B  v v (sv; yv) for  2 f g m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 cP large positive constant depending only on P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 c ( ; )  -Whittaker coe cient of  at cusp  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 c( ) ( ; n) n-Whittaker ideal coe cient of  at cusp  for  2 O =V . . . . . . . . . . . . . . . . . . . . 47 C  eld of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 DF absolute discriminant of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 D fundamental domain for  nX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 D(y) deep in a cusp, D(y) = f(x;y) 2 D : Ny 2 (y;1)g . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 D absolute di erent ideal of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 e(x) e(x) = e2 ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33 f mock P-Hecke multiplicative function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 fP P-Hecke multiplicative function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 F number  eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Fv completion of F at place v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 F1 F1 = Q vj1 Fv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 F fundamental domain for  nX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 F(Y ) function associated to mock P-Hecke multiplicative function f . . . . . . . . . . . . . . . 50 g element of PGL(2; F ) such that g (1) =  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 G G = Q vj1Gv = PGL(2; F1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Gv Gv = PGL(2; Fv) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 hF class number of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 H2 hyperbolic 2-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 H3 hyperbolic 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 viH Hamilton’s quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 I (y) I-Bessel function,  2 C; y 2 R>0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 Im imaginary part of H2;H3 or X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 J well chosen positive integer based on F(Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 K Q vj1Kv, maximal compact subgroup of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Kv maximal compact subgroup of Gv for v j 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 K (y) K-Bessel function,  2 C; y 2 R>0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 m number of archimedean places of F , i.e. m = r1 + r2 . . . . . . . . . . . . . . . . . . . . . . . . . 11 M congruence locally symmetric space, i.e. M =  nX . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 n degree of F , i.e. n = [F : Q] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 N integral ideal of F representing level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 N global norm, i.e. N = NFQ, extended to X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Nv local norm at v j 1, i.e. N = NFvR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 N P-friable ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 NN PN-friable ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 O ring of integers of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 O group of integral units of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 O 1 roots of unity of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 p prime ideal of F , often principal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 P subset of unrami ed prime ideals not dividing N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 PN set of unrami ed principal prime ideals not dividing N . . . . . . . . . . . . . . . . . . . . . . . 41 P(Y ) fp 2 P : Np 2 [ p Y =2; p Y ]g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Pj suitably chosen subset of P(Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57 P1(F ) projective linear space of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Q  eld of rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 QUE Quantum Unique Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 r1 number of real places of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 r2 number of complex places of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 R  eld of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 sv  v = sv(1 sv) 2 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 s s = (sv)vj1 2 Cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 S(y) compact centre of  nX , grows as y !1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 T (n) n-Hecke operator (usually for n 2 NN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 T (y) y-coordinate of D(y), i.e. fy 2 Rm>0 : by 2 V;Ny 2 (y;1)g . . . . . . . . . . . . . . . . . . . . 36 Tr global trace, i.e. Tr = TrFQ, extended to F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Trv local trace for v j 1, i.e. Trv = Tr Fv R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 viiU fundamental domain for (O 1 \ V )nF1=N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 v j 1 archimedean place of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 V V = (1 + N) \ O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 V fundamental domain for V n bY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 vol(z) volume measure for z 2 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 V1(x) volume measure for x-coordinate in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 V2(y) volume measure for y-coordinate in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 W (s; z) full Whittaker function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Wv(sv; zv) local Whittaker function at v j 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 x x = (xv)vj1 2 F1, coordinate of z = (x;y) 2 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 X symmetric space of G, equals (H2)r1  (H3)r2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 y0 (small) positive constant depending only on F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 y y = (yv)vj1 2 Rm>0, coordinate of z = (x;y) 2 X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 by by = (yv=(Ny)1=n)vj1 2 bY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Y0 (large) positive constant depending only on F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 bY fy 2 Rm>0 : Ny = 1g, hyperplane in R m >0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 z element of X , alternate coordinates z = (x;y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Z ring of rational integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1  v Laplacian at place v j 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  (z) Hecke-Maa cusp form on  nX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43   (z)   (z) =  (g z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 c  (y; )  -Fourier coe cient of  at cusp  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  congruence group of level N, often  =  0(N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  (N) principal congruence subgroup of level N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  0(N) speci c congruence group of level N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41  ( )  ( ) = g 1  g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17  F  F = PGL(2;O) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  v  v-eigenvalue of Maa form  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31   = ( v)vj1 2 Cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31   (n) T (n)-Hecke eigenvalue of  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43  subgroup of  ( )-stabilizer of 1 2 P1(F ) for any  2 P1(F ) . . . . . . . . . . . . . . . . . . 18   measure associated to a Hecke-Maa cusp form  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66  set of inequivalent representatives of  nP1(F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27  ( ; z) function measuring closeness of z 2 X to  2 P1(F ) . . . . . . . . . . . . . . . . . . . . . . . . . . 22  element of P1(F ) regarded as cusp of  , often  = ( :  ) . . . . . . . . . . . . . . . . . . . . 15 h ;  i trace form billinear pairing on F1  F1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 viiiAcknowledgments First and foremost, I am deeply indebted to Professor Lior Silberman for providing me with extraordinary amounts of time and patience, sharing his seemingly endless bounty of mathematical knowledge, and challenging me with an exciting and multi-faceted problem. I have learned a great deal from our many conversations and feel privileged to have studied under his supervision. I am also very grateful for the funding I have received from NSERC, the department, and Professor Silberman. In addition, I would like to thank Professor William Casselman and Professor Julia Gordon for their generous mathematical and professional advice, kindness, and continuous support. With Professor Silberman, they organized many seminars which were both enriching and enter- taining. I would like to specially acknolwedge Dr. Gordon for reading my thesis and providing many helpful comments. Studying at the Department of Mathematics at UBC has been wonderful with the ex- tremely helpful and friendly department sta , particularly Lee Yupitun and Marlowe Dickson. Involvement in the graduate student and number theory community has been a great personal and educational experience, punctuated with deep (non-)mathmatical discussions, for which I would like to thank Carmen Bruni, Vince Chan, Robert Fraser, Mario Garcia Armas, Amir Ghadermarzi, Kyle Hambrook, Alia Hamieh, Nick Harland, Jay Heumann, Zheng Li, Guillermo Mantilla-Soler, Gourab Ray, Jim Richardson, Lance Robson, Malcolm Rupert, Vasu Tewari, Alex Tomberg, and Erick Wong amongst many others. Simply put, I am only here because of my parents, who have been sel essly supportive throughout my life, let alone my education. I dedicate this thesis to them, as their hardships and struggles have always been for my future well-being. Also, I am grateful for my wonderful in-laws who have become a new source of encouragement and support. Finally, I would like to thank my wife, Mubnii, for her constant love, understanding, patience and inspiration; you are always more than I can ever expect. ixTo my parents xIntroduction The methodologies and techniques employed in this thesis are primarily number theoretic in nature, but the fundamental phenomenon in question, namely \escape of mass", is a com- monplace concept in real analysis. Let M be a topological measure space, and f jg1j=1 be a sequence of measures on M . There exists many notions of convergence of measures, but in this thesis, we are concerned with a speci c type, namely weak- convergence. We say the sequence  j converges to a measure  in the weak- topology if for every compact set A  M , we have Z A d j ! Z A d ; as j !1: More simply, we say  j weak- converges to  , or write  j wk-  !  . If one assumes all  j are probability measures, i.e. R M d j = 1 and  j wk-  !  , then it is a natural question to ask: Is  still a probability measure? In other words, is R M d = 1? In these general terms, the question is a simple exercise in real analysis. If M is compact, then it follows immediately from the de nition of weak- convergence that  is indeed a probability measure. However, if M is not compact, then the answer is: not necessarily. Since f jg1j=1 are probability measures, it follows that R M d  1, but it is certainly possible that this inequality is strict, or in other words, there may be escape of mass. A simple example demonstrating this phenomenon would be taking M := R and d j := 1[j;j+1](x)dx for j  1 where dx is the Lebesgue measure. With this choice, one can easily see that  j weak- converges to the zero measure, and so have \lost mass". Thus, in general, the set of probability measures is not closed in the weak- topology. Our intention is to study this phenomenon known as \escape of mass" in a setting with number theoretic origins and implications, where further restrictions are placed on the measures  j and manifold M . Beginning with the classical case, let  be a congruence group (e.g. 1SL(2;Z)) and H2 denote the upper half plane of complex numbers endowed with the hyperbolic metric. The negatively curved manifold M =  nH2 is called a congruence surface. It is well known that M is non-compact and so, in general, escape of mass can occur. However, we restrict our attention to probability measures of number-theoretic importance, namely those associated to Hecke-Maa cusp forms on M (see Chapter 3 for details). A Hecke-Maa cusp form is a function  2 L2( nH2), which is a simultaneous eigenfunction of the hyperbolic Laplacian  , and every Hecke operator T (n); n 2 Z+. Further, to  , we can associate the probability measure d  := j (z)j2dvol(z) jj jj2L2 where dvol(z) is the hyperbolic area measure on H2. Then one may ask if escape of mass can occur for such probability measures, which is addressed in the following theorem: Theorem (Soundararajan, 2010 [Sou10]). Let   SL(2;Z) be a congruence group. Suppose f jg1j=1 is a sequence of Hecke-Maa cusp forms on  nH 2, and   j wk-  !  for some measure  . Then  is a probability measure. In other words, no escape of mass occurs. Utilizing the multiplicative properties of the Hecke eigenvalues of  inherited from the Hecke operators fT (n) : n 2 Z+g, Soundararajan was able to eliminate the possibility of escape of mass by applying deceptively simple yet ingenious elementary number theory. This key result and its methods are the basis of this thesis work and the related results. Now, the primary application of Soundararajan’s result is completing the proof of Arithmetic Quantum Unique Ergodicity (AQUE) and in fact, the problem of AQUE was the motivation to prove such a result. To describe AQUE, let us begin with the initial conjecture of Quantum Unique Ergodicity (QUE), which has been widely studied since it was  rst stated by Rudnick and Sarnak in 1994. Conjecture (Quantum Unique Ergodicity, [RS94]). Let M be a compact negatively curved manifold M with Laplacian  and volume measure vol. Suppose f jg1j=1  L 2(M) is a sequence of eigenfunctions of  with eigenvalues  j !1 and with associated probability measures d  j = j j(z)j2dvol(z) jj j jj2L2 : If   j wk-  !  , then  = vol. As stated in these general terms, QUE remains elusive without further restrictions on M , and seemingly has little to do with number theory. Instead, we are concerned with the same 2conjecture, but further assume M is a congruence surface, giving rise to the simpler conjecture with connections to number theory: Conjecture (Arithmetic Quantum Unique Ergodicity, [RS94]). Let M be a congruence hy- perbolic surface with Laplacian  and volume measure vol. Suppose f jg1j=1  L 2(M) is a sequence of eigenfunctions of  with associated probability measures   j . If   j wk-  !  , then  = vol. Not long after this conjecture, the combined works of Luo and Sarnak [LS95], and Jakobson [Jak94] on rmed AQUE for the continuous spectrum of  . However, AQUE for the discrete spectrum remained unknown until in 2006, Lindenstrauss [Lin06] proved AQUE for Hecke-Maa cusp forms  except for the possibility of escape of mass. Theorem (Lindenstrauss, 2006 [Lin06]). Let M be a congruence surface. Suppose f jg1j=1  L2(M) is a sequence of Hecke-Maa cusp forms with Laplace eigenvalues  j !1. If   j wk-  !  , then  = c  vol for some c 2 [0; 1]. If M is a compact surface, one can conclude c = 1 by the de nition of weak- convergence, thus completing the proof of AQUE for compact congruence hyperbolic surfaces M . However, in the non-compact case, one must necessarily guarantee  is a probability measure, i.e. eliminate escape of mass, to conclude c = 1 and hence complete the proof of AQUE. Several years later, the aforementioned theorem of Soundarajan proves exactly this fact yielding Corollary (Lindenstrauss [Lin06], Soundararajan [Sou10]). AQUE holds for Hecke-Maa cusp forms on congruence surfaces. For a more detailed overview of the QUE conjecture, one should see an article by Sarnak [Sar11], and for further details on AQUE, there are notes by Einsiedler and Ward [EW10] or an article by Marklof [Mar06]. Now, this thesis is concerned with generalizations of Soundararajan’s theorem to higher dimensional analogues of congruence hyperbolic surfaces M =  nH2. A natural and well known analogue is a Hilbert modular variety. For a totally real number  eld F of degree n, let O denote the ring of integers. Then the group SL(2;O) acts discretely on the n-fold product of upper half planes, (H2)n via the n embeddings of F (see Chapter 2). The  nite volume manifold M = SL(2;O)n(H2)n is known as the Hilbert modular variety of F . In the case F = Q, the  eld of rational numbers, this is simply the modular surface. As in Chapter 3, Hecke-Maa cusp forms can be de ned more generally on Hilbert modular varieties, and so one can conjecture AQUE holds for Hecke-Maa cusp forms on Hilbert modular varieties. 3In fact, one can allow F to be an arbitrary number  eld with r1 real embeddings and r2 complex embeddings so n = r1 + 2r2. Replacing (H2)n by X := (H2)r1  (H3)r2 where H3 is hyperbolic 3-space, we may again de ne a  nite volume manifold  F nX where  F = PGL(2;O). Even more generally, one can replace  F by congruence subgroups  of PGL(2;O) and study the manifolds M =  nX , which we call congruence locally symmetric spaces. Again, Hecke-Maa cusp forms can be de ned in this case, so we arrive at the following generalized conjecture. Conjecture 1. AQUE holds for Hecke-Maa cusp forms on congruence locally symmetric spaces, and hence on Hilbert modular varieties as well. In principle, as remarked by Sarnak [Sar11], a theorem of the form of Lindenstrauss’ [Lin06] can be established for this conjecture by following methods of [Lin06], [EKL06] and [BL03]. Therefore, in order to positively answer the above conjecture, one must eliminate escape of mass on congruence locally symmetric spaces as Soundararajan did for congruence surfaces. This elimination of escape of mass for congruence locally symmetric spaces is the central result of this thesis. Theorem. Let M be a congruence locally symmetric space, and let f jg1j=1  L 2(M) be Hecke- Maa cusp forms of M with associated probability measures   j . If   j wk-  !  for some measure  , then  is a probability measure. In other words, there is no escape of mass. In proving the main theorem above, we will very closely follow the approach of Soundarara- jan [Sou10] in the classical case except for one main distinction. For a Hecke-Maa cusp form  on the modular surface SL(2;Z)nH2, the Whittaker coe cients fc ( ;n) : n 2 Zg at a cusp  of  can be identi ed with its Hecke eigenvalues   (n); namely, for all n 2 Z, c ( ;n) = C    (n) where C 2 C is some constant. As a result, c ( ;n) inherits the multiplicative properties of   (n), on which Soundararajan’s result critically relies. Now, in our case, for a Hecke-Maa cusp form  on a congruence locally symmetric space  nX of level N, this identi cation is no longer true. Instead, the Whittaker coe cients fc ( ; ) :  2 (DN) 1g at a cusp  possess a less restrictive relation with the Hecke eigen- values f  (n) : n  O ideal g. Namely, suppose an integral ideal n is composed of unrami ed principal prime ideals not dividing the level of  , so n = ( ) is itself principal. If  2 (DN) 1 is a unit modulo n, then c ( ;  ) = c ( ; )    (n): 4This thesis introduces objects related to this weaker identity, called mock P-Hecke multiplicative functions, and adapts Soundararajan’s argument to this scenario, producing the key technical result analogous to Theorem 3 of Soundararajan [Sou10]: Theorem. Let P be a set of unrami ed prime ideals of a number  eld F not dividing the integral ideal N, and f be a mock P-Hecke multiplicative function of level N. If P has positive natural density, then X Nn y=Y jf(n)j2  P  1 + log Y p Y  X Nn y jf(n)j2: for 1  Y  y. On a separate note, while Soundararajan’s proof is written for SL(2;Z)nH2, it is apparent that the argument can be easily adjusted to any congruence surface  nH2. The proof we provide shall explicitly take into account the level of  and hence any congruence locally symmetric space  nX . Before embarking on the proof eliminating escape of mass, we summarize the contents of this thesis. First, in Chapter 1, we review necessary material about hyperbolic 2- and 3-space. Second, in Chapter 2, we precisely de ne a congruence locally symmetric space M , and describe its structure as a  nite number of cusps and a compact centre. In Chapter 3, we de ne Hecke- Maass cusp forms on M , provide their Whittaker expansion, and show the relations between coe cients and Hecke eigenvalues. Then in Chapter 4, we establish the key technical result on mock P-Hecke multiplicative functions. Finally, Chapter 5 is the culmination of the proof eliminating escape of mass. 5Chapter 1 Hyperbolic 2- and 3-space This section reviews necessary and well-known material regarding the geometry of hyperbolic 2- and 3-space and related group actions. The information is drawn from [Iwa02] for hyperbolic 2-space, and from [EGM98] for hyperbolic 3-space. General theory for hyperbolic n-space can be found in [BH99], [Rat94], [BP92] and [CFKP97]. 1.1 Hyperbolic 2-space and PGL(2;R) To model hyperbolic 2-space, i.e. the maximally symmetric, simply connected, 2-dimensional Riemannian manifold with constant sectional curvature  1, we shall use the upper half space model, namely H2 := fz = (x; y) : x 2 R; y 2 R>0g with the metric ds2 = y 2(dx2 + dy2): We may regard H2 as a subset of the complex numbers C by identifying (x; y) 7! x + iy, i.e. i = (0; 1). We also de ne the map Im : H2 ! R>0 (x; y) 7! y: The associated volume measure is given by dvol(z) = dA(x)dy y2 where dA(x) = dx; (1.1) 6and the associated Laplace-Beltrami operator is given by  = y2  @2 @x2 + @2 @y2  : (1.2) Closely related to H2 is the group PGL(2;R) := GL(2;R)=f I :  2 R g which acts transitively on the upper half plane H2  C via fractional linear transformations: g  z := 8 >>< >>: az + b cz + d det g > 0 az + b cz + d det g < 0 ; for g =  a b c d ! 2 PGL(2;R) and z 2 H2; Note that this action is independent of the choice of representative for g, since scalar matrices act trivially. Since the PGL(2;R)-action on H2 is transitive, we have that the orbit of i = (0; 1) 2 H2 is PGL(2;R)  (0; 1) = H2: By direct calculations, one can verify that the stabilizer of (0; 1) 2 H2 is PO(2;R) := fA 2 GL(2;R) : ATA = Ig=f Ig; so we may identify the quotient group PGL(2;R)=PO(2;R) with H2 via the map gPO(2;R) 7! g  (0; 1). Equipping the quotient group with the quotient topology, this identi cation is in fact a di eomorphism. We summarize our conclusions in the following proposition. Proposition 1.1. The space H2, equipped with the metric ds, is di eomorphic to PGL(2;R)=PO(2;R), equipped with the natural quotient topology. Further, the di eomorphism may be chosen so that the PGL(2;R) action on H2 transfers to an action by left multiplication on PGL(2;R)=PO(2;R). Now, consider the space of functions C1(H2), on which the Laplace-Beltrami operator  acts naturally. The group PGL(2;R) also acts on naturally on this space: for g 2 PGL(2;R) and  (z) 2 C1(H2), we de ne g   (z) :=  (g  z): Either by direct computation or realizing  as an element of the center of the universal en- veloping algebra of the Lie algebra associated to PGL(2;R), one may verify that the action of 7 commutes with the action of PGL(2;R) on C1(H2). Proposition 1.2. The Laplace-Beltrami operator  commutes with the action of PGL(2;R) on C1(H2), which is de ned by g   (z) =  (g  z) for g 2 PGL(2;R) and  (z) 2 C1(H2). As a  nal note, the PGL(2;R)-action on H2 preserves the volume measure dvol(z), which one can show through direct computation. Proposition 1.3. The volume measure dvol(z) of H2 is invariant under the action of PGL(2;R), i.e. dvol(g  z) = dvol(z) for all g 2 PGL(2;R). 1.2 Hyperbolic 3-space and PGL(2;C) To model hyperbolic 3-space, i.e. the maximally symmetric, simply connected, 3-dimensional Riemannian manifold with constant sectional curvature  1, we shall utilize the set H3 := fz = (x; y) : x 2 C; y 2 R>0g with the metric ds2 = y 2(dx21 + dx 2 2 + dy 2) where x = x1 + ix2. We may regard H3 as a subset of the Hamilton’s quaternions H by identifying (x; y) 7! x+ jy, i.e. j = (0; 1). We also de ne the map Im : H3 ! R>0 (x; y) 7! y: The associated volume measure is given by dvol(z) = dA(x)dy y3 where dA(x) = dx1dx2; (1.3) and the associated Laplace-Beltrami operator is given by  = y2  @2 @x21 + @2 @x22 + @2 @y2   y @ @y : (1.4) 8Closely related to H3 is the group PGL(2;C) := GL(2;C)=f I :  2 C g which acts on H3  H via fractional linear transformations: g  z := (az + b)(cz + d) 1; for g =  a b c d ! 2 PGL(2;C) and z 2 H3; where one considers the point at 1 in a natural limiting sense. One can verify (or force by de nition) that scalar matrices act trivially, and so the action is independent of the choice of representative of g. Since the PGL(2;C)-action on H3 is transitive, we have that the orbit of (0; 1) 2 H3 is PGL(2;C)  (0; 1) = H3: By direct calculations, one can verify that the stabilizer of (0; 1) 2 H3 is PU(2;C) := fA 2 GL(2;C) : AHA = Ig=f I :  2 C ; j j = 1g; so we may identify the quotient group PGL(2;C)=PU(2;R) with H3 via the map gPU(2;C) 7! g  (0; 1). Equipping the quotient group with the quotient topology, this identi cation is in fact a di eomorphism. We summarize our conclusions in the following proposition. Proposition 1.4. The space H3, equipped with the metric ds, is di eomorphic to PGL(2;C)=PU(2;C), equipped with the natural quotient topology. Further, the di eomorphism may be chosen so that the PGL(2;C) action on H3 transfers to an action by left multiplication on PGL(2;C)=PU(2;C). Now, consider the space of functions C1(H3), on which the Laplace-Beltrami operator  acts naturally. The group PGL(2;C) also acts on naturally on this space: for g 2 PGL(2;C) and  (z) 2 C1(H3), we de ne g   (z) :=  (g  z): Either by direct computation or realizing  as an element of the center of the universal en- veloping algebra of the Lie algebra associated to PGL(2;C), one may deduce that the action of  commutes with the action of PGL(2;C) on C1(H3). Proposition 1.5. The Laplace-Beltrami operator  commutes with the action of PGL(2;C) 9on C1(H3), which is de ned by g   (z) =  (g  z) for g 2 PGL(2;C) and  (z) 2 C1(H3). As a  nal note, the PGL(2;C)-action on H3 preserves the volume measure dvol(z), which one can show through direct computation or general theory. Proposition 1.6. The volume measure dvol(z) of H3 is invariant under the action of PGL(2;C), i.e. dvol(g  z) = dvol(z) for all g 2 PGL(2;R). 10Chapter 2 Congruence Locally Symmetric Spaces This chapter is dedicated to generalizing our understanding of a fundamental domain for the classical modular surface SL(2;Z)nH2 (see Figure 2.1) to a congruence locally symmetric space M =  nX . We describe in detail a congruence locally symmetric space M =  nX utilizing the models of hyperbolic 2- and 3-space, and  xing notation for various parametrizations of M . Much of the discussion will be focused on the cusps of M and the precise structure of the cusp stabilizer fundamental domain. Finally, we provide a complete description of a fundamental domain for M as a disjoint union consisting of a compact centre, and a  nite number of cusps. The material on congruence locally symmetric spaces is derived from Chapter 1 of [Gee88]; other sources include [Hir73] and [Fre90]. The classical case F = Q is discussed in [Miy89], [Iwa02] and [IK04]. Any necessary algebraic number theory can be found in [Neu99], [Lan94], [Mar77], or [ME05]. 2.1 Symmetric Space of G = PGL(2; F1) For the remainder of this thesis,  x a number  eld F of degree n (i.e. [F : Q] = n) with r1 real embeddings and 2r2 complex embeddings, so n = r1 + 2r2. For each place v of F , denote Fv to be the completion of F with respect to v. De ne the groups Gv := PGL(2; Fv) := GL(2; Fv) .  