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, 2012 Abstract Let F be a number field, G = PGL(2, F∞ ), and K be a maximal compact subgroup of G. We eliminate the possibility of escape of mass for measures associated to Hecke-Maaß 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)\H2 , and follows his approach closely. First, we define M , a congruence locally symmetric space covered by G/K, and articulate the details of its structure. Then we define Hecke-Maass cusp forms and provide their Whittaker expansion along with identities regarding the Whittaker coefficients. 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-Maaß cusp forms on M . ii Table 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, F∞ ) . . . . . . . . . . . . . . . . . . . . . . . . . 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 Γ\X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Hecke-Maaß Cusp Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Maaß Forms and their Fourier Expansion . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Maaß Cusp Forms and their Whittaker Expansion . . . . . . . . . . . . . . . . . 33 3.3 Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Hecke-Maaß Cusp Forms and their Whittaker Coefficients . . . . . . . . . . . . . 43 iii 4 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 iv List of Figures Figure 2.1 A fundamental domain for SL(2, Z)\H2 . . . . . . . . . . . . . . . . . . . . . 12 Figure 2.2 Depiction of Γ\X with 3 cusps . . . . . . . . . . . . . . . . . . . . . . . . . . 29 v List of Symbols a Av AQUE integral ideal associated to σ ∈ P1 (F ), except in Chapter 4 . . . . . . . . . . . . . . . . . . . . 16 area measure of Fv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Arithmetic Quantum Unique Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Bv± (sv ; yv ) special function composed of Bessel functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 δv B δ (s; y) v|∞ Bv (sv ; yv ) for δ ∈ {±} cP large positive constant depending only on P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 cσ (φ; α) α-Whittaker coefficient of φ at cusp σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (ξ) cσ (φ; n) n-Whittaker ideal coefficient of φ at cusp σ for ξ ∈ O× /V . . . . . . . . . . . . . . . . . . . . 47 field of complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 C DF D absolute discriminant of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 fundamental domain for Λ\X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 D(y) deep in a cusp, D(y) = {(x, y) ∈ D : Ny ∈ (y, ∞)} . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 D f absolute different ideal of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 e(x) = e2πix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 mock P-Hecke multiplicative function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 fP P-Hecke multiplicative function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 F F∞ number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 completion of F at place v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 F∞ = v|∞ Fv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 F fundamental domain for Γ\X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 F(Y ) function associated to mock P-Hecke multiplicative function f . . . . . . . . . . . . . . . 50 gσ G element of PGL(2, F ) such that gσ (∞) = σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16 G = v|∞ Gv = PGL(2, F∞ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Gv Gv = PGL(2, Fv ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 hF H2 class number of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 hyperbolic 2-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 H3 hyperbolic 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 e(x) Fv vi H Hamilton’s quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Iν (y) I-Bessel function, ν ∈ C, y ∈ R>0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34 Im K imaginary part of H2 , H3 or X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 well chosen positive integer based on F(Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 v|∞ Kv , maximal compact subgroup of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Kv maximal compact subgroup of Gv for v | ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Kν (y) K-Bessel function, ν ∈ C, y ∈ R>0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 m number of archimedean places of F , i.e. m = r1 + r2 . . . . . . . . . . . . . . . . . . . . . . . . . 11 M congruence locally symmetric space, i.e. M = Γ\X . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 n degree of F , i.e. n = [F : Q] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 N N integral ideal of F representing level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 global norm, i.e. N = NFQ , extended to X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Nv local norm at v | ∞, i.e. N = NFRv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 N O P-friable ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 PN -friable ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ring of integers of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 O× group of integral units of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 O1× roots of unity of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 p prime ideal of F , often principal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 P subset of unramified prime ideals not dividing N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 PN Pj set of unramified principal prime ideals not dividing N . . . . . . . . . . . . . . . . . . . . . . . 41 √ √ {p ∈ P : Np ∈ [ Y /2, Y ]} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 suitably chosen subset of P(Y ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 P1 (F ) projective linear space of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Q QUE field of rational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Quantum Unique Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 r1 number of real places of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 r2 number of complex places of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 R s field of real numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 λv = sv (1 − sv ) ∈ C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 s = (sv )v|∞ ∈ Cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 S(y) compact centre of Γ\X , grows as y → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 T (n) T (y) n-Hecke operator (usually for n ∈ NN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 y-coordinate of D(y), i.e. {y ∈ Rm >0 : y ∈ V, Ny ∈ (y, ∞)} . . . . . . . . . . . . . . . . . . . . 36 Tr global trace, i.e. Tr = TrFQ , extended to F∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 Trv local trace for v | ∞, i.e. Trv = TrFRv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 J NN P(Y ) sv vii v|∞ fundamental domain for (O1× ∩ V )\F∞ /N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 archimedean place of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 V V = (1 + N) ∩ O× . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 V fundamental domain for V \Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 volume measure for z ∈ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 U vol(z) V1 (x) V2 (y) volume measure for x-coordinate in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 volume measure for y-coordinate in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 W (s; z) full Whittaker function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Wv (sv ; zv ) local Whittaker function at v | ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 x = (xv )v|∞ ∈ F∞ , coordinate of z = (x, y) ∈ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 x X y0 y symmetric space of G, equals (H2 )r1 × (H3 )r2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 (small) positive constant depending only on F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 y = (yv )v|∞ ∈ Rm >0 , coordinate of z = (x, y) ∈ X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Y y = (yv /(Ny)1/n )v|∞ ∈ Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 (large) positive constant depending only on F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 m {y ∈ Rm >0 : Ny = 1}, hyperplane in R>0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 z element of X , alternate coordinates z = (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Z ring of rational integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 ∆v Laplacian at place v | ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 φ(z) Hecke-Maaß cusp form on Γ\X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 φσ (z) φσ (z) = φ(gσ z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 φσ (y; α) α-Fourier coefficient of φ at cusp σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Γ congruence group of level N, often Γ = Γ0 (N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Γ(N) principal congruence subgroup of level N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Γ0 (N) specific congruence group of level N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Γ(σ) Γ(σ) = gσ−1 Γgσ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ΓF = PGL(2, O) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 y Y0 ΓF λv λ ∆v -eigenvalue of Maaß form φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 λ = (λv )v|∞ ∈ Cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 λφ (n) T (n)-Hecke eigenvalue of φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Λ µφ subgroup of Γ(σ) -stabilizer of ∞ ∈ P1 (F ) for any σ ∈ P1 (F ) . . . . . . . . . . . . . . . . . . 18 measure associated to a Hecke-Maaß cusp form φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Ω set of inequivalent representatives of Γ\P1 (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ρ(σ; z) function measuring closeness of z ∈ X to σ ∈ P1 (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . 22 σ element of P1 (F ) regarded as cusp of Γ, often σ = (α : β) . . . . . . . . . . . . . . . . . . . . 15 trace form billinear pairing on F∞ × F∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ·, · viii Acknowledgments 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 entertaining. 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 extremely helpful and friendly department staff, 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 selflessly 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. ix To my parents x Introduction 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 commonplace concept in real analysis. Let M be a topological measure space, and {µj }∞ j=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 specific type, namely weak-∗ convergence. We say the sequence µj converges to a measure µ in the weak-∗ topology if for every compact set A ⊆ M , we have dµj → A dµ, as j → ∞. A wk-∗ More simply, we say µj weak-∗ converges to µ, or write µj −→ µ. If one assumes all µj are probability measures, i.e. wk-∗ M dµj = 1 and µj −→ µ, then it is a natural question to ask: Is µ still a probability measure? In other words, is M dµ = 1? In these general terms, the question is a simple exercise in real analysis. If M is compact, then it follows immediately from the definition of weak-∗ convergence that µ is indeed a probability measure. However, if M is not compact, then the answer is: not necessarily. Since {µj }∞ j=1 are probability measures, it follows that M dµ ≤ 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 dµj := 1[j,j+1] (x)dx for j ≥ 1 where dx is the Lebesgue measure. With this choice, one can easily see that µj 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 µj and manifold M . Beginning with the classical case, let Γ be a congruence group (e.g. 1 SL(2, Z)) and H2 denote the upper half plane of complex numbers endowed with the hyperbolic metric. The negatively curved manifold M = Γ\H2 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-Maaß cusp forms on M (see Chapter 3 for details). A Hecke-Maaß cusp form is a function φ ∈ L2 (Γ\H2 ), which is a simultaneous eigenfunction of the hyperbolic Laplacian ∆, and every Hecke operator T (n), n ∈ Z+ . Further, to φ, we can associate the probability measure dµφ := |φ(z)|2 dvol(z) ||φ||2L2 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 Γ ≤ SL(2, Z) be a congruence group. Suppose wk-∗ 2 {φj }∞ j=1 is a sequence of Hecke-Maaß cusp forms on Γ\H , and µφj −→ µ for some measure µ. Then µ is a probability measure. In other words, no escape of mass occurs. Utilizing the multiplicative properties of the Hecke eigenvalues of φ inherited from the Hecke operators {T (n) : n ∈ Z+ }, 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 first stated by Rudnick and Sarnak in 1994. Conjecture (Quantum Unique Ergodicity, [RS94]). Let M be a compact negatively curved 2 manifold M with Laplacian ∆ and volume measure vol. Suppose {φj }∞ j=1 ⊆ L (M ) is a sequence of eigenfunctions of ∆ with eigenvalues λj → ∞ and with associated probability measures dµφj = |φj (z)|2 dvol(z) . ||φj ||2L2 wk-∗ If µφj −→ µ, then µ = 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 2 conjecture, 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 hy2 perbolic surface with Laplacian ∆ and volume measure vol. Suppose {φj }∞ j=1 ⊆ L (M ) is a wk-∗ sequence of eigenfunctions of ∆ with associated probability measures µφj . If µφj −→ µ, then µ = vol. Not long after this conjecture, the combined works of Luo and Sarnak [LS95], and Jakobson [Jak94] onfirmed AQUE for the continuous spectrum of ∆. However, AQUE for the discrete spectrum remained unknown until in 2006, Lindenstrauss [Lin06] proved AQUE for Hecke-Maaß cusp forms φ except for the possibility of escape of mass. Theorem (Lindenstrauss, 2006 [Lin06]). Let M be a congruence surface. Suppose {φj }∞ j=1 ⊆ wk-∗ L2 (M ) is a sequence of Hecke-Maaß cusp forms with Laplace eigenvalues λj → ∞. If µφj −→ µ, then µ = c · vol for some c ∈ [0, 1]. If M is a compact surface, one can conclude c = 1 by the definition 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 µ 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-Maaß 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 = Γ\H2 . A natural and well known analogue is a Hilbert modular variety. For a totally real number field 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 finite volume manifold M = SL(2, O)\(H2 )n is known as the Hilbert modular variety of F . In the case F = Q, the field of rational numbers, this is simply the modular surface. As in Chapter 3, Hecke-Maaß cusp forms can be defined more generally on Hilbert modular varieties, and so one can conjecture AQUE holds for Hecke-Maaß cusp forms on Hilbert modular varieties. 3 In fact, one can allow F to be an arbitrary number field 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 define a finite volume manifold ΓF \X where ΓF = PGL(2, O). Even more generally, one can replace ΓF by congruence subgroups Γ of PGL(2, O) and study the manifolds M = Γ\X , which we call congruence locally symmetric spaces. Again, Hecke-Maaß cusp forms can be defined in this case, so we arrive at the following generalized conjecture. Conjecture 1. AQUE holds for Hecke-Maaß 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. 