I :  2 F v  We are interested in the m := r1 + r2 archimedean places v of F , denoted v j 1. For v j 1, 11Figure 2.1: A fundamental domain for SL(2;Z)nH2 Source: c Fropu , 2004, by permission. Retrieved 25 April 2012 from Wikipedia. http://en.wikipedia.org/wiki/File:ModularGroup-FundamentalDomain-01.png we choose an explicit maximal compact subgroup of Gv, namely Kv := 8 < : PO(2;R) v real PU(2;C) v complex ; where PO(2;R) = O(2;R)=Z(O(2;R)) =  A 2 GL(2;R) : ATA = I  =f Ig; PU(2;C) = U(2;C)=Z(U(2;C)) = fA 2 GL(2;C) : A A = Ig =   I :  2 C ; j j = 1  : De ne G := PGL(2; F1) = Y vj1 Gv; K := Y vj1 Kv; 12where F1 = Q vj1 Fv. This yields the following di eomorphism: G=K = Y vj1 Gv=Kv  = r1Y j=1 H2  r2Y j=1 H3 such that through the above map, the action of G on G=K by left multiplication transfers to a component-wise action of G on (H2)r1  (H3)r2 as described in Chapter 1. The space X = XF := (H2)r1  (H3)r2 will be referred to as the symmetric space of G. We collect the facts and notation from Chapter 1 to provide a su cient and consistent description of X . Elements of X will be written in the following coordinates: z = (zv)vj1 2 X where zv = (xv; yv) 2 8 < : H2 v real H3 v complex : Often, we will decompose these coordinates so that z = (x;y) 2 X where x = (xv)vj1 2 F1; and y = (yv)vj1 2 R m >0: With this parametrization, we de ne the map Im : X ! Rm>0 (x;y) 7! y: so Im(z) = (Imv(zv))vj1. The volume measure associated to X is simply the product measure dvol(z) = ^vj1dvolv(zv), or in (x;y)-coordinates is given by dvol(z) = dV1(x) dV2(y) where dV1(x) := ^ vj1 dAv(xv) and dV2(y) := ^ vj1 1 Nvyv  dyv yv : (2.1) Note Nv extends the local norm N Fv R of a place v j 1 to H 2 for v real and to H3 for v complex by Nvzv = 8 < : zv v real zvzv v complex : 13We shall also extend the global norm N = NFQ to z 2 X via the usual product formula: Nz = Y vj1 Nvzv: As a simple remark, for r 2 R>0, the substitution y 7! r  y := (ryv)vj1 yields dV2(r  y) = ^ vj1 1 Nv(ryv) dyv yv =  Y vj1 1 r[Fv :R]  dV2(y) = r ndV2(y): (2.2) Finally,  v shall denote the Laplacian associated to the zv-coordinate of z 2 X . 2.2 Congruence Subgroups and their Cusps Consider the group PGL(2; F ) := GL(2; F )=   I :  2 F  which can be naturally embedded PGL(2; F ) ,! Y vj1 PGL(2; Fv) = G (2.3) via the usual embeddings F ,! Q vj1 Fv. Through this embedding, PGL(2; F ) acts on G=K by left multiplication, and hence acts on X . We aim to understand the action of a class of discrete subgroups of PGL(2; F ) acting on X . Let O denote the ring of integers of F , and de ne the distinguished group  F := PGL(2;O) =  A 2M2 2(O) : detA 2 O   =   I :  2 O  : De nition 2.1. Let N be an integral ideal of O. De ne a subgroup  of  F = PGL(2;O) to be a congruence subgroup of level N if  contains  (N) := fA 2 GL(2;O) : A  I (mod N)g =   I :  2 (1 + N) \ O  : the principal congruence subgroup of level N. Note  (N) is the kernel of the reduction map PGL(2;O)! PGL(2;O=N): If the level of  is unspeci ed, we simply call  a congruence subgroup. 14Remark. The Hilbert modular group PSL(2;O) = fA 2M2 2(O) : detA = 1g . f Ig : is a congruence subgroup of level O. Proposition 2.2. Every congruence subgroup  of level N is  nite index in  F = PGL(2;O), and discrete in G = Q vj1 PGL(2; Fv). Proof. The  nite index property follows immediately from the reduction map, since [ F :  (N)] = #PGL(2;O=N) <1, so any intermediate subgroup is also  nite index. For the discreteness, it su ces to prove the result for  =  F by the  nite index property. This fact is immediate as  F = PGL(2;O) ,! Y vj1 PGL(2; Fv) = G via F ,! Q vj1 Fv, and noting O embeds discretely under this map. Suppose   PGL(2;O) is a congruence subgroup of level N, so from (2.3),  possesses an action on X . We aim to understand the structure of M =  nX , which is equivalent to the double coset space  nG=K. We call M a congruence locally symmetric space (covered by G=K). To begin, we introduce the notion of a cusp. Recall the group PGL(2; F ) possesses a natural left action on projective linear space P1(F ), and thus so does a congruence subgroup  . We remark that, via the n embeddings of F , this action is compatible with the action on X but we will not require this fact. De nition 2.3. A cusp  of congruence subgroup   PGL(2;O) is a  -orbit in P1(F ). One often identi es a cusp with a representative of its orbit. For  = ( :  ) 2 P1(F ), we shall always assume both  and  are integral. Note that this choice  = ( :  ) is unique up to multiplication in O, i.e.  = (  :   ) for  2 O. Also, the point (1 : 0) 2 P1(F ) is the point at in nity and is denoted 1. Proposition 2.4. There exists an bijective correspondence between the set of cusps of  F = PGL(2;O) and the ideal class group C ‘(F ) of F , namely  F  P1(F )! C ‘(F )  F ( :  ) 7! ( ;  ): In particular, the number of cusps of  F equals hF , the class number of F . 15Proof. [Gee88, p. 6] Suppose  ;  2 P1(F ) are contained in the same PGL(2;O) orbit. Then there exists  =  a b c d ! 2 PGL(2;O) such that    =  . Writing  = ( :  ) with  ;  2 O, we may assume that  6= 0. Then we see that  =    =   (  : 1) =  a  + b c  + d : 1 ! =  a + b c + d : 1  = (a + b : c + d ) ; so the ideal class associated to  is (a + b ; c + d ). Since  2 PGL(2;O), this ideal equals ( ;  ). Thus, the map is well-de ned. Since O is a Dedekind domain, every ideal of O is generated by at most 2 elements, implying the map is surjective. It remains to prove the map is injective. Suppose  = ( :  ) and  = ( :  ) possess the same associated ideal class, where  ;  ;  ;  2 O. Then by the de nition of equivalence in the ideal class group, we may multiply by a suitable element of F and assume ( ;  ) = ( ;  ) = a. Observe 1 2 O = aa 1 =  a 1 +  a 1 so there exists   ;   2 a 1 such that        = 1. Similarly, choose   ;   2 a 1 such that        = 1. In other words, the matrices g :=        ! ; g :=        ! have determinant 1, and transform the cusp 1 = (1 : 0) to  and  respectively. Notice g g 1 2 PGL(2;O) =  F and from our previous observations, g g  1  sends  to  . Hence,  and  belong to the same  F -orbit, as desired. In the above proof, the integral ideal a is dependent on the choice of  and  for which  = ( :  ) 2 P1(F ). Henceforth, to remove this dependence, for  2 P1(F ), always choose  ;  2 O such that  = ( :  ) and jNaj is minimum where a = ( ;  ). It follows that if  2 O divides both  and  then  2 O , so the pair  and  are now determined up to multiplication by a unit  2 O . Consequently, the integral ideal a of minimum norm is uniquely de ned for each  2 P1(F ), and shall be referred to as the ideal associated to  2 P1(F ) . From Proposition 2.4, we see 16that the same ideal a is associated to any element of the orbit  F   . As a separate remark, notice that even if we choose  ;  2 O such that a = ( ;  ) is the ideal associated to  2 P1(F ), the precise de nition of g still depends on the choice of  and  . Nonetheless, for the remainder of this paper, we will retain the de nition of g using  and  generating the ideal associated to  2 P1(F ). The dependence of g on the exact choice of  and  will not be relevant as the choice only depends on F . Corollary 2.5. Every  nite index subgroup  of  F , and in particular every congruence sub- group, has  nitely many cusps. Proof. Every cusp of  F decomposes into at most [ F :  ] cusps of  , and since  F has  nitely many cusps, so must  . From Proposition 2.2, the result therefore applies to congruence subgroups. For the remainder of this thesis, we shall  x the level to be the integral ideal N. 2.3 Cusp Stabilizer For  = ( :  ) 2 P1(F ) and a congruence subgroup  , de ne  ( ) := g 1  g , which is a subgroup of PGL(2; F ). Under this conjugation, the cusp  of  nX becomes the cusp 1 of  ( )nX since g (1) =  . By direct veri cation, we see that the  ( )-stabilizer of 1 = (1 : 0) is given by the upper triangular elements of  ( ). To obtain more detailed information, we shall explicitly describe the subgroup of upper triangular elements of  (N)( ) since it is  nite index in  ( ). Recall that det g = 1 and g is of the form  a a 1 a a 1 ! where a = ( ;  ) for which  = ( :  ) with speci ed  ;  2 O. By direct computations, one can verify that  (N)( ) = ( a b c d ! : a; d 2 1 + N; b 2 Na 2; c 2 Na2; ad bc 2 O ) . f I :  2 O g and so stab(1;  (N)( )) = (   0 1 ! :  2 (1 + N) \ O ;  2 Na 2 ) . f I :  2 O g: (2.4) 17We remark that now both of these de nitions are independent of our choice of  and  for which  = ( :  ), since the ideal a associated to  is also independent of these choices. For the sake of simplicity, we would like to restrict our attention to a  nite index subgroup of stab(1;  (N)( )) by restricting the additive subgroup over which b ranges. Proposition 2.6. Let  = ( :  ) 2 P1(F ) and  be a level N congruence subgroup. De ne  ( ) := g 1  g where g is as in Proposition 2.4. Then the  ( )-stabilizer of 1 2 P1(F ) contains the  nite- index subgroup, depending only on N,  =  N := (   0 1 ! :  2 V;  2 N ) . f I :  2 O g: where V = VN = (1 + N) \ O is a  nite index subgroup of O . Equivalently, the  -stabilizer of  2 P1(F ) contains the subgoup g  g 1 . Proof. From (2.4) and the preceding discussion, it is follows that  is a  nite-index subgroup of stab(1;  ( )) since the additive group N is  nite index in Na 2 with index equal to Na2. To see that V has  nite index in O , it su ces to note that V is the pre-image of f1g in the reduction map O ! (O=N) and f1g is obviously  nite index in the  nite group (O=N) . With this  nite index subgroup g  g 1 of the cusp stabilizer stab( ;  ), we may  nd a domain which projects onto the fundamental domain of stab( ;  )nX , and thus obtain an understanding of its structure and parametrization. Theorem 2.7. De ne  =  N  PGL(2;O) as in Proposition 2.6. Then there exists pre- compact domains U = UN  F1 and V = VN  bY such that the set D = DN := f(x;y) 2 X : x 2 U ; by 2 V;Ny 2 (0;1)g is a fundamental domain for  nX , where by :=  yv (Ny)1=n  vj1 ; Ny := Y vj1 Nvyv; bY := fy 2 Rm>0 : Ny = 1g: 18Proof. [Gee88, p. 9{11] Without loss, every element  2  N may be written as  =   0 0 1 ! 1  0 1 !  0 0 1 ! with  2 VN;  2 N and  2 VN \ O 1 , where O  1 are the roots of unity of O  . The matrix involving  is redundant, but helpful for our purposes. We shall construct the desired fundamental domain by considering the individual action of each element. First, recall that we write z = (x;y) 2 X where x 2 F1 and y 2 Rm>0. Notice that Nby = 1 for any (x;y) 2 X. We begin by anlayzing the action of the diagonal element for  2 VN:   0 0 1 !  (x;y) = ( x; j jy): where j j = (j vj)vj1. Note that the coordinate j jy = (j vjyv)v is simply component-wise multiplication, and so if  is a root of unity, j jy = y. We wish to articulate the VN-action on the y-component. Consider the surjective map cIm : X ! bY = n y 2 Rm>0 : Ny = 1 o (x;y) 7! by =  yv (Ny)1=n  vj1 : Observe every point (x;y) 2 X is uniquely de ned by the triple (x; by;Ny). Since jN j = 1 for  2 O , the group O acts multiplicatively on bY via  7! j j = (j vj)v: Since the kernel of this map are the roots of unity, denoted O 1 , this action may be viewed as a faithful action of O =O 1 . Moreover, one should note that jN j = 1 for any  2 O  . The multiplicative action of O on bY transfers to an additive action via the bijective logarithm map log : bY ! log bY = n (av)vj1 2 R m : X vj1 Trvav = 0 o y 7! log y := (log yv)vj1 since log(j jy) = (log j vj + log yv)vj1. By Dirichlet’s Unit theorem, O  , and hence the  nite index subgroup V = VN, is a lattice in log bY, so we may choose a pre-compact fundamental domain logV for the additive action V n log bY. The exponential map exp : log bY ! bY is 19a topological isomorphism, under which logV becomes a pre-compact fundamental domain V = VN for the multiplicative action of V n bY. Thus, for any (x;y) 2 X we may choose  2 V such that j jby 2 V. Now, we consider the action of the unipotent element: for  2 N,  1  0 1 !  ( x; j jy) = ( x +  ; j jy): Since N is a lattice in F1, a fundamental domain for the additive action F1=N is pre-compact. If we also include the multiplicative action of the group O 1 \ V , then we have   0 0 1 !  ( x +  ; j jy) = (  x +   ; j jy): since j j = (j jv)v = (1)v. We see that the element only acts in the  rst coordinate on F1. Since O 1 \ V is a  nite group and a fundamental domain for F1=N is pre-compact , it follows that a fundamental domain U = UN for (O  1 \ V )nF1=N is pre-compact. Thus, for  x 2 F1, we may choose  2 O 1 \ V and  2 N so that   x +   2 UN. To conclude, we may choose  2  such that   z 2 D. Finally, we prove that distinct points z; z0 2 D are not  -equivalent. Suppose there exists  2  such that   z = z0. Using the decomposition of  as before, this implies (  x +   ; j jy) = (x0;y0) for some  2 V;  2 N and  2 O 1 \V . By comparing coordinates, we see that j jy = y 0, which implies j jby = by0. However, because by; by0 2 V, it must be that j vj = 1 for all v j 1, or in other words,  2 O 1 \ VN is a root of unity. Considering the  rst coordinate, we see that x0 =   x +   2 (O 1 \ VN)  x  N since  ;  2 O 1 \V and  2 N. As x;x 0 2 U , it must be that  = 0 and   = 1. In other words,  is the identity motion, as desired. Remark. Since  is a  nite-index subgroup of stab(1;  ( )), the fundamental domain D = DN for  nX is composed of  nitely many copies of the fundamental domain for stab(1;  ( ))nX. The above theorem then suggests that the notion of depth into the cusp 1 is measured by the single parameter Ny, i.e. a large value of Ny implies y is \close to in nity". More crudely, there is only \one way to in nity". 20Corollary 2.8. Keep notation as in Theorem 2.7. Then via the map  nX  stab(1;  ( ))nX ; the domain D projects onto a fundamental domain for stab(1;  ( ))nX . Equivalently, via the map g  g  1  nX  stab( ;  )nX ; the domain g D projects onto a fundamental domain for stab( ;  )nX . Proof. This follows immediately from Proposition 2.6 and Theorem 2.7 since  is a  nite index subgroup of stab(1;  ( )). 2.4 Distance to Cusps As in the remark following Theorem 2.7, each cusp  2 P1(F ) of  possesses a one- dimensional parameter which measures the \closeness" of a point z 2 X . Motivated by and using the notation of Theorem 2.7, we de ne D(y) = DN(y) := f(x;y) 2 D : Ny 2 (y;1)g ; for y  0. We note some simple properties:  D(0) = D.  D(y)  D(y0) for 0 < y  y0.  D(y)! ? as y !1.  D(y) is a fundamental domain for the action of  on X (y) := f(x;y) 2 X : Ny 2 (y;1)g: From Corollary 2.8, elements z 2 g D(y) are \close" to  2 P1(F ) when y is large. Equivalently, we have g 1 z 2 D(y), which implies y < N(Im(g 1 z)) = Ny jN( z +  )j2 : where  = ( :  ) and a = ( ;  ) is the ideal associated to  . We see that the expression on 21the righthand side measures the closeness of a point z to  2 P1(F ), so we de ne  ( ; z) := Ny jN(  z +  )j2 for the same choice of  and  . As a special example, we have  (1; z) = Ny since a = O and ( :  ) = (1 : 0). Observe that, since  and  generate the ideal associated to  , the expression  ( ; z) does not depend on the choice of  ;  , which are determined up to multiplication by an integral unit. Because suppose we write  = (  :   ) for some  2 O , then Ny jN(   z +   )j2 = Ny j(N )N(  z +  )j2 = Ny jN(  z +  )j2 as jN j = 1. Thus,  ( ; z) is well-de ned for z 2 X and  2 P1(F ). Notice, by de nition of  ( ; z), it follows that g D(y) = fz 2 g D :  ( ; z) > yg (2.5) for each  2 P1(F ). The purpose of this section is to prove several useful lemmas regarding  ( ; z). First, we prove a simple invariance property. Lemma 2.9. For  2 P1(F ) and z 2 X , we have  (   ;   z) =  ( ; z) for all  2  F = PGL(2;O). Proof. [Gee88, p. 7] Write  = ( :  ) with a = ( ;  ) and write  =  a b c d ! for a; b; c; d 2 O. By direct computations, we see that Im(  z) = y jcz + dj2 ;   z = az + b cz + d 22from which it follows that  (c + d )(  z) + a + b =  (c + d ) az + b cz + d + a + b =  (c + d )(az + b) + (a + b )(cz + d) cz + d =  bc  ad z + ad + bc z cz + d = (  z +  ) det  cz + d Combining these calculations, we obtain the desired result for  2  F . Next, we show that every point z 2 X must be uniformly close to at least one  2 P1(F ). Lemma 2.10. There exists a positive constant y0 = y0(F ) such that, for every z 2 X , there exists some  2 P1(F ) with  ( ; z) > y0. Speci cally, one may take y0 = (2nDF ) 1 where DF is the absolute discriminant of F . Proof. [Gee88, p. 8] We claim it su ces to choose y0 > 0, depending only of F , such that for every z 2 X, there exists a solution  ;  2 O to the inequality jN(  z +  )j2 Ny < 1 y0 : (2.6) If this is the case, then we show  = ( :  ) 2 P1(F ) is the desired element. Note that the LHS of (2.6) is not necessarily equal to  ( ; z) since  and  may not generate the ideal associated to  . However, suppose  = ( 0 :  0) and a = ( 0;  0) is the ideal associated to  , then  =  0 and  =  0 for some  2 O, and so y0 < Ny jN(  z +  )j2 = Ny j(N )N(  0z +  0)j2   ( ; z) because jN j  1. This proves the claim, and so it su ces to prove a solution  ;  2 O exists to (2.6) for some y0 > 0 depending only on F . Before proving the claim, recall for each place v j 1, there is a non-canonically associated embedding  v : F ,! Fv giving rise to the place v. Further, the collection of f vg [ f vgvj1 gives the entire list of n embeddings, Hom(F;C). With this indexing, we de ne the injective map X ! Cn z = (zv)vj1 7! (z ) 2Hom(F;C) 23where z = 8 < : zv  =  v for some v zv  =  v for some v : Using this notation, we see that for z 2 X , Nz = Y vj1 Nvzv = Y  z since at complex v, Nvzv = zvzv and at real v, Nvzv = zv. Now, to prove the claim, we bound the LHS of (2.6) by considering the norm expressed as a product at each embedding  : F ,! C, as per the above. By the Triangle Inequality, j   z +   j y  1=2   j   x +   j y  1=2  + j  j y 1=2  : Thus, if we can choose c ; d 2 R>0, depending only on F , for each embedding  such that the set of 2n equalities j  x +   j y  1=2   c ; j  j y 1=2   d ;  2 Hom(F;C) (2.7) possesses a solution  ;  2 O, then by applying the previous inequality, we see that jN(  z +  )j2 Ny = Y  j    z +   j 2 y 1  Y vj1 (c + d ) 2 so we may take y0 = Q  (c + d )  2. To achieve a solution to (2.7), let f!(k)gnk=1 be an integral basis for O in F , and write  = nX k=1 a(k)!(k);  = nX k=1 b(k)!(k) where a(k); b(k) 2 Z for k = 1; : : : ; n are free variables. Substituting these sums in (2.7), we obtain 2n linear inequalities    nX k=1 a(k)  (y 1=2 ! (k)  ) + nX k=1 b(k)  ( x y  1=2  ! (k)  )     c ;  2 Hom(F;C)    nX k=1 a(k)  0 + nX k=1 b(k)  (y1=2 ! (k)  )     d ;  2 Hom(F;C) 24with 2n variables, fa(k); b(k) : k = 1; : : : ; ng. De ning the n n matrices A11 = (y  1=2  ! (k)  ) ;k; A12 = ( x y  1=2  ! (k)  ) ;k; A22 = (y 1=2  ! (k)  ) ;k; and n 1 vectors a = (a(k))k and b = (b(k))k, we see that the linear system is of the form  A11 A12 0 A22 ! a b ! : From Minkowksi’s Theorem on linear forms [Neu99, p. 27{28], a solution exists with a(k); b(k) 2 Z to the above system provided Y  c d  det  A11 A12 0 A22 ! : We can easily see that the RHS = det(A11A22) = det((! (k)  )k; ) 2 = DF where DF is the discriminant of F , so we may take c = d = D 1=2n F implying r0 = (2 nDF ) 1. Finally, we prove that a point z 2 X can be very close, in a uniform sense, to at most one  2 P1(F ). Lemma 2.11. There exists su ciently large Y0 = Y0(F ) > 0 such that if, for z 2 X , we have  ( ; z) > Y0 and  ( ; z) > Y0 for  ;  2 P1(F ), then  =  . Proof. [Gee88, p. 6] In a remark following Proposition 2.4, we noted that the ideal a associated to a point  2 P1(F ) is actually the same for any element in the orbit  F   . Since, by Proposition 2.4, there are only  nitely many  F -orbits of P1(F ), it follows that we may bound the norm of a by some constant C > 0 depending only on F . Suppose z = (x;y) and write  = ( :  ) where  ;  2 O generate the ideal associated to  , and similarly write and  = ( :  ) where  ;  2 O generate the ideal associated to  . Now, recall that by Dirichlet’s theorem, the embedding O ! Rm>0  7! (j vj)v makes the subgroup (O )2 a multiplicative lattice in the hyperplane bY = fy0 2 Rm>0 : Ny 0 = 1g. 25This implies for any y0 2 Rm>0, that we may  nd a unit  2 O  such that [j 2jy0 is contained in some fundamental domain for (O )2n bY. Since this domain is precompact in bY, it is bounded away from zero and in nity. This implies that each coordinate in [j 2jy0 is bounded by some constant c = c(F ) > 0. To summarize, for each y0 2 Rm>0, there exists some  2 O  such that j 2vy 0 vj  c  (Ny 0)1=n; v j 1 where c > 0 is some constant depending only on F . Thus, utilizing the above observation for y0 = Im(g 1 z)  1 =  j vzv +  vj 2 yv  vj1 we have, after multiplying y0 by an appropriate unit  2 O , for v j 1     2vy 0 v    =    y 1v (  v vzv +  v v) 2     c  (Ny0)1=n = c   ( ; z) 1=n < c  Y  1=n0 since  ( ; z) = (Ny0) 1 by de nition and  ( ; z) > Y0 by assumption. We may replace  and  by   and   since  ( ; z) is independent of this choice. Then the above inequality yields two inequalities: j vxv +  vj y  1=2 v  c 1=2  Y  1=n0 v j 1 j vj y 1=2 v  c 1=2  Y  1=n0 v j 1 (2.8) by writing zv = (xv; yv). Similarly, we may obtain inequalities for  and  . j vxv +  vj y  1=2 v  c 1=2  Y  1=n0 v j 1 j vj y 1=2 v  c 1=2  Y  1=n0 v j 1 (2.9) On the other hand,  v v   v v = (  vxv +  v)y  1=2 v   vy 1=2 v  (  vxv +  v)y  1=2 v   vy 1=2 v from which it follows by the Triangle Inequality, (2.8), and (2.9) that jN(     )j < cnY  10 : 26Thus, for Y0 > cn, we see that the algebraic integer      has norm whose absolute value is less than 1, and hence      = 0 implying  =  . Since c > 0 depends only on F , this completes the proof. 2.5 Fundamental Domain of  nX With a relatively complete understanding of the structure of stab( ;  )nX and the distance of points in X to cusps of  , we may now su ciently describe  nX for our purposes. Proposition 2.12. Let  be a congruence subgroup of level N, and let   P1(F ) be a set of inequivalent representatives of  nP1(F ). Then the image of the set G(y) := [  2 g D(y)  X under the map X !  nX , (a) for 0 < y  y0, surjects onto  nX , where y0 is as in Lemma 2.10. (b) for y > Y0, injects into  nX , where Y0 is as in Lemma 2.11, and further the union over  2  is disjoint. Proof. [Gee88, p. 9{11] (a) It su ces to show that for z 2 X , there exists  2  and  2  such that   z 2 g D(y0): From Lemma 2.10, we have that there exists  0 2 P1(F ) such that  ( 0; z) > y0. Applying an appropriate element  0 2    F , by Lemma 2.9, we have  ( ;  0  z) > y0 for  =  0   0 2  . Then, from the proof Theorem 2.7, we may choose an appropriate element  00 2  such that (g  00g 1  0)  z 2 g D(y0): Since g  g 1   (N)   , we may take  := g  00g 1  0 2  to obtain the desired result. (b) From (2.5), we see that z 2 g D(y) implies  ( ; z) > y. Then by Lemma 2.11, the union over  2  is necessarily disjoint. To see that the set injects, suppose z;   z 2 G(y) for some  2  . Thus, again by (2.5), we have  ( ; z) > y and  ( ;   z) > y 27for some  ;  2  . Using Lemma 2.9, we see that  (  1   ; z) > y  Y0. Applying Lemma 2.11, we conclude   1  =  , and so  =  as  is a set of inequivalent representatives of  nP1(F ). The decomposition described in Theorem 2.13 below is best accompanied with the visual aid in Figure 2.2. Theorem 2.13. Let  be a congruence subgroup of level N, and let   P1(F ) be a set of inequivalent representatives of  nP1(F ). For y su ciently large, depending only on F , there exists a set S(y)  X such that (i) The set S(y) t G  2 g D(y)  X surjects onto a fundamental domain F for  nX . (ii) The set G  2 g D(y)  X injects into a fundamental domain F for  nX . (iii) S(y) is compact (iv) S(y)  S(y0) for y  y0  y0. (v) As y !1, the image of S(y), under the map X !  nX , approaches  nX . 28Figure 2.2: Depiction of  nX with 3 cusps Proof. [Gee88, p. 9{11] For y0 as in Lemma 2.10, de ne S(y) :=  G(y0) n G(y)  = [  2 g (D(y0) n D(y)): We claim this choice of S(y) has the desired properties. (i) The surjectivity follows by noting S(y) [ [  2 g D(y)  [  2 g D(y0) and applying part (a) of Proposition 2.12. The disjoint union follows by de nition of S(y) and part (b) of Proposition 2.12. 29(ii) This is a restatement of Proposition 2.12(b). (iii) Observe D(y0) n D(y) = f(x;y) 2 X : x 2 U ; by 2 V;Ny 2 [y0; y]g where U  F1 and V  bY are precompact sets de ned in Proposition 2.6. The above subset of X is then topologically of the form U  V  [y0; y] and is hence compact; therefore, S(y), the closure of a  nite union of these subsets, is itself compact. (iv) This is immediate from the de nition of S(y) and noting D(y)  D(y0) for y  y0. (v) The limit of S(y) as y !1 is well-de ned by (iv). The desired result follows from (i), and noting D(y)! ? as y !1. 30Chapter 3 Hecke-Maa Cusp Forms In this chapter, we will de ne a Hecke-Maa cusp form in several stages, and ultimately provide its Whittaker expansion with proof. We will conclude by demonstrating crucial relations satis ed by the Whittaker coe cients which form the basis of the following chapter. Sources for this discussion include [Fre90] and [Gee88], and in terms of the ad eles, [Hid90]. The case F = Q, which is fairly similar, can be found in [Iwa02], [IK04], [Miy89], or [DS05]. 3.1 Maa Forms and their Fourier Expansion Let   PGL(2; O) be a congruence subgroup of level N. De nition 3.1 (Automorphic form). A function  : X ! C is an automorphic form (with respect to  ) provided (i)  (  z) =  (z) for all  2  (ii)  is an eigenfunction of the Laplacian  v for every v j 1, i.e.  v =  v where  v = sv(1 sv) 2 C. Denote  = ( v)v ands = (sv)v. Our interest lies in automorphic forms which satisfy certain growth conditions, namely belong to the L2-space of  nX . Recall that, for C-valued functions  and  on  nX , the L2-inner product on  nX is given by h ;  iL2 = Z F  (z) (z)dz 31where F  X is a fundamental domain for  nX . We denote the L2-norm of  by jj jjL2( nX ), or more simply as jj jjL2 when  is understood. De nition 3.2 (Maa form). A function  : X ! C is a Maa form (with respect to  ) provided  is an automorphic form, and also  2 L2( nX ). Remark. By Elliptic Regularity of  v, it follows that  2 C1( nX ). See [Eva10, x6.3] for details. Henceforth, let  : X ! C be a Maa form of  . Now, let  be a cusp of  , and de ne   (z) :=  (g  z) where g 2 G is as de ned in Proposition 2.4, satisfying g (1) =  . We claim   is now a Maa form of  ( ) = g  g 1 . This follows from the fact that  v commutes with the Gv-action on  and also since dvol(z) is invariant under the G-action. Roughly speaking, studying  at the cusp  amounts to studying   at the cusp 1. Recall from Proposition 2.6 that  := (   0 1 ! :  2 V;  2 N ) . f I :  2 V g: is contained in the  ( )-stabilizer of 1, or more generally,    ( ). In particular, we have that   is invariant by  , which by setting  = 1 2 V , implies   (x +  ;y) =   (x;y); for  2 N: where we have written   (z) =   (x;y) for z = (x;y). Under this additive action, N is a lattice in F1 = Q vj1 Fv, the x-coordinate. De ne the standard trace form bilinear pairing h  ;  i : F1  F1 ! C hw;xi 7! Tr(wx) := X vj1 Trv(wvxv) where Tr extends the global trace TrFQ to F1 by the above formula, and Trv extends the local trace TrFvR at v j 1 by Trvxv = 8 < : xv v real xv + xv v complex This pairing yields the dual lattice N_ = (DN) 1 where D is the absolute di erent ideal of F 32[Lan94, x3.1]. Hence, the Fourier series expansion of  at the cusp  is given by   (x;y) = X  2(DN) 1 c  (y; )e(hx;  i) (3.1) where e(x) = exp(2 ix) and c  (y; ) := Z U   (x;y)e( hx;  i)dV1(x); with U being a fundamental domain for F1=N. Note c  (y; ) is the  -Fourier coe cient of  at cusp  . Since the Maa form   (z) is an eigenfunction for  v for each v j 1, it follows by Elliptic Regularity that   (z) 2 C1(X ), i.e. it possesses in nitely many derivatives in the coordinates xv and yv for all v j 1. From general Fourier analysis, it follows that the Fourier series (3.1) converges absolutely and uniformly on compacta. We are concerned with Maa cusp forms, whose de nition depends upon this Fourier expansion. 3.2 Maa Cusp Forms and their Whittaker Expansion In this section, we aim to elaborate further on the Fourier expansion at the cusp  of a Maa form  . Material on partial di erential equations will be skipped but can be found in [Eva10]. De nition 3.3 (Maa cusp form). A function  : X ! C is a Maa cusp form (with respect to  ) if  is a Maa form of  and additionally, c  (y; 0)  0 for every cusp  of  . Henceforth, we shall assume  is a Maa cusp form of  . From this de nition and (3.1), it follows that the Fourier expansion at a cusp  of  is   (x;y) = X  2(DN) 1nf0g c  (y; )e(hx;  i): (3.2) Since   (z) is smooth and an eigenfunction for  v, whose derivatives involve yv, we are able to yield more information about the precise form of the Fourier coe cients c  (y; ). For v j 1, applying  v to both sides of (3.2), we  nd  v  (x;y) = X  2(DN) 1nf0g  v(c  (y; )e(hx;  i)): Note we may commute the derivative with the in nite sum due to the uniform convergence of 33the Fourier series. Now, comparing Fourier coe cients of both sides, and using the form of the Laplace operator  v given in (1.2) and (1.4), one obtains a partial di erential equation for each v j 1. Therefore, the coe cient c  (y; ) with  6= 0 satis es the following separable linear system of partial di erential equations: 8 >>< >>: y2v  @2 @y2v (y) 4 2 2v (y)  =  v (y); v real y2v  @2 @y2v (y) 16 2j vj2 (y)   yv @ @yv (y) =  v (y); v complex (3.3) where  : Rm>0 ! C. Remark. If  were more generally a Maa form, the zeroth Fourier coe cient   (y; 0) satis es the same PDE but  v = 0, which results in a very di erent solution. In this sense, the analysis could continue along the same path, but is unnecessary for our purposes. We present the solution to the system in (3.3) with the following lemma and proposition. Lemma 3.4. Let av 2 R>0 and  v = sv(1 sv) 2 C. Suppose  v : yv ! C satis es 8 < : y2v   00v (yv) a 2 v v(yv)  =  v v(yv); v real y2v   00v (yv) a 2 v v(yv)   yv 0 v(yv) =  v v(yv); v complex then for some constants c1; c2 2 C, depending on  v, we have  v(yv) = c1 p Nv(avyv)K1=2 sv(avyv) + c2 p Nv(avyv)I1=2 sv(avyv) where K (z) and I (z) are the K and I Bessel functions. Proof. See [Zwi98, x45] or verify using computer algebra package Maple. Information about the Bessel functions can be found in [Iwa02, Appendix B] or [Bow10]. For non-zero  = ( v) 2 F1, set av = 2 Trv(j vj) = 8 < : 2 j vj v real 4 j vj v complex : Then, with this choice, the above lemma corresponds to the system described in (3.3). Moti- vated by this choice, we shall de ne B v (sv; yv) := p Nvyv  K1=2 sv(2 Trv(yv)) B+v (sv; yv) := p Nvyv  I1=2 sv(2 Trv(yv)) 34so then by Lemma 3.4, the set of functions fB+v (sv; j vjyv); B  v (sv; j vjyv)g forms a basis of solutions for the corresponding ordinary di erential equation from the system in (3.3). Proposition 3.5. Let  = ( v)v 2 F1; = ( v)v 2 Cm. Write  v = sv(1  sv) and set s = (sv)vj1. For the sequence of signs  = ( v)vj1 2 f g m, de ne B (s; y) := Y vj1 B vv (sv; yv) where y 2 Rm>0. Then the set of 2 m functions n B (s; j jy) :  2 f gm o forms a basis of smooth solutions over C for the system of partial di erential equations given in (3.3). Proof. This is a direct consequence of Lemma 3.4 and the fact that the system (3.3) may be solved by separation of variables. See [Zwi98] or [Eva10] for details on this method. With Proposition 3.5, we already have a strong understanding of the coe cient c  (y; ) as it satis es (3.3); in particular, c  (y; ) may be written as a linear combination of B (s; j jy) for  2 f gm. This fact combined with invariance properties of   provides a relation between di erent Fourier coe cients, which is articulated in the lemma below. Lemma 3.6. Let  be a Maa cusp form of  , and  be a cusp. By Proposition 3.5, we may write for  2 (DN) 1 n f0g, c  (y; ) = X  2f gm c ( )B  (s; j jy) where c ( ) 2 C also depends on  . Then de ning V  O as in Proposition 2.6, we have c ( ) = c (  ) for all  2 V and  2 f gm. 35Proof. Since   0 0 1 ! 2    ( ) for  2 V , we have   (  z) =   (z). Applying the Fourier expansion (3.1) with the form above for c  (y; ), and sending  7!    1, we see for all z 2 X ,  (  z) = X  2(DN) 1nf0g X  2f gm c ( )B  (s; j  jy)  e(h x;  i) = X  2(DN) 1nf0g X  2f gm c (   1)B (s; j jy)  e(hx;  i) by noting h x;  i = hx;   i. Comparing the RHS with the same expansion for  (z), and noting the linear independence of B (s; j jy) as a function of y from Proposition 3.5, we conclude that c ( ) = c (   1) as required. Further, since   2 L2( ( )nX), we obtain the simple lemma below bounding the mass of a Fourier coe cient deep in a cusp. Lemma 3.7. Let  be a Maa cusp form of  and  be a cusp. Then for y su ciently large, depending only on F , Z T (y) jc  (y; )j2dV2(y)  jj  jjL2( ( )nX ) <1 where  2 (DN) 1 n f0g;  is a cusp of  , and using notation from Theorem 2.7, T (y) = TN(y) := fy 2 Rm>0 : by 2 V;Ny 2 (y;1)g: Proof. By Parseval’s Formula, we have Z U j  (x;y)j2dV1(x) = X  2(DN) 1 jc  (y; )j2  jc  (y; )j2: for y 2 Rm>0. For y su ciently large, D(y) injects into a fundamental domain for  ( )nX by Theorem 2.13. Since   2 L2( ( )nX ), it follows that we may integrate both sides of the above inequality over the remainder of D(y) given explicitly in Theorem 2.7, namely over the set T (y) = fy 2 Rm>0 : by 2 V;Ny 2 (r;1)g. Thus, we obtain the simple bound Z T (y) jc  (y; )j2dV2(y)  Z D(y) j  (z)j2dvol(z)  jj  jjL2( ( )nX ) <1: 36for y su ciently large, depending only on F . Utilizing these lemmas and the asymptotics of B v , we can completely characterize the form of c  (y; ) for  6= 0. The asymptotics of B v are known to be B v (sv; yv)  sv exp( 2 Trv(yv)) (3.4) as yv !1; see [Iwa02, Appendix B]. Theorem 3.8. Let  be a Maa cusp form of  with Laplace eigenvalues  = ( v)vj1. Writing  v = sv(1 sv), we have for  2 (DN) 1 n f0g and cusp  of  , that c  (y; ) = c Y vj1 B v (sv; j vjyv) for some constant c 2 C depending on  ;  and  . Proof. As in Lemma 3.6, we know c  (y; ) is of the form: c  (y; ) = X  2f gm c B  (s; j jy) where c = c ( ) 2 C. Our goal is to show that c is non-zero only when  = ( ; ; : : : ; ). By Lemma 3.6, we have c  (y;  ) = X  2f gm c B  (s; j  jy); for  2 V: Our goal will be achieved by choosing  2 V appropriately and analyzing the asymptotics of c  (y;  ) in T (y) for y  1 su ciently large. De ning  + := fv j 1 :  v = (+)g   := fv j 1 :  v = ( )g; for  2 f gm, we choose  such that # + is maximum and c 6= 0. Evidently, the choice may not be unique. Suppose, for a contradiction, that  + 6= ?. Now, since the set V  Rm>0 is compact by Proposition 2.6, the constants a := minfyv : y 2 V; v j 1g; b := maxfyv : y 2 V; v j 1g; 37exist and are positive. Then for y 2 T (y), we have by 2 V and so a(Ny)1=n  yv  b(Ny)1=n: (3.5) In particular, this shows that as Ny!1, every coordinate of y goes to in nity. Our aim is to choose  2 V appropriately with respect to  . If  = (+;+; : : : ;+), then choose  := 1 2 V . Otherwise, choose  2 V such that j vj > 1; v 2  + and j vj < 1; v 2   ; which is possible because by Dirichlet’s Unit Theorem, V is an (m  1)-dimensional lattice in fy 2 Rm>0 : Ny = 1g. See [Lan94, p. 104{108] or [Neu99, x1.7] for details on this choice. Observe for  2 f gm from (3.4), we have B (s; j  jy)  s; exp  2  X v2 + Trv(j v vjyv) X v2  Trv(j v vjyv)   (3.6) as Ny!1 for y 2 T (y). From (3.5), it follows that B (s; j  jy) B (s; j  jy)  s; exp  4  X v2 +n + Trv(j v vjyv) X v2  n  Trv(j v vjyv)    s; exp  4 (Ny)1=n  X v2 +n + Trv(j v vjb) X v2  n  Trv(j v vja) | {z } ( )   If  6=  and c 6= 0, then   n      , and, by maximality of  , the set   n   6= ?. Then by choice of  , if we replace  by a su ciently large power of itself (independent of y), we may assume the quantity in ( ) is negative for all  6=  . In other words, for  6=  such that c 6= 0, we have shown B (s; j  jy) = B (s; j  jy)  o(1) as Ny!1. Thus, we may write c  (y;   ) = B (s; j  jy)  c + X  6= c  o(1)  as Ny ! 1 for y 2 T (y). From this equation, it follows by (3.6) and (3.5) that for Ny 38su ciently large, jc  (y;   )j  jc j 2 jB (s; j  jy)j  s; jc j exp  2 X v2 + Trv(j v vjyv) 2 X v2  Trv(j v vjyv)   s; jc j exp  2 (Ny)1=n  X v2 + Trv(j v vja) X v2  Trv(j v vjb) | {z } (  )   Again, by choice of  , if we replace  by a su ciently large power of itself (independent of y), we may assume the quantity in (  ) is positive. More simply, for some constant  > 0, we have jc  (y;   )j  s; jc j exp( (Ny)1=n): for y 2 T (y) and Ny su ciently large. Using this bound, we see that for y su ciently large, Z T (y) jc  (y; )j2dV2(y) s; jc j Z T (y) exp(2 (Ny)1=n)dV2(y)  s; jc j exp(2  y 1=n) Z T (y) dV2(y): (3.7) Finally, sending y = (yv)v 7! y1=n  y = (y1=n  yv)v, the set T (y) is mapped bijectively to T (1) and from (2.2), we have dV2(y) 7! y 1  dV2(y). Thus, the RHS of (3.7) is  s; jc j exp(2  y 1=n)y 1 Z T (1) dV2(y) s; jc j exp(2  y1=n)y 1: Taking y !1, we deduce from (3.7) that Z T (y) jc  (y; )j2dV2(y)!1 contradicting Lemma 3.7. This completes the proof. In light of Theorem 3.8 and (3.1), we make the following de nitions. De nition 3.9 (Local Whittaker Function). For v j 1, de ne the local Whittaker function at v j 1 to be Wv(sv; zv) := p yvK1=2 sv(2 Trv(yv))  e(Trv(xv)) 39where sv 2 C and zv = (xv; yv) 2 Fv  R>0. Observe that Wv(sv; zv) = B  v (sv; yv)e(Trv(xv)): De nition 3.10 (Whittaker Function). De ne the (unrami ed) Whittaker function of G to be W (s; z) := Y vj1 Wv(sv; zv) where s = (sv)vj1 2 Cm; z = (zv)vj1 2 X . From the previous de nition and Theorem 3.8, we may de ne c ( ; ) 2 C to be such that c  (y; ) = c ( ; )jN j 1=2W (s;  z); for z 2 X where   z := ( vxv; j vjyv)vj1 2 F1  Rm>0. The constant c ( ; ) shall be called the  - Whittaker coe cient of  at the cusp  . Note the normalization factor jN j 1=2 is not necessary, but chosen to better suit later multiplicative relations. Corollary 3.11 (Whittaker Expansion of Maa cusp forms). Let  be a Maa cusp form of  with Laplace eigenvalues  = ( v)vj1, and let  be a cusp of  . Writing  v = sv(1 sv), we have that for z 2 X ,   (z) = X  2(DN) 1nf0g c ( ; )jN j 1=2W (s;  z) where c ( ; ) 2 C and   z := ( vxv; j vjyv)vj1 2 X for  2 F . The in nte sum is absolutely and uniformly convergent. Proof. This is an immediate consequence of Theorem 3.8 and (3.1); the convergence properties come from the fact that this sum is the Fourier expansion of a smooth function. Remark. The expansion of   (z) given in Corollary 3.11 is known as the Whittaker expansion of  at the cusp  . 403.3 Hecke Operators We wish to de ne Hecke operators acting on a Hecke-Maa cusp form  , so by Atkin-Lehner theory, we may assume that  is a Hecke-Maa cusp form with respect to  0(N) := ( a b c d ! 2 GL(2;O) : c  0 (mod N) ) . f I :  2 O g: for some ideal N. Relevant discussion can be found in [Iwa02, x8.5] and [Hid90]. As a result, we will assume without loss that  =  0(N) is our  xed congruence subgroup of level N for the rest of this chapter. De ning Hecke operators can take several approaches with some more natural than others; for example, Hida [Hid90] takes an ad elic perspective. In our case, we shall only need Hecke operators de ned a speci c subset of ideals, namely NN := fn  O : p j n =) p 2 PNg where PN := fp  O : p unrami ed principal prime ideal and p - Ng: To begin, we shall de ne Hecke operators for powers of prime ideals p 2 PN. Since the primes are principal, we may take an explicit approach for de ning Hecke operators, which has little di erence from the usual Hecke operators de ned over F = Q. As a result, we will quote material from sources discussing classical Hecke operators over Q, such as [IK04] and [Iwa02]. More thorough and general proofs over Q can be found in [Miy89]. De nition 3.12 (pk-Hecke operator). Let k  0 and p 2 PN have uniformizer $, i.e. p = ($). Suppose  :  0(N)nX ! C. Then we de ne T (pk) : X ! C by (T (pk) )(z) := 1 p Npk kX j=0 X  2O=pj    $k j  0 $j !  z  where the inner sum over  is over a set of inequivalent representatives of O=pj . The operator T (pk) is called the pk-Hecke operator. Remark. This de nition is independent of the choice of representatives for O=pj and uniformizer $ as  =  0(N) contains all elements of the form    0 1 ! ; where  2 O ;  2 O: 41First, we note that the Hecke operator is in fact an action on  0(N)-invariant functions, and further on the space of Maa cusp forms. Proposition 3.13 (x14.6 of [IK04]). The Hecke operator T (p) for p 2 PN acts on the space of  0(N)-invariant functions, i.e. f :  0(N)nX ! Cg. Corollary 3.14 (x8.5 of [Iwa02]). The Hecke operator T (n) for n 2 NN is an L2-bounded linear operator acting on the space of Maa cusp forms of  0(N). Now, we describe the multiplicative relations of Hecke operators on the space of Maa cusp forms of  0(N). We shall  rst discuss the collection of fT (pk)g1k=0 for a  xed prime ideal p 2 PN. Note that the normalization factor (Npk) 1=2 a ects these relations, and in other sources, is often replaced by 1 or (Npk) 1. Our choice is primarily to retain consistentcy with [Sou10]. It is a simple elementary number theory exercise to adjust the relations according to these factors. Proposition 3.15 (Proposition 14.9 of [IK04]). Let p 2 PN be given. Then for k  2, T (pk) = T (pk 1)T (p) T (pk 2) on the space of Maa cusp forms of  0(N). The second relation demonstrates that Hecke operators for distinct prime ideals p; q 2 PN commute. Proposition 3.16 (Proposition 14.9 of [IK04]). Let p; q 2 PN be distinct prime ideals. Then T (p)T (q) = T (q)T (p) on the space of Maa cusp forms of  0(N). Thus, we may extend our de nition of Hecke operators to all ideals of NN multiplicatively. De nition 3.17 (n-Hecke operators). For m; n 2 NN are relatively prime, i.e. (m; n) = (1) = O, de ne T (mn) := T (m)T (n): With this de nition, we may collect our results, and summarize them in the following theorem. Theorem 3.18 (Proposition 14.9 of [IK04], or x8.5 of [Iwa02]). On the space of Maa cusp forms of  0(N), the Hecke operators fT (n) : n 2 NNg satisfy the following multiplicative rela- tions: 42(i) T (O) = id. (ii) T (m)T (n) = X dj(m;n) T  mn d2  for m; n 2 NN. 3.4 Hecke-Maa Cusp Forms and their Whittaker Coe cients We are now in a position to de ne the key object of interest: Hecke-Maa cusp forms. In order to possess well-de ned Hecke operators, recall that we have assumed  =  0(N) for some ideal N. De nition 3.19 (Hecke-Maa cusp form). A function  : X ! C is a Hecke-Maa cusp form (with respect to  0(N)) provided   is a Maa cusp form   is an eigenfunction of every Hecke operator T (n) for all ideals n. More explicitly,  satis es all of the following: (i)  is  0(N)-invariant, i.e.  (  z) =  (z) for all  2  0(N) and z 2 X . (ii)  is an eigenfunction of  v for every v j 1, i.e.  v (z) =  v (z) where  v = sv(1 sv) 2 C. Denote  = ( v)v and s = (sv)v. (iii)  2 L2( 0(N)nX), i.e. jj jj2L2 = Z F j (z)j2dvol(z) <1 where F is a fundamental domain for  0(N)nX . (iv) The zeroth Fourier coe cient of  at every cusp  of  0(N) vanishes, i.e. c  (y; 0) = 0 for all cusps  of  0(N). (v)  is an eigenfunction of every Hecke operator T (n) for all ideals n, i.e. T (n) (z) =   (n) (z) where   (n) 2 C is the n-Hecke eigenvalue of  . 