2 Theorem. Let M be a congruence locally symmetric space, and let {φj }∞ j=1 ⊆ L (M ) be Heckewk-∗ Maaß cusp forms of M with associated probability measures µφj . If µφj −→ µ for some measure µ, then µ 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 Soundararajan [Sou10] in the classical case except for one main distinction. For a Hecke-Maaß cusp form φ on the modular surface SL(2, Z)\H2 , the Whittaker coefficients {cσ (φ; n) : n ∈ Z} at a cusp σ of φ can be identified with its Hecke eigenvalues λφ (n); namely, for all n ∈ Z, cσ (φ; n) = Cφ · λφ (n) where Cφ ∈ C is some constant. As a result, cσ (φ; n) inherits the multiplicative properties of λφ (n), on which Soundararajan’s result critically relies. Now, in our case, for a Hecke-Maaß cusp form φ on a congruence locally symmetric space Γ\X of level N, this identification is no longer true. {cσ (φ; α) : α ∈ (DN)−1 } Instead, the Whittaker coefficients at a cusp σ possess a less restrictive relation with the Hecke eigen- values {λφ (n) : n ⊆ O ideal }. Namely, suppose an integral ideal n is composed of unramified principal prime ideals not dividing the level of Γ, so n = (η) is itself principal. If α ∈ (DN)−1 is a unit modulo n, then cσ (φ; αη) = cσ (φ; α) · λφ (n). 4 This 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 unramified prime ideals of a number field 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 |f (n)|2 Nn≤y/Y P 1 + log Y √ Y |f (n)|2 . Nn≤y for 1 ≤ Y ≤ y. On a separate note, while Soundararajan’s proof is written for SL(2, Z)\H2 , it is apparent that the argument can be easily adjusted to any congruence surface Γ\H2 . The proof we provide shall explicitly take into account the level of Γ and hence any congruence locally symmetric space Γ\X . 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 define a congruence locally symmetric space M , and describe its structure as a finite number of cusps and a compact centre. In Chapter 3, we define HeckeMaass cusp forms on M , provide their Whittaker expansion, and show the relations between coefficients 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. 5 Chapter 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 := {z = (x, y) : x ∈ R, y ∈ R>0 } with the metric ds2 = y −2 (dx2 + dy 2 ). We may regard H2 as a subset of the complex numbers C by identifying (x, y) → x + iy, i.e. i = (0, 1). We also define the map Im : H2 → R>0 (x, y) → y. The associated volume measure is given by dvol(z) = dA(x)dy y2 where dA(x) = dx, 6 (1.1) and the associated Laplace-Beltrami operator is given by ∆ = y2 ∂2 ∂2 + 2 . 2 ∂x ∂y (1.2) Closely related to H2 is the group PGL(2, R) := GL(2, R)/{λI : λ ∈ R∗ } which acts transitively on the upper half plane H2 ⊆ C via fractional linear transformations: az + b cz + d g · z := az + b cz + d det g > 0 , for g = det g < 0 a b c d ∈ PGL(2, R) and z ∈ 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) ∈ H2 is PGL(2, R) · (0, 1) = H2 . By direct calculations, one can verify that the stabilizer of (0, 1) ∈ H2 is PO(2, R) := {A ∈ GL(2, R) : AT A = I}/{±I}, so we may identify the quotient group PGL(2, R)/PO(2, R) with H2 via the map gPO(2, R) → g · (0, 1). Equipping the quotient group with the quotient topology, this identification is in fact a diffeomorphism. We summarize our conclusions in the following proposition. Proposition 1.1. The space H2 , equipped with the metric ds, is diffeomorphic to PGL(2, R)/PO(2, R), equipped with the natural quotient topology. Further, the diffeomorphism 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 C ∞ (H2 ), on which the Laplace-Beltrami operator ∆ acts naturally. The group PGL(2, R) also acts on naturally on this space: for g ∈ PGL(2, R) and φ(z) ∈ C ∞ (H2 ), we define g · φ(z) := φ(g · z). Either by direct computation or realizing ∆ as an element of the center of the universal enveloping 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 C ∞ (H2 ). Proposition 1.2. The Laplace-Beltrami operator ∆ commutes with the action of PGL(2, R) on C ∞ (H2 ), which is defined by g · φ(z) = φ(g · z) for g ∈ P GL(2, R) and φ(z) ∈ C ∞ (H2 ). As a final 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 ∈ 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 := {z = (x, y) : x ∈ C, y ∈ R>0 } with the metric ds2 = y −2 (dx21 + dx22 + dy 2 ) where x = x1 + ix2 . We may regard H3 as a subset of the Hamilton’s quaternions H by identifying (x, y) → x + jy, i.e. j = (0, 1). We also define the map Im : H3 → R>0 (x, y) → y. The associated volume measure is given by dvol(z) = dA(x)dy y3 where dA(x) = dx1 dx2 , (1.3) and the associated Laplace-Beltrami operator is given by ∆ = y2 ∂2 ∂2 ∂2 ∂ + + −y . 2 2 2 ∂y ∂x1 ∂x2 ∂y 8 (1.4) Closely related to H3 is the group PGL(2, C) := GL(2, C)/{λI : λ ∈ C× } which acts on H3 ⊆ H via fractional linear transformations: g · z := (az + b)(cz + d)−1 , for g = a b c d ∈ PGL(2, C) and z ∈ H3 , where one considers the point at ∞ in a natural limiting sense. One can verify (or force by definition) 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) ∈ H3 is PGL(2, C) · (0, 1) = H3 . By direct calculations, one can verify that the stabilizer of (0, 1) ∈ H3 is PU(2, C) := {A ∈ GL(2, C) : AH A = I}/{λI : λ ∈ C× , |λ| = 1}, so we may identify the quotient group PGL(2, C)/PU(2, R) with H3 via the map gPU(2, C) → g · (0, 1). Equipping the quotient group with the quotient topology, this identification is in fact a diffeomorphism. We summarize our conclusions in the following proposition. Proposition 1.4. The space H3 , equipped with the metric ds, is diffeomorphic to PGL(2, C)/PU(2, C), equipped with the natural quotient topology. Further, the diffeomorphism 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 C ∞ (H3 ), on which the Laplace-Beltrami operator ∆ acts naturally. The group PGL(2, C) also acts on naturally on this space: for g ∈ PGL(2, C) and φ(z) ∈ C ∞ (H3 ), we define g · φ(z) := φ(g · z). Either by direct computation or realizing ∆ as an element of the center of the universal enveloping 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 C ∞ (H3 ). Proposition 1.5. The Laplace-Beltrami operator ∆ commutes with the action of PGL(2, C) 9 on C ∞ (H3 ), which is defined by g · φ(z) = φ(g · z) for g ∈ P GL(2, C) and φ(z) ∈ C ∞ (H3 ). As a final 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 ∈ PGL(2, R). 