43Remark. In the previous section, Hecke operators T (n) were de ned for certain ideals n, namely n 2 NN, albeit one can de ne them for all ideals n. In the literature, Hecke-Maa cusp forms are simultaneous eigenforms of all Hecke operators, but as already noted, we shall only require T (n) for n 2 NN. The Hecke eigenvalues   (n) for n 2 NN naturally inherit the multiplicative properties of their Hecke operators as described in the previous section. Proposition 3.20. Let  be a Hecke-Maa cusp form with respect to  0(N). Then (i)   (O) = 1. (ii)   (pk) =   (pk 1)  (p)   (pk 2) for p 2 PN and k  2. (iii)   (m)  (n) = X dj(m;n)    mn d2  for m; n 2 NN. Proof. This is immediate from Proposition 3.15 and Theorem 3.18. Note (ii) is a speci c case of (iii). For a Maa cusp form  , recall that we have the following Whittaker expansion from Corollary 3.11.   (z) = X  2(DN) 1nf0g c ( ; )jN j 1=2W (s;  z) where  is a cusp of  0(N). If we additionally assume that  is a Hecke-Maa cusp form, then the Whittaker coe cients c ( ; ) are closely related to the Hecke eigenvalues of  . Proposition 3.21. Let  be a Hecke-Mass cusp form of  0(N) and  be a cusp. For p 2 PN, let $ be a uniformizer for p. If  2 (DN) 1 is a unit modulo p, then c ( ; $ k) = c ( ; )    (pk) for k  1: Proof. We will follow the argument structure of [IK04, x14.6]. Applying T (pk) and then g to  (z), we utilize the Whittaker expansion from Corollary 3.11 to deduce (Np)k=2(g  T (pk)) (z) = kX j=0 X  2O=pj    $k jz +  $j  = X  2(DN) 1nf0g c ( ; ) jN j1=2 kX j=0 X  2O=pj W  s;  $k jz +  $j  (3.8) 44We prove that the RHS of (3.8) is actually the Whittaker expansion of T (pk)  (z). To do this, we determine the non-zero terms of (3.8) by considering the inner sum over  for a  xed non-zero  2 (DN) 1 and 0  j  k. We remark that, in the following arguments, the re- arranging of terms is valid, as the Whittaker expansion is absolutely convergent due to the in nite di erentiability of   . Notice by De nition 3.9 and De nition 3.10, we have X  2O=pj W  s;  $k jz +  $j  = X  2O=pj W  s; j $k 2j j  y  e  Tr    $k jx +  $j   = W  s; j $k 2j jy  e  Tr   $k 2jx   X  2O=pj e  Tr    $j   (3.9) For this last sum, we claim X  2O=pj e  Tr    $j   = 8 < : (Np)j  p( )  j 0 else (3.10) where  p : F ! Z is the p-adic valuation. Again, we note that O=pj is a set of inequivalent representatives rather than a set of cosets. Since p - N we may assume, without loss, that the representatives  belong to N. Observe that the map O ! C 1  7! e(Tr(   $j )) is a character of O=pj . If  p( )  j, then  $ j 2 (DN) 1 implying Tr( $ j ) 2 Z for all  2 N by de nition of the absolute di erent. Thus, the character is trivial for  2 N and hence for all the representatives  in the sum. Further noting #O=pj = (Np)j gives the result in this case. If  p( ) < j, then since p 2 PN is unrami ed and p - N, the character is non-trivial (evaluate at any  62 p), so by orthgonality of characters, the sum is zero. Thus, from (3.10), the terms involving e(Tr( $k 2jx)) in (3.9) are non-zero only if  $k 2j 2 (DN) 1. As a result, the RHS of (3.8) is indeed the Whittaker expansion of T (pk)  (z). Now, our goal is to collect terms in (3.9) and determine the coe cient of e(Tr( x)) for a given  2 (DN) 1 which is a unit modulo p. For  2 (DN) 1, de ne A = f( ; j) :  =  $ k 2j ;  2 (DN) 1pjg: 45Then by (3.8), (3.9) and orthogonality of Fourier coe cients, we see that (Np)k=2 \(g T (pk) )(y; ) = W  s; j jy  X ( ;j)2A c ( ; ) jN j1=2  Npj ; (3.11) so it remains to determine the pairs ( ; j) 2 A . Evidently, for a given  2 (DN) 1, the value of j determines  2 (DN) 1 by the formula  :=  $2j k. Since  is a p-adic unit by assumption, we see that  p( ) = 2j  k: On the other hand, from our previous observations,  p( )  j, and so 2j  k  j implying j = k. Thus, A consists of exact one element ( $k; k). Substituting this result into (3.11), we see that \(g T (pk) )(y; ) = c ( ; $k)jN j 1=2W (s; j jy): On the other hand, g T (pk) =   (pk)  , and so we also have \(g T (pk) )(y; ) =   (pk)c ( ; )jN j 1=2W (s; j jy): Comparing the previous two equations, we have the desired result. Corollary 3.22. Let  be a Hecke-Mass cusp form of  0(N) and  be a cusp. For n 2 NN, choose  2 n such that n = ( ). If  2 (DN) 1 is a unit modulo n, then c ( ;  ) = c ( ; )    (n): Proof. This follows immediately from Proposition 3.21 and Proposition 3.20. Corollary 3.22 provides signi cant information about the Whittaker coe cients. In the classical case of F = Q for [Sou10], it is the statement of Weak Multiplicity One; in other words, all Whittaker coe cients correspond directly to Hecke eigenvalues, up to a constant depending only on  . However, for a general number  eld F , the Whittaker coe cients c ( ; ) behave di erently for Hecke operators of rami ed ideals or ideals dividing N, so the coe cients cannot be exactly identi ed with Hecke eigenvalues. Nonetheless, the relation in Corollary 3.22 between Whittaker coe cients and Hecke eigenvalues will fundamentally drive the material in the following chapter and the proof of the main result. In order to provide natural de nitions for the following chapter, we rearrange the Whittaker expansion in a more practical form via the following lemma. 46Lemma 3.23. Let  be a Hecke-Mass form of  0(N) and  be a cusp. Then c ( ; ) = c ( ;   ) for  2 V and  2 (DN) 1, where V  O is as in Proposition 2.6. Proof. Since   0 0 1 ! 2    0(N)( ) for  2 V , we have   (  z) =   (z). Applying the Whittaker expansion, and sending  7!    1, we see  (  z) = X  2(DN) 1nf0g c ( ; )W (s;   z) = X  2(DN) 1nf0g c ( ;   1)W (s;  z) Comparing with  (z), we deduce c ( ; ) = c ( ;   1). Fix a  nite set of representatives  for O =V and a single generator  2 (DN) 1 for every principal fractional ideal n of (DN) 1. Then, by Lemma 3.23 we may de ne the n-Whittaker ideal coe cient (of  at cusp  for  ) c( ) ( ; n) := c ( ;   ) yielding the following corollary. Corollary 3.24. Let  be a Hecke-Mass form of  0(N) and  be a cusp. Then   (z) = X  2O =V X n (DN) 1 n=( )6=(0) c( ) ( ; n)  X  2V W (s;     z)  where the in nite sums are absolutely and uniformly convergent. Further, if n 2 NN and m is a fractional ideal of (DN) 1 such that m and n are coprime, then c( ) ( ;mn) = c ( )  ( ;m)  (n) for  2 O  =V: Proof. The Whittaker ideal expansion is immediate from Lemma 3.23. The absolute and uni- form convergence is inherited from Corollary 3.11. The last property is a restatement of Corol- lary 3.22. 47Chapter 4 Mock P-Hecke Multiplicative Functions The purpose of this chapter is to abstract functions satisfying the properties of Whittaker coe cients and Hecke eigenvalues derived in Proposition 3.20 and Corollary 3.24, and then analyze the relevant growth measures of these functions which shall arise in the proof eliminating escape of mass. The structure and approach will closely follow Soundararajan [Sou10] with which we will provide direct comparisons to our results. 4.1 Statement of Main Theorem Throughout this chapter, the level N is an integral ideal, and P shall denote a  xed subset of unrami ed prime ideals of F not dividing N. Denote the set of P-friable ideals by N = N (P) := fn  O : p j n =) p 2 Pg: Note the ideal (1) = O 2 N vacuously and (0) 62 N . We begin with a de nitions motivated by the properties from Proposition 3.20. De nition 4.1 (P-Hecke Multiplicative). For a subset P of unrami ed prime ideals of F , a function fP : N ! C is P-Hecke multiplicative (of level N) if (i) fP(O) = 1 (ii) fP(m)fP(n) = X dj(m;n) fP  mn d2  48Remark. If P = PN = fp  O : p unrami ed principal prime ideals; p - Ng, then for a Hecke- Maa cusp form of  0(N), the Hecke eigenvalues   : N ! C are P-Hecke multiplicative by Proposition 3.20. Remark. In the proof of the main result, we shall only require two special cases of property (ii): namely, when m = pk and n = p 2 P, fP(pk+1) = fP(pk)fP(p) fP(pk 1): (4.1) and when (m; n) = O, we have fP(m)fP(n) = fP(mn). The next de nition is motivied by the property of Whittaker coe cients in Corollary 3.24, and the fact that Hecke eigenvalues of a Maa form are P-Hecke multiplicative for an appro- priate choice of P. De nition 4.2 (Mock P-Hecke Multiplicative). Let P be a set of unrami ed prime ideals not dividing N. Suppose f is a C-valued function on the fractional ideals of (DN) 1. Then we say f is mock P-Hecke Multiplicative (of level N) if there exists a P-Hecke multiplicative function such that f(mn) = f(m)fP(n) for n 2 N and m a fractional ideal of (DN) 1 with m and n coprime. Remark. Let  be a Hecke-Maa cusp form of  0(N),  be a cusp, and  2 O =V . If we de ne P = PN = fp  O : p unrami ed principal prime ideal; p - Ng as before, and set f(n) := 8 < : c( ) ( ; n) n principal fractional ideal of (DN) 1 0 else then, by Corollary 3.24, f is mock P-Hecke mutliplicative where fP(n) =   (n) for n 2 NN. For a  xed mock P-Hecke multiplicative function f of level N, our aim is to understand the decay of X Nn y=Y jf(n)j2 where 1  Y  y. We shall consider Y to vary, and y to be  xed. We present the main technical theorem. 49Theorem 4.3. Let P be a set of unrami ed prime ideals of F not dividing N, and f be a mock P-Hecke multiplicative function of level N. If P has positive natural density, then X Nn y=Y jf(n)j2  P  1 + log Y p Y  X Nn y jf(n)j2: for 1  Y  y. This chapter is dedicated to the proof of this result, which is the analogue to Theorem 3 of [Sou10]. Let y > 1 be a  xed value henceforth. For Y  1, de ne F(Y ) = F(Y ; y) := P Nn y=Y jf(n)j 2 P Nn y jf(n)j 2 Theorem 4.3 then equivalently asserts that F(Y ) P 1 + log Y p Y (4.2) provided P has positive natural density. This chapter’s goal is to establish this fact. Observe  F(Y )  1 for all Y  1.  F(Y ) = 0 for Y > y.  F is a decreasing function of Y . For convenience, we shall extend the domain of f to all fractional ideals of F by de ning f(t) := 0 if t is not a fractional ideal of (DN) 1. 4.2 Preliminary Lemmas We begin with a simple lemma, utilized frequently by Soundararajan throughout [Sou10]; this is essentially an immediate consequence of the Hecke relations of fP . Lemma 4.4. Let f be mock P-Hecke multiplicative of level N. Suppose p 2 P and n is a fractional ideal of (DN) 1. Then f(n)fP(p) = 8 < : f(np) p - n f(np) + f(np 1) p j n (4.3) so in particular jf(n)fP(p)j  jf(np)j+ jf(np 1)j; (4.4) 50and jf(np)j  jf(n)fP(p)j+ jf(np 1)j: (4.5) Similarly, f(n)fP(p2) = 8 >>>< >>>: f(np2) p - n f(np2) + f(n) p k n f(np2) + f(n) + f(np 2) p2 j n (4.6) so in particular jf(n)fP(p2)j  jf(np2)j+ jf(n)j+ jf(np 2)j: (4.7) and jf(np2)j  jf(n)fP(p2)j+ jf(n)j+ jf(np 2)j: (4.8) Proof. Note that since p - DN, the ideal n is integral in Op, so it is valid to only consider cases where p does or does not divide n. Write n = pkm where k  0 and m is a fractional ideal of (DN) 1 prime to p. Then the mock P-Hecke multiplicativity of f , and Hecke multiplicativity of fP from (4.1) gives f(n)fP(p) = f(m)fP(pk)fP(p) = f(m)  8 < : fP(p) k = 0 fP(pk+1) + fP(pk 1) k  1 = 8 < : f(np) k = 0 f(np) + f(np 1) k  1 as required. Similarly, we have f(n)fP(p2) = f(m)fP(pk)fP(p2) = f(m)  8 >>>< >>>: fP(p2) k = 0 fP(p3) + fP(p) k = 1 fP(pk+1) + fP(pk 1) k  2 = 8 >>>< >>>: f(np2) k = 0 f(np2) + f(n) k = 1 f(np) + f(np 1) k  2 as required. 51The next lemma gives an easy bound of the magnitude of fP in relation to F at prime ideals p 2 P. To be brief, the bound is derived from a combination of Lemma 4.4 and an application of Cauchy-Schwarz. Lemma 4.5. Suppose p 2 P. If Np  y, then jfP(p)j  2 F(Np)1=2 and, if Np  p y, then jfP(p)j  2 F(Np2)1=4 : Proof. Compare with Lemma 3.1 of [Sou10]. Let q := Np and n be any fractional ideal of (DN) 1. For the  rst bound, observe that (4.4) of Lemma 4.4 implies jfP(p)f(n)j2  2(jf(np)j2 + jf(np 1)j2) by Cauchy-Schwarz. Applying this inequality, we  nd jfP(p)j2 X Nn y=q jf(n)j2  2 X Nn y=q (jf(np)j2 + jf(np 1)j2): Since Np = q, notice Nnp  y and Nnp 1  y provided Nn  y=q, so the sum on the RHS is  4 X Nn y jf(n)j2 as required. Similarly, for the second bound, equation (4.4) of Lemma 4.4 implies jfP(p)f(n)j2  3(jf(np2)2+ jf(n)j2 + jf(np 2)j2) by Cauchy-Schwarz. Applying this inequality, we  nd jfP(p2)j2 X Nn y=q2 jf(n)j2  3 X Nn y=q2 (jf(np2)j2 + jf(n)j2 + jf(np 2)j2): Since Np = q, notice the norms of np2; n and np 2 are  y provided Nn  y=q2, so the sum on the RHS is  9 X Nn y jf(n)j2: Combining the two above inequalities, we conclude jfP(p2)j2  9 F(Np2) 52Then by the Hecke relations given in (4.1), jfP(p)j2  jfP(p2)j+ 1  3 F(Np2)1=2 + 1 Since F(Np2)  1, we have 1  F(Np2) 1=2, yielding the result. Proposition 4.6. Suppose a 2 N is a square-free integral ideal. Then X Nn y=Y ajn jf(n)j2   (a) Y pja (1 + jfP(p)j2)F(Y  Na) X Nn y jf(n)j2 where  (a) denotes the number of integral ideals dividing a. Furthermore, X Nn y=Y a2jn jf(n)j2   3(a) Y pja (2 + jfP(p)j2)F(Y  Na2) X Nn y jf(n)j2 where  3(a) denotes the number of ways of writing a as a product of three integral ideals. Proof. Compare with Proposition 3.2 of [Sou10]. Let m be a fractional ideal of (DN) 1. Using (4.5) from Lemma 4.4, we may  nd by induction that jf(ma)j  X st=a jfP(s)jjf(mt 1)j where the sum runs over integral ideals s and t. Note that in the above inductive argument, we utilize that a is square-free to simplify a product of distinct prime divisors of a into fP(s). Now, from the above inequality, it follows by Cauchy-Schwarz that jf(ma)j2   (a) X st=a jf(s)j2jf(mt 1)j2 where  (a) denotes the number of integral ideal divisors of a. Summing this inequality over all m  y=(Y  Na) and commuting the sums, we have X Nn y=Y ajn jf(n)j2 = X Nm y=(Y  Na) jf(ma)j2  X Nm y=(Y  Na)  (a) X st=a jfP(s)j2jf(mt 1)j2 =  (a) X st=a jfP(s)j2 X Nm y=(Y  Na) jf(mt 1)j2: 53Recall that f(mt 1) = 0 for t - m by convention. Thus, for t j a we have Nt  1 so we may bound the inner sum as follows: X Nm y=(Y  Na) jf(mt 1)j2 = X Nm y=(Y  Na) tjm jf(mt 1)j2 = X Nm0 y=(Y  Nat) jf(m0)j2  X Nm y=(Y  Na) jf(m)j2 = F(Y  Na) X Nn y jf(n)j2: which is independent of t. Using this bound in the previous inequality, we  nd X n y=Y ajn jf(n)j2   (a) 0 @ X sja jfP(s)j2 1 AF(Y  Na) X Nn y jf(n)j2: For the sum over s j a, we use the Hecke multiplicativity of fP to deduce X sja jfP(s)j2 = Y pja (1 + fP(p))2 because a is square-free. Substituting this into the former inequality, we have the desired  rst bound. Similarly, for the second bound, let m be a fractional ideal of (DN) 1. Using (4.8) from Lemma 4.4, we may  nd by induction that jf(ma2)j  X rst=a jfP(r2)jjf(mt 2)j where the sum runs over integral ideals r:s and t. Again, in the above inductive argument, we utilize that a is square-free to simplify a product of distinct prime divisors of a into fP(r2). Then from the above inequality, it follows by Cauchy-Schwarz that jf(ma2)j2   3(a) X rst=a jfP(r2)j2jf(mt 2)j2 where  3(a) denotes the number of integral ideal triples (r; s; t) such that rst = a. Summing 54this inequality over all m  y=(Y  Na2) and commuting the sums, we have X Nn y=Y a2jn jf(n)j2 = X Nm y=(Y  Na2) jf(ma2)j2  X Nm y=(Y  Na2)  3(a) X rst=a jfP(r2)j2jf(mt 2)j2 =  3(a) X rst=a jfP(r2)j2 X Nm y=(Y  Na2) jf(mt 2)j2: Recall that f(mt 2) = 0 for t2 - m by convention. Thus, for t j a we have Nt  1 so we may bound the inner sum as follows: X Nm y=(Y  Na2) jf(mt 2)j2 = X Nm y=(Y  Na2) t2jm jf(mt 2)j2 = X Nm0 y=(Y  N(at)2) jf(m0)j2  X Nm y=(Y  Na2) jf(m)j2 = F(Y  Na2) X Nn y jf(n)j2: which is independent of t. Using this bound in the previous inequality, we  nd X n y=Y a2jn jf(n)j2   3(a)  X rst=a jfP(r2)j2 ! F(Y  Na2) X Nn y jf(n)j2: (4.9) For the sum over triples rst = a, we see that X rst=a jfP(r2)j2 = X rja jfP(r2)j2 (ar 1) Since fP is Hecke-multiplicative and a is square-free, we may write each term in the above sum as jfP(r2)j2 (ar 1) = Y pjr jfP(p2)j2 Y pjar 1 2: 55Using this identity and the former equation. we deduce X rst=a jfP(r2)j2 = Y pja (2 + jfP(p2)j2): Substituting this into the RHS of (4.9), we have the desired result. 