10 Chapter 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)\H2 (see Figure 2.1) to a congruence locally symmetric space M = Γ\X . We describe in detail a congruence locally symmetric space M = Γ\X utilizing the models of hyperbolic 2- and 3-space, and fixing 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 finite 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, F∞ ) For the remainder of this thesis, fix a number field 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. Define the groups Gv := PGL(2, Fv ) := GL(2, Fv ) ηI : η ∈ Fv× We are interested in the m := r1 + r2 archimedean places v of F , denoted v | ∞. For v | ∞, 11 Figure 2.1: A fundamental domain for SL(2, Z)\H2 Source: c Fropuff, 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 PO(2, R) Kv := PU(2, C) v real , v complex where PO(2, R) = O(2, R)/Z(O(2, R)) = A ∈ GL(2, R) : AT A = I /{±I}, PU(2, C) = U(2, C)/Z(U(2, C)) = {A ∈ GL(2, C) : A∗ A = I} / ηI : η ∈ C× , |η| = 1 . Define G := PGL(2, F∞ ) = Gv , v|∞ 12 K := Kv , v|∞ where F∞ = v|∞ Fv . This yields the following diffeomorphism: r1 Gv /Kv ∼ = G/K = r2 2 j=1 v|∞ H3 H × j=1 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 sufficient and consistent description of X . Elements of X will be written in the following coordinates: z = (zv )v|∞ ∈ X H2 where zv = (xv , yv ) ∈ H3 v real . v complex Often, we will decompose these coordinates so that z = (x, y) ∈ X where x = (xv )v|∞ ∈ F∞ , and y = (yv )v|∞ ∈ Rm >0 . With this parametrization, we define the map Im : X → Rm >0 (x, y) → y. so Im(z) = (Imv (zv ))v|∞ . The volume measure associated to X is simply the product measure dvol(z) = ∧v|∞ dvolv (zv ), or in (x, y)-coordinates is given by dvol(z) = dV1 (x) dV2 (y) where dV1 (x) := dAv (xv ) and v|∞ dV2 (y) := v|∞ 1 dyv · . Nv yv yv (2.1) Note Nv extends the local norm NFRv of a place v | ∞ to H2 for v real and to H3 for v complex by z v Nv zv = z z v v 13 v real v complex . We shall also extend the global norm N = NFQ to z ∈ X via the usual product formula: N v zv . Nz = v|∞ As a simple remark, for r ∈ R>0 , the substitution y → r · y := (ryv )v|∞ yields dV2 (r · y) = v|∞ 1 dyv = Nv (ryv ) yv 1 v|∞ r[Fv :R] dV2 (y) = r−n dV2 (y). (2.2) Finally, ∆v shall denote the Laplacian associated to the zv -coordinate of z ∈ X . 2.2 Congruence Subgroups and their Cusps Consider the group PGL(2, F ) := GL(2, F )/ ηI : η ∈ F × which can be naturally embedded PGL(2, F ) → PGL(2, Fv ) = G (2.3) v|∞ via the usual embeddings F → v|∞ 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 define the distinguished group ΓF := PGL(2, O) = A ∈ M2×2 (O) : det A ∈ O× / ηI : η ∈ O× . Definition 2.1. Let N be an integral ideal of O. Define a subgroup Γ of ΓF = PGL(2, O) to be a congruence subgroup of level N if Γ contains Γ(N) := {A ∈ GL(2, O) : A ≡ I (mod N)} / ηI : η ∈ (1 + N) ∩ O× . the principal congruence subgroup of level N. Note Γ(N) is the kernel of the reduction map PGL(2, O) → PGL(2, O/N). If the level of Γ is unspecified, we simply call Γ a congruence subgroup. 14 Remark. The Hilbert modular group PSL(2, O) = {A ∈ M2×2 (O) : det A = 1} {±I} . is a congruence subgroup of level O. Proposition 2.2. Every congruence subgroup Γ of level N is finite index in ΓF = PGL(2, O), and discrete in G = v|∞ PGL(2, Fv ). Proof. The finite index property follows immediately from the reduction map, since [ΓF : Γ(N)] = #PGL(2, O/N) < ∞, so any intermediate subgroup is also finite index. For the discreteness, it suffices to prove the result for Γ = ΓF by the finite index property. This fact is immediate as ΓF = PGL(2, O) → PGL(2, Fv ) = G v|∞ via F → v|∞ Fv , and noting O embeds discretely under this map. Suppose Γ ≤ PGL(2, O) is a congruence subgroup of level N, so from (2.3), Γ possesses an action on X . We aim to understand the structure of M = Γ\X , which is equivalent to the double coset space Γ\G/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 Γ. 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. Definition 2.3. A cusp σ of congruence subgroup Γ ≤ PGL(2, O) is a Γ-orbit in P1 (F ). One often identifies a cusp with a representative of its orbit. For σ = (α : β) ∈ P1 (F ), we shall always assume both α and β are integral. Note that this choice σ = (α : β) is unique up to multiplication in O, i.e. σ = (µα : µβ) for µ ∈ O. Also, the point (1 : 0) ∈ P1 (F ) is the point at infinity and is denoted ∞. Proposition 2.4. There exists an bijective correspondence between the set of cusps of ΓF = P GL(2, O) and the ideal class group C (F ) of F , namely ΓF · P1 (F ) → C (F ) ΓF (α : β) → (α, β). In particular, the number of cusps of ΓF equals hF , the class number of F . 15 Proof. [Gee88, p. 6] a there exists γ = c we may assume that Suppose σ, τ ∈ P1 (F ) are contained in the same P GL(2, O) orbit. Then b ∈ P GL(2, O) such that γ · σ = τ . Writing σ = (α : β) with α, β ∈ O, d β = 0. Then we see that τ = γ · σ = γ · ( αβ : 1) = = a αβ + b c αβ + d :1 aα + bβ :1 cα + dβ = (aα + bβ : cα + dβ) , so the ideal class associated to τ is (aα + bβ, cα + dβ). Since γ ∈ P GL(2, O), this ideal equals (α, β). Thus, the map is well-defined. 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 σ = (α : β) and τ = (δ : ρ) possess the same associated ideal class, where α, β, δ, ρ ∈ O. Then by the definition of equivalence in the ideal class group, we may multiply by a suitable element of F and assume (α, β) = (δ, ρ) = a. Observe 1 ∈ O = aa−1 = αa−1 + βa−1 so there exists α∗ , β ∗ ∈ a−1 such that αβ ∗ − α∗ β = 1. Similarly, choose δ ∗ , ρ∗ ∈ a−1 such that δρ∗ − δ ∗ ρ = 1. In other words, the matrices gσ := α α∗ β β∗ , gτ := δ δ∗ ρ ρ∗ have determinant 1, and transform the cusp ∞ = (1 : 0) to σ and τ respectively. Notice gσ gτ−1 ∈ P GL(2, O) = ΓF and from our previous observations, gσ gτ−1 sends τ to σ. Hence, σ and τ belong to the same ΓF -orbit, as desired. In the above proof, the integral ideal a is dependent on the choice of α and β for which σ = (α : β) ∈ P1 (F ). Henceforth, to remove this dependence, for σ ∈ P1 (F ), always choose α, β ∈ O such that σ = (α : β) and |Na| is minimum where a = (α, β). It follows that if divides both α and β then ∈ by a unit ∈ O× , ∈O so the pair α and β are now determined up to multiplication O× . Consequently, the integral ideal a of minimum norm is uniquely defined for each σ ∈ P1 (F ), and shall be referred to as the ideal associated to σ ∈ P1 (F ) . From Proposition 2.4, we see 16 that the same ideal a is associated to any element of the orbit ΓF · σ. As a separate remark, notice that even if we choose α, β ∈ O such that a = (α, β) is the ideal associated to σ ∈ P1 (F ), the precise definition of gσ still depends on the choice of α and β. Nonetheless, for the remainder of this paper, we will retain the definition of gσ using α and β generating the ideal associated to σ ∈ P1 (F ). The dependence of gσ on the exact choice of α and β will not be relevant as the choice only depends on F . Corollary 2.5. Every finite index subgroup Γ of ΓF , and in particular every congruence subgroup, has finitely many cusps. Proof. Every cusp of ΓF decomposes into at most [ΓF : Γ] cusps of Γ, and since ΓF has finitely many cusps, so must Γ. From Proposition 2.2, the result therefore applies to congruence subgroups. For the remainder of this thesis, we shall fix the level to be the integral ideal N. 2.3 Cusp Stabilizer For σ = (α : β) ∈ P1 (F ) and a congruence subgroup Γ, define Γ(σ) := gσ−1 Γgσ , which is a subgroup of PGL(2, F ). Under this conjugation, the cusp σ of Γ\X becomes the cusp ∞ of Γ(σ) \X since gσ (∞) = σ. By direct verification, we see that the Γ(σ) -stabilizer of ∞ = (1 : 0) is given by the upper triangular elements of Γ(σ) . To obtain more detailed information, we shall explicitly describe the subgroup of upper triangular elements of Γ(N)(σ) since it is finite index in Γ(σ) . Recall that det gσ = 1 and gσ is of the form a a−1 a a−1 where a = (α, β) for which σ = (α : β) with specified α, β ∈ O. By direct computations, one can verify that Γ(N)(σ) = a b c d : a, d ∈ 1 + N, b ∈ Na−2 , c ∈ Na2 , ad − bc ∈ O× {ηI : η ∈ O× } and so stab(∞; Γ(N)(σ) ) = θ 0 1 : ∈ (1 + N) ∩ O× , θ ∈ Na−2 17 {ηI : η ∈ O× }. (2.4) We remark that now both of these definitions are independent of our choice of α and β for which σ = (α : β), since the ideal a associated to σ is also independent of these choices. For the sake of simplicity, we would like to restrict our attention to a finite index subgroup of stab(∞; Γ(N)(σ) ) by restricting the additive subgroup over which b ranges. Proposition 2.6. Let σ = (α : β) ∈ P1 (F ) and Γ be a level N congruence subgroup. Define Γ(σ) := gσ−1 Γgσ where gσ is as in Proposition 2.4. Then the Γ(σ) -stabilizer of ∞ ∈ P1 (F ) contains the finiteindex subgroup, depending only on N, θ Λ = ΛN := 0 1 : ∈ V, θ ∈ N {ηI : η ∈ O× }. where V = VN = (1 + N) ∩ O× is a finite index subgroup of O× . Equivalently, the Γ-stabilizer of σ ∈ P1 (F ) contains the subgoup gσ Λgσ−1 . Proof. From (2.4) and the preceding discussion, it is follows that Λ is a finite-index subgroup of stab(∞; Γ(σ) ) since the additive group N is finite index in Na−2 with index equal to Na2 . To see that V has finite index in O× , it suffices to note that V is the pre-image of {1} in the reduction map O× → (O/N)× and {1} is obviously finite index in the finite group (O/N)× . With this finite index subgroup gσ Λgσ−1 of the cusp stabilizer stab(σ; Γ), we may find a domain which projects onto the fundamental domain of stab(σ; Γ)\X , and thus obtain an understanding of its structure and parametrization. Theorem 2.7. Define Λ = ΛN ⊆ PGL(2, O) as in Proposition 2.6. Then there exists precompact domains U = UN ⊆ F∞ and V = VN ⊆ Y such that the set D = DN := {(x, y) ∈ X : x ∈ U, y ∈ V, Ny ∈ (0, ∞)} is a fundamental domain for Λ\X , where y := yv (Ny)1/n v|∞ , Ny := N v yv , v|∞ 18 Y := {y ∈ Rm >0 : Ny = 1}. Proof. [Gee88, p. 9–11] Without loss, every element γ ∈ ΛN may be written as γ= with ζ 0 1 θ 0 0 1 0 1 0 1 ∈ VN , θ ∈ N and ζ ∈ VN ∩ O1× , where O1× are the roots of unity of O× . The matrix involving ζ 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) ∈ X where x ∈ F∞ and y ∈ Rm >0 . Notice that Ny = 1 for any (x, y) ∈ X. We begin by anlayzing the action of the diagonal element for 0 0 1 ∈ VN : · (x, y) = ( x, | |y). where | | = (| v |)v|∞ . Note that the coordinate | |y = (| v |yv )v is simply component-wise multiplication, and so if is a root of unity, | |y = y. We wish to articulate the VN -action on the y-component. Consider the surjective map → Y = y ∈ Rm >0 : Ny = 1 yv (x, y) → y = . (Ny)1/n v|∞ Im : X Observe every point (x, y) ∈ X is uniquely defined by the triple (x, y, Ny). Since |N | = 1 for ∈ O× , the group O× acts multiplicatively on Y via → | | = (| v |)v . Since the kernel of this map are the roots of unity, denoted O1× , this action may be viewed as a faithful action of O× /O1× . Moreover, one should note that |N | = 1 for any ∈ O× . The multiplicative action of O× on Y transfers to an additive action via the bijective logarithm map log : Y → log Y = (av )v|∞ ∈ Rm : Trv av = 0 v|∞ y → log y := (log yv )v|∞ since log(| |y) = (log | v | + log yv )v|∞ . By Dirichlet’s Unit theorem, O× , and hence the finite index subgroup V = VN , is a lattice in log Y, so we may choose a pre-compact fundamental domain log V for the additive action V \ log Y. The exponential map exp : log Y → Y is 19 a topological isomorphism, under which log V becomes a pre-compact fundamental domain V = VN for the multiplicative action of V \Y. Thus, for any (x, y) ∈ X we may choose ∈V such that | |y ∈ V. Now, we consider the action of the unipotent element: for θ ∈ N, 1 θ 0 1 · ( x, | |y) = ( x + θ, | |y). Since N is a lattice in F∞ , a fundamental domain for the additive action F∞ /N is pre-compact. If we also include the multiplicative action of the group O1× ∩ V , then we have ζ 0 · ( x + θ, | |y) = (ζ x + ζθ, | |y). 0 1 since |ζ| = (|ζ|v )v = (1)v . We see that the element only acts in the first coordinate on F∞ . Since O1× ∩ V is a finite group and a fundamental domain for F∞ /N is pre-compact , it follows that a fundamental domain U = UN for (O1× ∩ V )\F∞ /N is pre-compact. Thus, for x ∈ F∞ , we may choose ζ ∈ O1× ∩ V and θ ∈ N so that ζ x + ζθ ∈ UN . To conclude, we may choose γ ∈ Λ such that γ · z ∈ D. Finally, we prove that distinct points z, z ∈ D are not Λ-equivalent. Suppose there exists γ ∈ Λ such that γ · z = z . Using the decomposition of γ as before, this implies (ζ x + ζθ, | |y) = (x , y ) for some ∈ V, θ ∈ N and ζ ∈ O1× ∩ V . By comparing coordinates, we see that | |y = y , which implies | |y = y . However, because y, y ∈ V, it must be that | v | = 1 for all v | ∞, or in other words, ∈ O1× ∩ VN is a root of unity. Considering the first coordinate, we see that x = ζ x + ζθ ∈ (O1× ∩ VN ) · x · N since ζ, ∈ O1× ∩ V and θ ∈ N. As x, x ∈ U, it must be that θ = 0 and ζ = 1. In other words, γ is the identity motion, as desired. Remark. Since Λ is a finite-index subgroup of stab(∞; Γ(σ) ), the fundamental domain D = DN for Λ\X is composed of finitely many copies of the fundamental domain for stab(∞; Γ(σ) )\X. The above theorem then suggests that the notion of depth into the cusp ∞ is measured by the single parameter Ny, i.e. a large value of Ny implies y is “close to infinity”. More crudely, there is only “one way to infinity”. 20 Corollary 2.8. Keep notation as in Theorem 2.7. Then via the map Λ\X stab(∞; Γ(σ) )\X , the domain D projects onto a fundamental domain for stab(∞; Γ(σ) )\X . Equivalently, via the map gσ Λgσ−1 \X stab(σ; Γ)\X , the domain gσ D projects onto a fundamental domain for stab(σ; Γ)\X . Proof. This follows immediately from Proposition 2.6 and Theorem 2.7 since Λ is a finite index subgroup of stab(∞; Γ(σ) ). 2.4 Distance to Cusps As in the remark following Theorem 2.7, each cusp σ ∈ P1 (F ) of Γ possesses a onedimensional parameter which measures the “closeness” of a point z ∈ X . Motivated by and using the notation of Theorem 2.7, we define D(y) = DN (y) := {(x, y) ∈ D : Ny ∈ (y, ∞)} , for y ≥ 0. We note some simple properties: • D(0) = D. • D(y) ⊇ D(y ) for 0 < y ≤ y . • D(y) → ∅ as y → ∞. • D(y) is a fundamental domain for the action of Λ on X (y) := {(x, y) ∈ X : Ny ∈ (y, ∞)}. From Corollary 2.8, elements z ∈ gσ D(y) are “close” to σ ∈ P1 (F ) when y is large. Equivalently, we have gσ−1 z ∈ D(y), which implies y < N(Im(gσ−1 z)) = Ny . |N(βz + α)|2 where σ = (α : β) and a = (α, β) is the ideal associated to σ. We see that the expression on 21 the righthand side measures the closeness of a point z to σ ∈ P1 (F ), so we define ρ(σ; z) := Ny |N(−βz + α)|2 for the same choice of α and β. As a special example, we have ρ(∞; z) = Ny since a = O and (α : β) = (1 : 0). Observe that, since α and β generate the ideal associated to σ, the expression ρ(σ; z) does not depend on the choice of α, β, which are determined up to multiplication by an integral unit. Because suppose we write σ = (µα : µβ) for some µ ∈ O× , then Ny Ny Ny = = 2 2 |N(−µβz + µα)| |(Nµ)N(−βz + α)| |N(−βz + α)|2 as |Nµ| = 1. Thus, ρ(σ; z) is well-defined for z ∈ X and σ ∈ P1 (F ). Notice, by definition of ρ(σ; z), it follows that gσ D(y) = {z ∈ gσ D : ρ(σ; z) > y} (2.5) for each σ ∈ P1 (F ). The purpose of this section is to prove several useful lemmas regarding ρ(σ; z). First, we prove a simple invariance property. Lemma 2.9. For σ ∈ P1 (F ) and z ∈ X , we have ρ(γ · σ; γ · z) = ρ(σ; z) for all γ ∈ ΓF = PGL(2, O). Proof. [Gee88, p. 7] Write σ = (α : β) with a = (α, β) and write γ = By direct computations, we see that Im(γ · z) = y , |cz + d|2 22 γ·z= az + b cz + d a b c d for a, b, c, d ∈ O. from which it follows that az + b + aα + bβ cz + d −(cα + dβ)(az + b) + (aα + bβ)(cz + d) = cz + d −bcα − adβz + adα + bcβz = cz + d (−βz + α) det γ = cz + d −(cα + dβ)(γ · z) + aα + bβ = −(cα + dβ) Combining these calculations, we obtain the desired result for γ ∈ ΓF . Next, we show that every point z ∈ X must be uniformly close to at least one σ ∈ P1 (F ). Lemma 2.10. There exists a positive constant y0 = y0 (F ) such that, for every z ∈ X , there exists some σ ∈ P1 (F ) with ρ(σ; z) > y0 . Specifically, one may take y0 = (2n DF )−1 where DF is the absolute discriminant of F . Proof. [Gee88, p. 8] We claim it suffices to choose y0 > 0, depending only of F , such that for every z ∈ X, there exists a solution α, β ∈ O to the inequality |N(−βz + α)|2 1 < . Ny y0 (2.6) If this is the case, then we show σ = (α : β) ∈ P1 (F ) is the desired element. Note that the LHS of (2.6) is not necessarily equal to ρ(σ; z) since α and β may not generate the ideal associated to σ. However, suppose σ = (α : β ) and a = (α , β ) is the ideal associated to σ, then α = α µ and β = β µ for some µ ∈ O, and so y0 < Ny Ny = ≤ ρ(σ; z) 2 |N(−βz + α)| |(Nµ)N(−β z + α )|2 because |Nµ| ≥ 1. This proves the claim, and so it suffices to prove a solution α, β ∈ O exists to (2.6) for some y0 > 0 depending only on F . Before proving the claim, recall for each place v | ∞, there is a non-canonically associated embedding τv : F → Fv giving rise to the place v. Further, the collection of {τv } ∪ {τ v }v|∞ gives the entire list of n embeddings, Hom(F, C). With this indexing, we define the injective map X → Cn z = (zv )v|∞ → (zτ )τ ∈Hom(F,C) 23 where z v zτ = z τ = τv for some v . τ = τ v for some v v Using this notation, we see that for z ∈ X , zτ N v zv = Nz = τ v|∞ since at complex v, Nv zv = zv z v and at real v, Nv zv = zv . Now, to prove the claim, we bound the LHS of (2.6) by considering the norm expressed as a product at each embedding τ : F → C, as per the above. By the Triangle Inequality, |−βτ zτ + ατ | yτ−1/2 ≤ |−βτ xτ + ατ | yτ−1/2 + |βτ | yτ1/2 . Thus, if we can choose cτ , dτ ∈ R>0 , depending only on F , for each embedding τ such that the set of 2n equalities |βτ xτ + ατ | yτ−1/2 ≤ cτ , |βτ | yτ1/2 ≤ dτ , τ ∈ Hom(F, C) (2.7) possesses a solution α, β ∈ O, then by applying the previous inequality, we see that |N(−βz + α)|2 = Ny so we may take y0 = τ (cτ | −βτ zτ + ατ |2 yτ−1 ≤ τ (cτ + dτ )2 v|∞ + dτ )−2 . To achieve a solution to (2.7), let {ω (k) }nk=1 be an integral basis for O in F , and write n n (k) (k) α= a ω , b(k) ω (k) β= k=1 k=1 where a(k) , b(k) ∈ Z for k = 1, . . . , n are free variables. Substituting these sums in (2.7), we obtain 2n linear inequalities n n a(k) · (yτ−1/2 ωτ(k) ) + k=1 n ≤ cτ , τ ∈ Hom(F, C) b(k) · (yτ1/2 ωτ(k) ) ≤ dτ , τ ∈ Hom(F, C) k=1 n (k) a k=1 b(k) · (−xτ yτ−1/2 ωτ(k) ) ·0 + k=1 24 with 2n variables, {a(k) , b(k) : k = 1, . . . , n}. Defining the n × n matrices A11 = (yτ−1/2 ωτ(k) )τ,k , A12 = (−xτ yτ−1/2 ωτ(k) )τ,k , A22 = (yτ1/2 ωτ(k) )τ,k , and n × 1 vectors a = (a(k) )k and b = (b(k) )k , we see that the linear system is of the form a A11 A12 0 A22 b . From Minkowksi’s Theorem on linear forms [Neu99, p. 27–28], a solution exists with a(k) , b(k) ∈ Z to the above system provided cτ dτ ≥ det τ A11 A12 0 A22 . We can easily see that the RHS = det(A11 A22 ) = det((ωτ(k) )k,τ )2 = DF 1/2n where DF is the discriminant of F , so we may take cτ = dτ = DF implying r0 = (2n DF )−1 . Finally, we prove that a point z ∈ X can be very close, in a uniform sense, to at most one σ ∈ P1 (F ). Lemma 2.11. There exists sufficiently large Y0 = Y0 (F ) > 0 such that if, for z ∈ X , we have ρ(σ; z) > Y0 and ρ(τ ; z) > Y0 for σ, τ ∈ P1 (F ), then σ = τ . Proof. [Gee88, p. 6] In a remark following Proposition 2.4, we noted that the ideal a associated to a point σ ∈ P1 (F ) is actually the same for any element in the orbit ΓF · σ. Since, by Proposition 2.4, there are only finitely many ΓF -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 σ = (α : β) where α, β ∈ O generate the ideal associated to σ, and similarly write and τ = (µ : ν) where µ, ν ∈ O generate the ideal associated to τ . Now, recall that by Dirichlet’s theorem, the embedding O× → Rm >0 → (| v |)v makes the subgroup (O× )2 a multiplicative lattice in the hyperplane Y = {y ∈ Rm >0 : Ny = 1}. 25 This implies for any y ∈ Rm >0 , that we may find a unit ∈ O× such that | 2 |y is contained in some fundamental domain for (O× )2 \Y. Since this domain is precompact in Y, it is bounded away from zero and infinity. This implies that each coordinate in | 2 |y is bounded by some constant c = c(F ) > 0. To summarize, for each y ∈ Rm >0 , there exists some | 2v yv | ≤ c · (Ny )1/n , ∈ O× such that v|∞ where c > 0 is some constant depending only on F . Thus, utilizing the above observation for y = Im(gσ−1 z)−1 = |βv zv + αv |2 yv ∈ O× , for v | ∞ we have, after multiplying y by an appropriate unit 2 v yv = yv−1 (− v βv zv + v|∞ 2 v αv ) ≤ c · (Ny )1/n = c · ρ(σ; z)−1/n −1/n < c · Y0 since ρ(σ; z) = (Ny )−1 by definition and ρ(σ; z) > Y0 by assumption. We may replace α and β by α and β since ρ(σ; z) is independent of this choice. Then the above inequality yields two inequalities: −1/n v|∞ −1/n v|∞ |βv xv + αv | yv−1/2 ≤ c1/2 · Y0 |βv | yv1/2 ≤ c1/2 · Y0 (2.8) by writing zv = (xv , yv ). Similarly, we may obtain inequalities for µ and ν. −1/n v|∞ −1/n v|∞ |νv xv + µv | yv−1/2 ≤ c1/2 · Y0 |νv | yv1/2 ≤ c1/2 · Y0 On the other hand, αv νv − βv µv = (−βv xv + αv )yv−1/2 · νv yv1/2 − (−νv xv + µv )yv−1/2 · βv yv1/2 from which it follows by the Triangle Inequality, (2.8), and (2.9) that |N(αν − βµ)| < cn Y0−1 . 26 (2.9) Thus, for Y0 > cn , we see that the algebraic integer αν − βµ has norm whose absolute value is less than 1, and hence αν − βµ = 0 implying σ = τ . Since c > 0 depends only on F , this completes the proof. 2.5 Fundamental Domain of Γ\X With a relatively complete understanding of the structure of stab(σ; Γ)\X and the distance of points in X to cusps of Γ, we may now sufficiently describe Γ\X for our purposes. Proposition 2.12. Let Γ be a congruence subgroup of level N, and let Ω ⊆ P1 (F ) be a set of inequivalent representatives of Γ\P1 (F ). Then the image of the set G(y) := gσ D(y) ⊆ X σ∈Ω under the map X → Γ\X , (a) for 0 < y ≤ y0 , surjects onto Γ\X , where y0 is as in Lemma 2.10. (b) for y > Y0 , injects into Γ\X , where Y0 is as in Lemma 2.11, and further the union over σ ∈ Ω is disjoint. Proof. [Gee88, p. 9–11] (a) It suffices to show that for z ∈ X , there exists γ ∈ Γ and σ ∈ Ω such that γ · z ∈ gσ D(y0 ). From Lemma 2.10, we have that there exists σ ∈ P1 (F ) such that ρ(σ ; z) > y0 . Applying an appropriate element γ ∈ Γ ⊆ ΓF , by Lemma 2.9, we have ρ(σ; γ · z) > y0 for σ = γ · σ ∈ Ω. Then, from the proof Theorem 2.7, we may choose an appropriate element γ ∈ Λ such that (gσ γ gσ−1 γ ) · z ∈ gσ D(y0 ). Since gσ Λgσ−1 ⊆ Γ(N) ⊆ Γ, we may take γ := gσ γ gσ−1 γ ∈ Γ to obtain the desired result. (b) From (2.5), we see that z ∈ gσ D(y) implies ρ(σ; z) > y. Then by Lemma 2.11, the union over σ ∈ Ω is necessarily disjoint. To see that the set injects, suppose z, γ · z ∈ G(y) for some γ ∈ Γ. Thus, again by (2.5), we have ρ(σ; z) > y and ρ(τ ; γ · z) > y 27 for some σ, τ ∈ Ω. Using Lemma 2.9, we see that ρ(γ −1 · τ ; z) > y ≥ Y0 . Applying Lemma 2.11, we conclude γ −1 ·τ = σ, and so σ = τ as Ω is a set of inequivalent representatives of Γ\P1 (F ). The decomposition described in Theorem 2.13 below is best accompanied with the visual aid in Figure 2.2. Theorem 2.13. Let Γ be a congruence subgroup of level N, and let Ω ⊆ P1 (F ) be a set of inequivalent representatives of Γ\P1 (F ). For y sufficiently large, depending only on F , there exists a set S(y) ⊆ X such that (i) The set S(y) gσ D(y) ⊆ X σ∈Ω surjects onto a fundamental domain F for Γ\X . (ii) The set gσ D(y) ⊆ X σ∈Ω injects into a fundamental domain F for Γ\X . (iii) S(y) is compact (iv) S(y) ⊇ S(y ) for y ≥ y ≥ y0 . (v) As y → ∞, the image of S(y), under the map X → Γ\X , approaches Γ\X . 