4.3 A Large Set of Prime Ideals Thus far, we have not made any assumption about the size of P, our set of prime ideals. In this section, we shall assume P has positive natural density in the set of prime ideals of F . Speci cally, we assume that the following limit  =  (P) := lim t!1 #fp 2 P : Np  tg #fp prime ideal : Np  tg exists and is positive, i.e.  > 0. It is well known [Lan94, p. 315] that this is equivalent to #fp 2 P : Np  tg   t log t : In Chapter 5, we will require the natural density of PN. Proposition 4.7. The set of ideals PN = fp  O : p unrami ed principal prime ideal with p - Ng has natural density equal to 1=hF where hF is the class number of F . Proof. Without loss, we may show the set of principal prime ideals of F has natural density 1=hF as this di ers from PN by only  nitely many prime ideals. Let L be the Hilbert class  eld of F . Recall a prime ideal p is principal in F if and only if it splits completely in L [Neu99, p. 409], which occurs if and only if its associated Frobenius element Frobp 2 Gal(L=F ) is trivial. Thus, by Chebatorev’s density theorem [Hei67], the set of principal prime ideals of F has natural density equal to #f1g#Gal(L=F ) = 1 hF . Now, we return to our generic set of prime ideals P with positive natural density. For Y  1, de ne P(Y ) := fp 2 P : Np 2 [ p Y =2; p Y ]g: 56Since P has positive density  , we have #P(Y )   p Y 2 log Y ; for Y  cP (4.10) where cP is some positive constant. In other words, we have a large set of primes, P(Y ). We wish to appropriately divide this large set P(Y ) according to values related to our mock P-Hecke multiplicative function f . The following construction mimics the discussion in [Sou10] preceding Proposition 3.3. We recall from Lemma 4.5 that for Np  p Y , jfP(p)j  2 F(Y )1=4 : Therefore, setting J :=  1 4 log 2 log(1=F(Y ))  + 3 (4.11) we see that 0  jfP(p)j  2J 1 for all p 2 P(Y ). We may thus partition P(Y ) into sets P0;P1; : : : ;PJ such that P0 = P0(Y ) = fp 2 P(Y ) : jfP(p)j  2 1g and for 1  j  J , Pj = Pj(Y ) = fp 2 P(Y ) : 2j 2 < jfP(p)j  2j 1g: For k  1, de ne N0(k) to be the set of fractional ideals of (DN) 1 divisible by at most k distinct squares of prime ideals in P0. For 1  j  J , de ne Nj(k) to be the set of fractional ideals of (DN) 1 divisible by at most k distinct prime ideals in Pj . The notion of divisibilty is well-de ned as P consists of unrami ed prime ideals. Proposition 4.8. Keep the notations above. For 2  k  jP0j=4, we have X Nn y=Y n2N0(k) jf(n)j2  4k jP0j X Nn y jf(n)j2: Further, if 1  j  J , and 1  k  jPj j=4 1, we have X Nn y=Y n2Nj(k) jf(n)j2  212k2 24j jPj j2 X Nn y jf(n)j2: 57Proof. Compare with Proposition 3.3 of [Sou10]. Since jfP(p)j  1=2 for p 2 P0, we have by the Hecke relations jfP(p2)j = jfP(p2) 1j  3=4. Thus, X Nn y=Y n2N0(k) jf(n)j2  X p2P0 p2-n jfP(p2)j2   X Nn y=Y n2N0(k) jf(n)j2  X p2P0 p2-n 9 16   9 16 (jP0j  k) X Nn y=Y n2N0(k) jf(n)j2  27 64 jP0j X Nn y=Y n2N0(k) jf(n)j2: (4.12) In the last step, we have noted k  jP0j=4. To achieve an upper bound on the LHS of (4.12), we  rst claim that for p 2 P0(n) and p2 - n, we have jf(n)fP(p2)j  jf(np2)j. For p - n, we have equality by (4.6) of Lemma 4.4. For p k n, again by (4.6), we see f(n)(fP(p2)  1) = f(np2). Since jfP(p2)j  1=2, it follows that jf(n)fP(p2)j  jf(n)(fP(p2)  1)j = jf(np2)j. This proves the claim. Thus, the LHS of (4.12) is  X Nn y=Y n2N0(k) X p2P0 p2-n jf(np2)j2: In the above sum, p 2 P0, we have Np2  Y , and so Nnp2  y. Also, n 2 N0(k) implies that the product np2 is divisible by at most k + 1 distinct squares of prime ideals of P0. Thus, in the sum, the terms are of the form jf(m)j2 where Nm  y and m is divisible by at most k + 1 distinct squares of prime ideals in P0; furthermore, each such term appears at most k+1 times. Therefore, the above is = X Nm y jf(m)j2  X m=np2 Nn y=Y;n2N0(k) p2P0;p2-n 1   (k + 1) X Nm y jf(m)j2: (4.13) For the second assertion, we argue similarly. Since jfP(p)j  2j 2 for p 2 Pj , by Hecke mul- tiplictativity, we have jfP(p1p2)j = jfP(p1)fP(p2)j  22j 4 for distinct p1; p2 2 Pj . Therefore, X Nn y=Y n2Nj(k) jf(n)j2  1 2 X p1;p22Pj p1 6=p2 p1-n; p2-n jfP(p1p2)j2   X Nn y=Y n2Nj(k) jf(n)j2  X p1;p22Pj p1 6=p2 p1-n; p2-n 24j 9  (4.14) 58The inner sum on the RHS counts the number of ordered pairs of distinct prime ideals in Pj not dividing n. By de nition, n 2 Nj(k) is divisible by at most k distinct prime ideals of Pj , and so there are at least jPj j  k prime ideals of Pj not dividing n. Thus, there are 2  jPj j k 2  terms in the inner sum, so the RHS  22j 9  2  jPj j  k 2  X Nn y=Y n2Nj(k) jf(n)j2 = 22j 9(jPj j  k)(jPj j  k  1) X Nn y=Y n2Nj(k) jf(n)j2  22j 9  9 16 jPj j 2 X Nn y=Y n2Nj(k) jf(n)j2 (4.15) since 2  k  jPj j=4 1. As for an upper bound, notice that since f is mock P-Hecke multiplicative, the LHS of (4.14) = 1 2 X Nn y=Y n2Nj(k) X p1;p22Pj p1 6=p2 p1-n; p2-n jf(np1p2)j2: In the above sum, since p1; p2 2 Pj , we have Np‘  p Y for ‘ = 1; 2, and so Nnp1p2  y. Also, n 2 Nj(k) implies that the product np1p2 is divisible by at most k + 2 distinct prime ideals of Pj . Thus, in the sum, the terms are of the form jf(m)j2 where Nm  y and m is divisible by at most k + 2 distinct prime ideals in Pj ; furthermore, each such term appears at most 2  k+2 2  times. Therefore, the above expression is   k + 2 2  X Nm y jf(m)j2  3k2 X Nm y jf(m)j2: Combining this result with (4.15), we have X Nn y=Y n2Nj(k) jf(n)j2  213k2 3  22j jPj j2 X Nm y jf(m)j2  212k2 22j jPj j2 X Nm y jf(m)j2 as desired. We have thus established the necessary facts to prove Theorem 4.3, the key technical result. 594.4 Proof of Theorem 4.3 Proof of Theorem 4.3. Compare with section 4 of [Sou10]. We shall utilize the notation of this chapter, and in particular, for a  xed value y  1, recall that we de ned F(Y ) = F(Y ; y) := P Nn y=Y jf(n)j 2 P Nn y jf(n)j 2 for Y  1. We again remark that F(Y ) is a decreasing function of Y , F(Y )  1, and F(Y ) = 0 for Y  y. We aim to show (4.2) holds. We claim it su ces to show F(Y )  C  1 + log Y p Y  for Y  y0, where C and y0 are positive constants depending only on P. Replacing C by maxfC; p y0g, the above inequality holds for Y  1 since F(Y )  1 for Y  1, thus proving the claim. To show the desired inequality, we shall take y0 := cP + 2 and C := 224  2 (4.16) where  =  P 2 (0; 1] is the natural density of P, and cP is as chosen in (4.10). Suppose, for a contradiction, that there exists Y  y0 such that F(Y ) > C  1 + log Y p Y  : (4.17) Since F (Y ) = 0 for Y > y, we may choose Y 2 [1; y] maximal with respect to the property that Y satis es the above inequality and no value larger than Y +1 does. Since Y  y0  cP , we have #P(Y )   p Y =(2 log Y ) by (4.10). As in the previous section, we divide P(Y ) into the sets Pj for 0  j  J where J is de ned in (4.11), so P(Y ) = tJj=0Pj . It follows by the Pigeonhole Principle, either #P0   p Y =(4 log Y ) or #Pj   p Y =(4J log Y ) for some 1  j  J . The arguments below are divided by these two distinguished cases. Case 1: #P0   p Y =(4 log Y ). Set K := [(#P0)F(Y )=8]. Then since Y satis es (4.17), we have K   p Y 4 log Y  C  1 + log Y p Y   1 8  1   C 32  1  218 since C  224. On the other hand, we can easily see that K  (#P0)=4 since F(Y )  1. Thus, 60we may apply Proposition 4.8 using K to conclude X Nn y=Y n2N0(K) jf(n)j2  1 2 F(Y ) X Nn y jf(n)j2 by noting 4K=(#P0)  12F(Y ) from the de nition of K. From this inequality, it follows that X Nn y=Y n62N0(K) jf(n)j2  1 2 F(Y ) X Nn y jf(n)j2: (4.18) If n 62 N0(K), then n must be divisible by at least K + 1 squares of prime ideals in P0. There are  #P0 K+1  integral ideals that are products of exactly K + 1 prime ideals from P0, and we denote this set of ideals as P0(K + 1). Each of these ideals has norm exceeding ( p Y =2)K+1 since P0  P(Y ). A fractional ideal n 62 N0(K) must be divisible by the square of one of these ideals, say a 2 P0(K + 1). To summarize, a2 j n and a is a square-free integral ideal composed entirely of prime ideals in P0  P. Thus, by Proposition 4.6, we have X Nn y=Y a2jn jf(n)j2   3(a) Y pja (2 + jfP(p)j2)F(Y  Na2) X Nn y jf(n)j2  3K+1  3K+1  F(Y  (Y=4)K+1) X Nn y jf(n)j2: In the above, we noted (i)  3(a) = 3K+1 as a is a product of exactly K + 1 distinct prime ideals, (ii) jfP(p)j2  1=2  1 by de nition of p 2 P0, and (iii) F is a decreasing function and Na2  (Y=4)K+1. Summing this inequality over all a 2 P0(K+1), by our previous observations, we deduce X Nn y=Y n62N0(K) jf(n)j2  X a2P0(K+1) X Nn y=Y a2jn jf(n)j2   #P0 K + 1  3K+1  3K+1F(Y  (Y=4)K+1) X Nn y jf(n)j2: (4.19) By some simple combinatorial bounds, Stirling’s formula, and our choice of K, we have  #P0 K + 1   (#P0)K+1 (K + 1)! <  e(#P0) K + 1  K+1 <  24 F(Y )  K+1 : 61Further, by the maximality of Y and (4.17), we know F(Y  (Y=4)K+1)  C  1 + log(Y K+2  4 (K+1)) Y (K+2)=2  4 (K+1)=2 !  2K+1C  1 + (K + 2) log Y Y (K+2)=2   2K+1C  1 + log Y Y 1=2  K+2  2K+1C  F(Y ) C  K+2 : Combining the last two inequalities into (4.19), we deduce that X Nn y=Y n62N0(K) jf(n)j2   24 F(Y )  K+1 3K+1  3K+1  2K+1C  F(Y ) C  K+2 X Nn y jf(n)j2   432 C  K+1 F(Y ) X Nn y jf(n)j2 < 1 2 F(Y ) X Nn y jf(n)j2 since C  224 and K  218, hence contradicting (4.18). Case 2: #Pj   p Y =(4J log Y ) for some 1  j  J . Set K := [22j 9(#Pj)F(Y )1=2]. By the contradiction assumption (4.17) and noting C  224, we see that J  3 + logF(Y ) 1 4 log 2  3 + log(Y 1=2=C) 4 log 2  log Y 4 : With this bound on J and (4.17), we have K  22j 9 p Y 4J log Y  C1=2  1 + log Y p Y  1=2  1  2 7 C1=2Y 1=4 (log Y )3=2  1: by also noting j  1. If we additionally observe that C = 224    2 and Y  2, we see that the RHS is  2 7  212  1 1  1  24; so K  24. 62On the other hand, for p 2 Pj , we have 22j 4  jfP(p)j2  4=F(Y )1=2 The left inequality follows by de nition of Pj , and the right inequality follows by Lemma 4.5 and noting Np2  Y . Rewriting this inequality, we see F(Y )1=2  2 2j+6, and so K  22j 9(#Pj)F(Y ) 1=2  (#Pj)=4: Thus, we may apply Proposition 4.8 using K to conclude X Nn y=Y n2Nj(K) jf(n)j2  1 2 F(Y ) X Nn y jf(n)j2: In the above, we have noted 2 12K2 24j(#Pj)2  12F(Y ) from the de nition of K. From this inequality, it follows that X Nn y=Y n62Nj(K) jf(n)j2  1 2 F(Y ) X Nn y jf(n)j2: (4.20) If n 62 Nj(K), then the fractional ideal n of (DN) 1 must be divisible by at least K+ 1 distinct prime ideals in Pj . There are  #Pj K+1  integral ideals that are products of exactly K + 1 prime ideals from Pj , and we denote this set of ideals by Pj(K + 1). Each of these ideals has norm exceeding ( p Y =2)K+1 since Pj  P(Y ). A fractional ideal n 62 Nj(K) must be divisible by one of these ideals, say a. To summarize, a j n and a 2 Pj(K + 1) is a square-free integral ideal composed entirely of prime ideals in Pj  P. Thus, by Proposition 4.6, we have X Nn y=Y ajn jf(n)j2   (a) Y pja (1 + jfP(p)j2)F(Y  Na)  2K+1  (22j 1)K+1F(Y  (Y=4)(K+1)=2) X Nn y jf(n)j2: In the above, we noted (i)  (a) = 2K+1 as a is a product of exactly K + 1 distinct prime ideals, (ii) jfP(p)j2  22j 2 by de nition of p 2 Pj , and (iii) F is a decreasing function and Na  (Y=4)2. Summing this inequality over all a 2 Pj(K + 1), by our previous observations, 63we deduce X Nn y=Y n62Nj(K) jf(n)j2  X a2Pj(K+1) X Nn y=Y ajn jf(n)j2   #P0 K + 1  22j(K+1)F(Y  (Y=4)(K+1)=2) X Nn y jf(n)j2: (4.21) By some simple combinatorial bounds, Stirling’s formula, and our choice of K, we have  #Pj K + 1   (#Pj)K+1 (K + 1)! <  e(#Pj) K + 1  K+1 <  3 22j 9F(Y )1=2  K+1 : Further, by the maximality of Y , we know F(Y  (Y=4)(K+1)=2)  C  1 + log(Y (K+3)=2  2 (K+1)) Y (K+3)=4  2 (K+1)=2 !  2(K+1)=2C  1 + 12(K + 3) log Y Y (K+3)=4 !  2(K+1)=2C  1 + log Y Y 1=2  (K+3)=2  2(K+1)=2C  F(Y ) C  (K+3)=2 : Combining the last two inequalities into (4.21), we deduce that X Nn y=Y n62Nj(K) jf(n)j2   3  2 2j+9 F(Y )1=2  K+1 22j(K+1)  2(K+1)=2C  F(Y ) C  (K+3)=2 X Nn y jf(n)j2   212 C  K+1 F(Y ) X Nn y jf(n)j2 < 1 2 F(Y ) X Nn y jf(n)j2 since C  224 and K  24, hence contradicting (4.20). This completes the proof. 64Remark. If one wishes to determine a constant C0 = C0(P) > 0 such that F(Y )  C0  1 + log Y p Y for all Y  1 (instead of just for su ciently large Y as per the above proof), then one may simply take C0 := maxf p y0; Cg, where y0 and C are as in (4.16), because F(Y )  1 for all Y  1. 65Chapter 5 Elimination of Escape of Mass We may now return to the motivating problem: elimination of escape of mass. Let F be a number  eld and let  =  0(N). As before, X is the product of hyperbolic 2- and 3-spaces associated to F , and our interest lies with the quotient space  nX . Suppose f jg1j=1 are a sequence of Hecke-Maa cusp forms of  , with associated probability measures d  j = j j(z)j2dvol(z) jj j jj2L2 on  nX . If the probability measures   j weak- converge to a measure  , then: Is  still a probability measure? In other words, is  ( nX ) = 1? In this chapter, we will retain this setup and answer this question in the a rmative. 