28 Figure 2.2: Depiction of Γ\X with 3 cusps Proof. [Gee88, p. 9–11] For y0 as in Lemma 2.10, define gσ (D(y0 ) \ D(y)). S(y) := G(y0 ) \ G(y) = σ∈Ω We claim this choice of S(y) has the desired properties. (i) The surjectivity follows by noting S(y) ∪ gσ D(y) ⊇ σ∈Ω gσ D(y0 ) σ∈Ω and applying part (a) of Proposition 2.12. The disjoint union follows by definition 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 ) \ D(y) = {(x, y) ∈ X : x ∈ U, y ∈ V, Ny ∈ [y0 , y]} where U ⊆ F∞ and V ⊆ Y are precompact sets defined 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 finite union of these subsets, is itself compact. (iv) This is immediate from the definition of S(y) and noting D(y) ⊆ D(y ) for y ≥ y . (v) The limit of S(y) as y → ∞ is well-defined by (iv). The desired result follows from (i), and noting D(y) → ∅ as y → ∞. 30 Chapter 3 Hecke-Maaß Cusp Forms In this chapter, we will define a Hecke-Maaß cusp form in several stages, and ultimately provide its Whittaker expansion with proof. We will conclude by demonstrating crucial relations satisfied by the Whittaker coefficients 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 Maaß Forms and their Fourier Expansion Let Γ ≤ PGL(2, O) be a congruence subgroup of level N. Definition 3.1 (Automorphic form). A function φ : X → C is an automorphic form (with respect to Γ) provided (i) φ(γ · z) = φ(z) for all γ ∈ Γ (ii) φ is an eigenfunction of the Laplacian ∆v for every v | ∞, i.e. ∆v φ = λ v φ where λv = sv (1 − sv ) ∈ C. Denote λ = (λv )v ands = (sv )v . Our interest lies in automorphic forms which satisfy certain growth conditions, namely belong to the L2 -space of Γ\X . Recall that, for C-valued functions φ and ψ on Γ\X , the L2 -inner product on Γ\X is given by φ, ψ L2 = φ(z)ψ(z)dz F 31 where F ⊆ X is a fundamental domain for Γ\X . We denote the L2 -norm of φ by ||φ||L2 (Γ\X ) , or more simply as ||φ||L2 when Γ is understood. Definition 3.2 (Maaß form). A function φ : X → C is a Maaß form (with respect to Γ) provided φ is an automorphic form, and also φ ∈ L2 (Γ\X ). Remark. By Elliptic Regularity of ∆v , it follows that φ ∈ C ∞ (Γ\X ). See [Eva10, §6.3] for details. Henceforth, let φ : X → C be a Maaß form of Γ. Now, let σ be a cusp of Γ, and define φσ (z) := φ(gσ · z) where gσ ∈ G is as defined in Proposition 2.4, satisfying gσ (∞) = σ. We claim φσ is now a Maaß form of Γ(σ) = gσ Γgσ−1 . This follows from the fact that ∆v commutes with the Gv -action on φ and also since dvol(z) is invariant under the G-action. Roughly speaking, studying φ at the cusp σ amounts to studying φσ at the cusp ∞. Recall from Proposition 2.6 that Λ := θ 0 1 : ∈ V, θ ∈ N {ηI : η ∈ V }. is contained in the Γ(σ) -stabilizer of ∞, or more generally, Λ ≤ Γ(σ) . In particular, we have = 1 ∈ V , implies that φσ is invariant by Λ, which by setting φσ (x + θ, y) = φσ (x, y), for θ ∈ N. where we have written φσ (z) = φσ (x, y) for z = (x, y). Under this additive action, N is a lattice in F∞ = v|∞ Fv , the x-coordinate. Define the standard trace form bilinear pairing · , · : F∞ × F∞ → C w, x → Tr(wx) := Trv (wv xv ) v|∞ where Tr extends the global trace TrFQ to F∞ by the above formula, and Trv extends the local trace TrFRv at v | ∞ by x v Trv xv = x + x v v real v v complex This pairing yields the dual lattice N∨ = (DN)−1 where D is the absolute different ideal of F 32 [Lan94, §3.1]. Hence, the Fourier series expansion of φ at the cusp σ is given by φσ (x, y) = φσ (y; α)e( x, α ) (3.1) α∈(DN)−1 where e(x) = exp(2πix) and φσ (x, y)e(− x, α )dV1 (x), φσ (y; α) := U with U being a fundamental domain for F∞ /N. Note φσ (y; α) is the α-Fourier coefficient of φ at cusp σ. Since the Maaß form φσ (z) is an eigenfunction for ∆v for each v | ∞, it follows by Elliptic Regularity that φσ (z) ∈ C ∞ (X ), i.e. it possesses infinitely many derivatives in the coordinates xv and yv for all v | ∞. From general Fourier analysis, it follows that the Fourier series (3.1) converges absolutely and uniformly on compacta. We are concerned with Maaß cusp forms, whose definition depends upon this Fourier expansion. 3.2 Maaß Cusp Forms and their Whittaker Expansion In this section, we aim to elaborate further on the Fourier expansion at the cusp σ of a Maaß form φ. Material on partial differential equations will be skipped but can be found in [Eva10]. Definition 3.3 (Maaß cusp form). A function φ : X → C is a Maaß cusp form (with respect to Γ) if φ is a Maaß form of Γ and additionally, φσ (y; 0) ≡ 0 for every cusp σ of Γ. Henceforth, we shall assume φ is a Maaß cusp form of Γ. From this definition and (3.1), it follows that the Fourier expansion at a cusp σ of Γ is φσ (x, y) = φσ (y; α)e( x, α ). (3.2) α∈(DN)−1 \{0} Since φσ (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 coefficients φσ (y; α). For v | ∞, applying ∆v to both sides of (3.2), we find λv φσ (x, y) = ∆v (φσ (y; α)e( x, α )). α∈(DN)−1 \{0} Note we may commute the derivative with the infinite sum due to the uniform convergence of 33 the Fourier series. Now, comparing Fourier coefficients of both sides, and using the form of the Laplace operator ∆v given in (1.2) and (1.4), one obtains a partial differential equation for each v | ∞. Therefore, the coefficient φσ (y; α) with α = 0 satisfies the following separable linear system of partial differential equations: ∂2ψ (y) − 4π 2 αv2 ψ(y) = λv ψ(y), v real yv2 ∂yv2 2 y 2 ∂ ψ (y) − 16π 2 |αv |2 ψ(y) − yv ∂ψ (y) = λv ψ(y), v complex v ∂yv2 ∂yv (3.3) where ψ : Rm >0 → C. Remark. If φ were more generally a Maaß form, the zeroth Fourier coefficient φσ (y; 0) satisfies the same PDE but αv = 0, which results in a very different 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 ∈ R>0 and λv = sv (1 − sv ) ∈ C. Suppose ψv : yv → C satisfies yv2 ψv (yv ) − a2v ψv (yv ) = λv ψv (yv ), v real y 2 ψ (y ) − a2 ψ (y ) − y ψ (y ) = λ ψ (y ), v v v v v v v v v v v v v complex then for some constants c1 , c2 ∈ C, depending on ψv , we have ψv (yv ) = c1 Nv (av yv )K1/2−sv (av yv ) + c2 Nv (av yv )I1/2−sv (av yv ) where Kν (z) and Iν (z) are the Kν and Iν Bessel functions. Proof. See [Zwi98, §45] or verify using computer algebra package Maple. Information about the Bessel functions can be found in [Iwa02, Appendix B] or [Bow10]. For non-zero α = (αv ) ∈ F∞ , set 2π|α | v av = 2πTrv (|αv |) = 4π|α | v v real . v complex Then, with this choice, the above lemma corresponds to the system described in (3.3). Motivated by this choice, we shall define Bv− (sv ; yv ) := Nv yv · K1/2−sv (2πTrv (yv )) Bv+ (sv ; yv ) := Nv yv · I1/2−sv (2πTrv (yv )) 34 so then by Lemma 3.4, the set of functions {Bv+ (sv ; |αv |yv ), Bv− (sv ; |αv |yv )} forms a basis of solutions for the corresponding ordinary differential equation from the system in (3.3). Proposition 3.5. Let α = (αv )v ∈ F∞ , λ = (λv )v ∈ Cm . Write λv = sv (1 − sv ) and set s = (sv )v|∞ . For the sequence of signs δ = (δv )v|∞ ∈ {±}m , define Bvδv (sv ; yv ) B δ (s; y) := v|∞ m where y ∈ Rm >0 . Then the set of 2 functions B δ (s; |α|y) : δ ∈ {±}m forms a basis of smooth solutions over C for the system of partial differential 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 coefficient φσ (y; α) as it satisfies (3.3); in particular, φσ (y; α) may be written as a linear combination of B δ (s; |α|y) for δ ∈ {±}m . This fact combined with invariance properties of φσ provides a relation between different Fourier coefficients, which is articulated in the lemma below. Lemma 3.6. Let φ be a Maaß cusp form of Γ, and σ be a cusp. By Proposition 3.5, we may write for α ∈ (DN)−1 \ {0}, cδ (α)B δ (s; |α|y) φσ (y; α) = δ∈{±}m where cδ (α) ∈ C also depends on σ. Then defining V ⊆ O× as in Proposition 2.6, we have cδ (α) = cδ (α ) for all ∈ V and δ ∈ {±}m . 