5.1 Decay High in the Cusp From Theorem 2.13, we have gained a very clear understanding of the structure of  nX . Using a single parameter y  1, we may divide a fundamental domain F of  nX into one compact set S(y), and a  nite collection of non-compact cusps fg D(y) :  2  nP1(F )g of the form fcompactg  R>0. This decomposition is best characterized by Figure 2.2. If the measure  supposedly lost mass, then the mass must have \escaped" into a cusp. In other words, one should expect that, for some cusp  , the quantity   j (g D(y)) does not go to zero as y !1, for all su ciently large j. Thus, in order to eliminate escape of mass, it su ces to show that, for  xed j, the measure   j possesses some uniform decay in every cusp g D(y) as y !1. The following proposition addresses this key issue by applying Theorem 4.3 to the Fourier coe cients of a Hecke-Maa cusp form. 66Proposition 5.1. Let F be a number  eld,  be a Hecke-Maa cusp form for  =  0(N), and  be a cusp of  nX . Then for Y  1, we have 1 jj jj2L2 Z g D(Y ) j (z)j2dvol(z) N 1 + log Y p Y : Proof. Compare with Proposition 2 of [Sou10]. Keep notation as in Chapters 2 and 3. We may assume that Y is su ciently large, say Y  Y0  1 where Y0 depends only on F , so that Theorem 2.13 holds for y  Y0. First, we apply Parseval’s Formula to   (z) =  (g z) using Corollary 3.11, yielding Z g D(Y ) j (z)j2dvol(z) = Z D(Y ) j  (z)j2dvol(z) = Z D(Y ) X  2(DN) 1nf0g jc ( ; )j 2 jW (s; z)j 2 jN j dvol(z): (5.1) The integral on the LHS converges since Y  Y0, and by Theorem 2.13, D(Y ) injects into a fundamental domain of  ( )nX on which   is L2-integrable. The sum on the RHS converges absolutely by Parseval’s formula, so we may rearrange terms arbitrarily. In particular, as in Corollary 3.24, we reindex the Whittaker coe cients by non-zero principal fractional ideals n = ( ) of (DN) 1, and a  nite set of representatives  of O =V . Thus, the RHS of (5.1) = Z D(Y ) X  2O =V X n (DN) 1 n=( ) 6=(0) jc( ) ( ; n)j 2W(s;   z) Nn dz (5.2) where W(s; z) = X  2V jW (s;  z)j2: Note jW (s; z)j = jW (s; y)j by De nition 3.10, so it follows that W(s; z) =W(s; y). Now, recall D(Y ) = f(x;y) 2 F1  Rm>0 : x 2 U ;y 2 T (Y )g where T (Y ) = fy 2 Rm>0 : by 2 V;Ny 2 (Y;1)g, so we may write (5.2) as an iterated integral dvol(z) = dV1(x)dV2(y) as in (2.1). Using this parametrization, we  nd that (5.2) is = Z U Z T (Y ) X  2O =V X n (DN) 1 n=( )6=(0) jc( ) ( ; n)j 2W(s; j  jy) Nn dV2(y) dV1(x): 67By Tonelli’s theorem, we may arbitrarily swap integrals and sums as we please. If we swap the integral dV2(y) and both sums, and send y = (yv)v 7! (yv=j v vj) = y=j  j, then T (Y ) maps bijectively to    T (Y ), and dV2(y)! Nn  dV2(y) similar to (2.2). Thus, the above expression = Z U X  2O =V X n (DN) 1 n=( ) 6=(0) jc( ) ( ; n)j 2 Z    T (Y ) W(s; y) dV2(y) dV1(x): (5.3) Observe that the quantity   is determined up to multiplication by a unit in V , but since W(s;   y) =W(s; y) for  2 V , this choice is irrelevant. Our general aim is swap back the same integral and sum, but the domain of integration is not immediately clear. Let us consider the inner most integral dV2(y) of (5.3) for a given  2 (DN) 1 n f0g and  2 O =V . By de nition of W, we have Z    T (Y ) W(s; y)dy = X  2V Z    T (Y ) jW (s;  y)j2dy: (5.4) Substituting y = (yv)v 7! (yv=j vj), we have    T (Y ) maps to     T (Y ) and dV2(y) 7! jN jdV2(y) = dV2(y). Thus, the RHS = X  2V Z     T (Y ) jW (s; y)j2dy: (5.5) Now notice, for  2 V ,     T (Y ) = fy 2 Rm>0 : by 2 b    V;Ny 2 (Y  Nn;1)g since b  =   . From the proof of Theorem 2.7, the collection of sets f  V :  2 V g are disjoint, and their union is the set bY = fy 2 Rm>0 : Ny = 1g. As b  has norm 1, and bY is a group under multiplication, it follows that the sets fb    V :  2 V g are also disjoint, and their union is bY. Thus, we have shown G  2V     T (Y ) = fy 2 Rm>0 : Ny 2 (Y  Nn;1)g = G  2V   T (Y  Nn): 68As a result, the expression in (5.5) and hence the LHS of (5.4) = X  2V Z   T (Y  Nn) jW (s; y)j2dV2(y) = X  2V Z T (Y  Nn) jW (s;  y)j2dV2(y) = Z T (Y  Nn) W(s; y)dV2(y); by doing the substitution y 7!  y and swapping the sum and integral again. With this observation, we see that (5.3) = Z U X  2O =V X n (DN) 1 n=( )6=(0) jc( ) ( ; n)j 2 Z T (Y  Nn) W(s; y) dV2(y) dV1(x): Swapping the sum over ideals and inner integral back, notice W(s; y) has a contribution in the amount jc( ) ( ; n)j2 if and only if y 2 T (Y  Nn). This occurs equivalently when y 2 T (Y0) and Ny  Y  Nn. Thus, the above equation = Z U X  2O =V Z T (Y0) W(s; y) X Nn Ny=Y n=( )6=(0) jc( ) ( ; n)j 2 dV2(y) dV1(x): (5.6) For each  2 O =V , de ne f and P = PN as in the remark following De nition 4.2, so P has natural density 1=hF according to Proposition 4.7. Thus, we may apply Theorem 4.3 (with y = Ny=Y0 and Y = Y=Y0) to  nd that (5.6) is  N log(eY=Y0) p Y=Y0 Z U X  2O =V Z T (Y0) W(s; y) X Nn Ny=Y0 n=( )6=(0) jc( ) ( ; n)j 2dV2(y)dV1(x): Notice the remaining double integral is the same expression as (5.6) with Y = Y0, so we may unwind all of our steps and see that the above, and hence (5.1) is  N log(eY ) p Y Z D(Y0) j  (z)j2dvol(z): Since Y0  1 depends only on F , it may be absorbed into the implicit constant. Finally, by Theorem 2.13, the set g D(Y0) injects into a fundamental domain for  0(N)nX, so the above integral is bounded by the L2-norm of  , as required. 695.2 No Escape of Mass As per the discussion in the previous section, with a uniform decay in the cusps, we may  nally eliminate the possibility of escape of mass for probability measures of Hecke-Maa cusp forms. The culmination of this thesis work is embodied in the following theorem and its proof. Theorem 5.2. Let F be a number  eld, and N be an integral ideal. Suppose f jg1j=1 are Hecke-Maa cusp forms on  0(N)nX with probability measures   j . Suppose   j !  weak- , that is to say, 1 jj j jj2L2 Z A j j(z)j2dvol(z)! Z A d : for every compact set A contained in a fundamental domain of  0(N)nX. Then  is a probability measure. In other words, no escape of mass occurs. Proof. Compare with Theorem 1 of [Sou10]. Keeping notation consistent with Theorem 2.13, we may choose Y su ciently large (de- pending on F ) such that F = S(Y ) t G  2 g D(Y ) where F is a fundamental domain for  0(N)nX. In other words, F may be written as a compact centre S(Y ) and a  nite union of cusps g D(Y ). We shall analyze the convergence of probability measures on each of these sets. On the compact set S(Y ), by de nition of weak- convergence, we have that Z S(Y ) d  j = Z S(Y ) d + oj(1) as j !1. On the other hand, from Proposition 5.1, we have Z g D(Y ) d  j  N log(eY ) p Y : Notice that this bound is independent of  j . Combining these two observations and noting   j are probability measures, we deduce 1 = Z F d  j = Z S(Y ) d  j + X  2 Z g D(Y ) d  j = Z S(Y ) d + oj(1) + X  2 ON  log(eY ) p Y  : 70where   P1(F ) is an inequivalent set of representatives of cusps of  0(N). Taking j ! 1, and Y !1, we see that 1 = lim Y!1 Z S(Y ) d = Z F d since by Theorem 2.13(v), S(Y )! F as Y !1. This completes the proof. Remark. In applying Proposition 5.1, for a  xed  j , we only require some decay in the cusp as Y !1, as long as it is uniform with respect to  j . 71Conclusion We have eliminated the possibility of escape of mass occurring for Hecke-Maa cusp forms on congruence locally symmetric spaces, and hence on Hilbert modular varieties. As intended, this result can be applied to become a complete proof of AQUE for congruence locally symmetric spaces with a proof of the following conjecture. Conjecture 2. Let M be a congruence locally symmetric space with volume measure vol. Sup- pose f jg1j=1  L 2(M) is a sequence of Hecke-Maa cusp forms with Laplace eigenvalues  (j) = ( (j)v )vj1 such that  (j) v ! 1 for some v j 1. If   j wk-  !  , then  = c  vol for some c 2 [0; 1]. Corollary. Assume Conjecture 2 holds. Then by Theorem 5.2, AQUE holds for Hecke-Maa cusp forms on congruence locally symmetric spaces. In other words, Conjecture 1 holds. Conjecture 2 is the desired analogue of Lindenstrauss’ result [Lin06] for congruence surfaces, and as already noted, should be able to be shown by following methods of [Lin06], [EKL06] and [BL03]. While any proof eliminating escape of mass will likely require knowledge of the structure of congruence locally symmetric spaces as in Chapter 2, the heart of our proof lies in the key tech- nical result, Theorem 4.3, on mock P-Hecke multiplicative functions. This mysterious argument about Whittaker coe cients and their multiplicative relations, pioneered by Soundararajan, is non-trivial but involves little more than elementary number theory techniques. It leaves an interested reader still feeling unaware of the \true" reason for no escape of mass on congruence locally symmetric spaces. An analogous proof written in terms of the ad eles may be potentially more revealing and would be a worthy investigation. Another direction of work is a generalization of AQUE to higher rank locally symmetric spaces such as PGL(n;R) for n  3. The equidistribution result analogous to Lindenstrauss’ [Lin06] has been established by Silberman and Venkatesh in [SV07] and [SV11]. However, it again remains to eliminate the possibility of escape of mass in the non-compact case. The methods employed in this thesis do not immediately extend to this scenario due to complications 72with the Whittaker expansion, and so further study is necessary. On the other hand, if one can produce an ad elic proof as previously mentioned, such an argument may more naturally extend to the higher rank case. These future research objectives have relevant and meaningful impacts in the pursuit and understanding of AQUE and its implications to a diverse set of  elds, such as number theory, ergodic theory, and dynamical systems. Signi cant and deeper investigations concerning these puzzling questions on escape of mass will certainly be required to achieve these goals. 73Bibliography [BH99] Martin Bridson and Andr e Hae iger. Metric Spaces of Non-Positive Curvature. Springer-Verlag, 1999. ! pages 6 [BL03] Jean Bourgain and Elon Lindenstrauss. Entropy of Quantum Limits. Comm. Math. Phys., 233(1):153{171, 2003. ! pages 4, 72 [Bow10] Frank Bowman. Introduction to Bessel Functions. Dover Publications, 2010. ! pages 34 [BP92] Riccardo Benedetti and Carlo Petronio. Lectures on Hyperbolic Geometry. Springer-Verlag, 1992. ! pages 6 [CFKP97] James Cannon, William Floyd, Richard Kenyon, and Walter Parry. Hyperbolic Geometry. In S. Levy, editor, Flavours of Geometry., volume 31, pages 59{115. MSRI Publications, 1997. ! pages 6 [DS05] Fred Diamond and Jerry Shurman. A First Course in Modular Forms. Springer, 2005. ! pages 31 [EGM98] J urgen Elstrodt, Fritz Grunewald, and Jens Mennicke. Groups Acting on Hyperbolic Space: Harmonic Analysis and Number Theory. Springer-Verlag, 1998. ! pages 6 [EKL06] Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss. Invariant measures and the set of exceptions to Littlewoods conjecture. Ann. of Math. (2), 164(2):513{560, 2006. ! pages 4, 72 [Eva10] Lawrence Evans. Partial Di erential Equations. Graduate Studies in Mathematics. American Mathematical Society, 2nd edition, 2010. ! pages 32, 33, 35 [EW10] Manfred Einsiedler and Thomas Ward. Arithmetic Quantum Unique Ergodicity for  nH. Lecture notes., 2010. ! pages 3 [Fre90] Eberhard Freitag. Hilbert Modular Forms. Springer-Verlag, 1990. ! pages 11, 31 [Gee88] Gerard van der Geer. Hilbert Modular Surfaces. Springer-Verlag, 1988. ! pages 11, 16, 19, 22, 23, 25, 27, 29, 31 74[Hei67] Hans Heilbronn. Zeta functions and L-functions. In J.W.S Cassels and A. Fr ohlich, editors, Algebraic Number Theory., pages 204{230. Academic Press, 1967. ! pages 56 [Hid90] Haruzo Hida. Hilbert Modular Forms and Iwasawa Theory. Oxford Mathematical Monographs. Oxford University Press, 1990. ! pages 31, 41 [Hir73] Friedrich Hirzebruch. Hilbert Modular Surfaces. L’Ens. Math., 71:183{281, 1973. ! pages 11 [IK04] Henryk Iwaniec and Emmanuel Kowalski. Analytic Number Theory, volume 53. American Mathematical Society, 2004. ! pages 11, 31, 41, 42, 44 [Iwa02] Henryk Iwaniec. Spectral Methods of Automorphic Forms, volume 53 of Graduate Studies in Mathematics. American Mathematical Society, 2 edition, 2002. ! pages 6, 11, 31, 34, 37, 41, 42 [Jak94] Dmitry Jakobson. Quantum unique ergodicity for Eisenstein series on PSL2(Z)nPSL2(R). Ann. Inst. Fourier, 44:1477{1504, 1994. ! pages 3 [Lan94] Serge Lang. Algebraic Number Theory. Springer-Verlag, 2nd edition, 1994. ! pages 11, 33, 38, 56 [Lin06] Elon Lindenstrauss. Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math., 163:165{219, 2006. ! pages 3, 4, 72 [LS95] Wenzhi Luo and Peter Sarnak. Quantum ergodicity of eigenfunctions on PSL2(Z)nH2. Inst. Hautes  Etudes Sci. Publ. Math., 81:207{237, 1995. ! pages 3 [Mar77] Daniel Marcus. Number Fields. Springer, 1977. ! pages 11 [Mar06] Jens Marklof. Arithmetic quantum chaos. In J.-P. Francoise, G.L. Naber, and S.T. Tsou, editors, Encyclopedia of Mathematical Physics., volume 1, pages 212{220. Elsevier, 2006. ! pages 3 [ME05] M. Ram Murty and Jody Esmonde. Problems in Algebraic Number Theory. Springer, 2nd edition, 2005. ! pages 11 [Miy89] Toshitsune Miyake. Modular Forms. Springer, 1989. ! pages 11, 31, 41 [Neu99] J urgen Neukirch. Algebraic Number Theory. Springer-Verlag, 1999. ! pages 11, 25, 38, 56 [Rat94] John Ratcli e. Foundations of Hyperbolic Manifolds. Springer-Verlag, 1994. ! pages 6 [RS94] Zeev Rudnick and Peter Sarnak. The behaviour of eigenstates of arithmetic hyperbolic manifolds. Comm. Math. Phys., 161:195{213, 1994. ! pages 2, 3 75[Sar11] Peter Sarnak. Recent Progress on the Quantum Unique Ergodicity Conjecture. Bull. Amer. Math. Soc., 48:211{228, 2011. ! pages 3, 4 [Sou10] Kannan Soundararajan. Quantum Unique Ergodicity for SL2(Z)nH. Ann. of Math. (2), 172(2):1529{1538, 2010. ! pages 2, 3, 4, 5, 42, 46, 48, 50, 52, 53, 57, 58, 60, 67, 70 [SV07] Lior Silberman and Akshay Venkatesh. On Quantum Unique Ergodicity for Locally Symmetric Spaces I. Geom. Funct. Anal., 17(3):960{998, 2007. ! pages 72 [SV11] Lior Silberman and Akshay Venkatesh. On Quantum Unique Ergodicity for Locally Symmetric Spaces II. To appear in GAFA. Available at http://arxiv.org/abs/1104.0074, 2011. ! pages 72 [Zwi98] Daniel Zwillinger. Handbook of Di erential Equations. Academic Press, 3rd edition, 1998. ! pages 34, 35 76

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