35 Proof. Since 0 0 1 ∈ Λ ⊆ Γ(σ) for ∈ V , we have φσ ( · z) = φσ (z). Applying the Fourier expansion (3.1) with the form above for φσ (y; α), and sending α → α −1 , we see for all z ∈ X , cδ (α)B δ (s; |α |y) · e( x, α ) φ( · z) = α∈(DN)−1 \{0} δ∈{±}m = cδ (α −1 )B δ (s; |α|y) · e( x, α ) α∈(DN)−1 \{0} δ∈{±}m by noting x, α = x, α . Comparing the RHS with the same expansion for φ(z), and noting the linear independence of B δ (s; |α|y) as a function of y from Proposition 3.5, we conclude that cδ (α) = cδ (α −1 ) as required. Further, since φσ ∈ L2 (Γ(σ) \X), we obtain the simple lemma below bounding the mass of a Fourier coefficient deep in a cusp. Lemma 3.7. Let φ be a Maaß cusp form of Γ and σ be a cusp. Then for y sufficiently large, depending only on F , T (y) |φσ (y; α)|2 dV2 (y) ≤ ||φσ ||L2 (Γ(σ) \X ) < ∞ where α ∈ (DN)−1 \ {0}, σ is a cusp of Γ, and using notation from Theorem 2.7, T (y) = TN (y) := {y ∈ Rm >0 : y ∈ V, Ny ∈ (y, ∞)}. Proof. By Parseval’s Formula, we have |φσ (x, y)|2 dV1 (x) = U |φσ (y; β)|2 ≥ |φσ (y; α)|2 . β∈(DN)−1 (σ) \X by for y ∈ Rm >0 . For y sufficiently large, D(y) injects into a fundamental domain for Γ Theorem 2.13. Since φσ ∈ L2 (Γ(σ) \X ), 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) = {y ∈ Rm >0 : y ∈ V, Ny ∈ (r, ∞)}. Thus, we obtain the simple bound |φσ (y; α)|2 dV2 (y) ≤ T (y) D(y) |φσ (z)|2 dvol(z) ≤ ||φσ ||L2 (Γ(σ) \X ) < ∞. 36 for y sufficiently large, depending only on F . Utilizing these lemmas and the asymptotics of Bv± , we can completely characterize the form of φσ (y; α) for α = 0. The asymptotics of Bv± are known to be Bv± (sv ; yv ) sv exp(±2πTrv (yv )) (3.4) as yv → ∞; see [Iwa02, Appendix B]. Theorem 3.8. Let φ be a Maaß cusp form of Γ with Laplace eigenvalues λ = (λv )v|∞ . Writing λv = sv (1 − sv ), we have for α ∈ (DN)−1 \ {0} and cusp σ of Γ, that Bv− (sv ; |αv |yv ) φσ (y; α) = c v|∞ for some constant c ∈ C depending on φ, σ and α. Proof. As in Lemma 3.6, we know φσ (y; α) is of the form: cδ B δ (s; |α|y) φσ (y; α) = δ∈{±}m where cδ = cδ (α) ∈ C. Our goal is to show that cδ is non-zero only when δ = (−, −, . . . , −). By Lemma 3.6, we have cδ B δ (s; |α |y), φσ (y; α ) = for ∈ V. δ∈{±}m Our goal will be achieved by choosing ∈ V appropriately and analyzing the asymptotics of φσ (y; α ) in T (y) for y ≥ 1 sufficiently large. Defining κ− := {v | ∞ : κv = (−)}, κ+ := {v | ∞ : κv = (+)} for κ ∈ {±}m , we choose κ such that #κ+ is maximum and cκ = 0. Evidently, the choice may not be unique. Suppose, for a contradiction, that κ+ = ∅. Now, since the set V ⊆ Rm >0 is compact by Proposition 2.6, the constants a := min{yv : y ∈ V, v | ∞}, b := max{yv : y ∈ V, v | ∞}, 37 exist and are positive. Then for y ∈ T (y), we have y ∈ V and so a(Ny)1/n ≤ yv ≤ b(Ny)1/n . (3.5) In particular, this shows that as Ny → ∞, every coordinate of y goes to infinity. Our aim is to choose ∈ V appropriately with respect to κ. := 1 ∈ V . If κ = (+, +, . . . , +), then choose Otherwise, choose ∈ V such that | v | > 1, v ∈ κ+ and | v | < 1, v ∈ κ− , which is possible because by Dirichlet’s Unit Theorem, V is an (m − 1)-dimensional lattice in {y ∈ Rm >0 : Ny = 1}. See [Lan94, p. 104–108] or [Neu99, §1.7] for details on this choice. Observe for δ ∈ {±}m from (3.4), we have B δ (s; | α|y) s,α Trv (| v αv |yv ) − exp 2π v∈δ + Trv (| v αv |yv ) v∈δ (3.6) − as Ny → ∞ for y ∈ T (y). From (3.5), it follows that B δ (s; | α|y) B κ (s; | α|y) s,α Trv (| v αv |yv ) − exp 4π v∈δ s,α + \κ+ Trv (| v αv |yv ) v∈δ exp 4π(Ny)1/n − \κ− Trv (| v αv |b) − Trv (| v αv |a) v∈δ − \κ− v∈δ + \κ+ (∗) If δ = κ and cδ = 0, then δ ± \ κ± ⊆ κ∓ , and, by maximality of κ, the set δ − \ κ− = ∅. Then by choice of , if we replace by a sufficiently large power of itself (independent of y), we may assume the quantity in (∗) is negative for all δ = κ. In other words, for δ = κ such that cδ = 0, we have shown B δ (s; | α|y) = B κ (s; | α|y) · o(1) as Ny → ∞. Thus, we may write φσ (y; α) = B κ (s; | α|y) cκ + cδ · o(1) δ=κ as Ny → ∞ for y ∈ T (y). From this equation, it follows by (3.6) and (3.5) that for Ny 38 sufficiently large, |φσ (y; α)| ≥ |cκ | κ |B (s; | α|y)| 2 s,α |cκ | exp 2π Trv (| v αv |yv ) − 2π s,α Trv (| v αv |yv ) v∈κ− v∈κ+ |cκ | exp 2π(Ny)1/n Trv (| v αv |a) − Trv (| v αv |b) v∈κ− v∈κ+ (∗∗) Again, by choice of , if we replace by a sufficiently large power of itself (independent of y), we may assume the quantity in (∗∗) is positive. More simply, for some constant η > 0, we have |φσ (y; α)| s,α |cκ | exp(η(Ny)1/n ). for y ∈ T (y) and Ny sufficiently large. Using this bound, we see that for y sufficiently large, |φσ (y; α)|2 dV2 (y) s,α exp(2η(Ny)1/n )dV2 (y) |cκ | T (y) T (y) s,α (3.7) |cκ | exp(2η · y 1/n ) dV2 (y). T (y) Finally, sending y = (yv )v → y 1/n · y = (y 1/n · yv )v , the set T (y) is mapped bijectively to T (1) and from (2.2), we have dV2 (y) → y −1 · dV2 (y). Thus, the RHS of (3.7) is s,α |cκ | exp(2η · y 1/n )y −1 dV2 (y) s,α |cκ | exp(2η · y 1/n )y −1 . T (1) Taking y → ∞, we deduce from (3.7) that |φσ (y; α)|2 dV2 (y) → ∞ T (y) contradicting Lemma 3.7. This completes the proof. In light of Theorem 3.8 and (3.1), we make the following definitions. Definition 3.9 (Local Whittaker Function). For v | ∞, define the local Whittaker function at v | ∞ to be Wv (sv ; zv ) := √ yv K1/2−sv (2πTrv (yv )) · e(Trv (xv )) 39 where sv ∈ C and zv = (xv , yv ) ∈ Fv × R>0 . Observe that Wv (sv ; zv ) = Bv− (sv ; yv )e(Trv (xv )). Definition 3.10 (Whittaker Function). Define the (unramified) Whittaker function of G to be Wv (sv ; zv ) W (s; z) := v|∞ where s = (sv )v|∞ ∈ Cm , z = (zv )v|∞ ∈ X . From the previous definition and Theorem 3.8, we may define cσ (φ; α) ∈ C to be such that φσ (y; α) = cσ (φ; α)|Nα|−1/2 W (s; α · z), where α · z := (αv xv , |αv |yv )v|∞ ∈ F∞ × Rm >0 . for z ∈ X The constant cσ (φ; α) shall be called the α- Whittaker coefficient of φ at the cusp σ. Note the normalization factor |Nα|−1/2 is not necessary, but chosen to better suit later multiplicative relations. Corollary 3.11 (Whittaker Expansion of Maaß cusp forms). Let φ be a Maaß cusp form of Γ with Laplace eigenvalues λ = (λv )v|∞ , and let σ be a cusp of Γ. Writing λv = sv (1 − sv ), we have that for z ∈ X , cσ (φ; α)|Nα|−1/2 W (s; α · z) φσ (z) = α∈(DN)−1 \{0} where cσ (φ; α) ∈ C and α · z := (αv xv , |αv |yv )v|∞ ∈ X for α ∈ F . The infinte 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 φσ (z) given in Corollary 3.11 is known as the Whittaker expansion of φ at the cusp σ. 40 3.3 Hecke Operators We wish to define Hecke operators acting on a Hecke-Maaß cusp form φ, so by Atkin-Lehner theory, we may assume that φ is a Hecke-Maaß cusp form with respect to a b Γ0 (N) := c d ∈ GL(2, O) : c ≡ 0 (mod N) {ηI : η ∈ O× }. for some ideal N. Relevant discussion can be found in [Iwa02, §8.5] and [Hid90]. As a result, we will assume without loss that Γ = Γ0 (N) is our fixed congruence subgroup of level N for the rest of this chapter. Defining 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 defined a specific subset of ideals, namely NN := {n ⊆ O : p | n =⇒ p ∈ PN } where PN := {p ⊆ O : p unramified principal prime ideal and p N}. To begin, we shall define Hecke operators for powers of prime ideals p ∈ PN . Since the primes are principal, we may take an explicit approach for defining Hecke operators, which has little difference from the usual Hecke operators defined 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]. Definition 3.12 (pk -Hecke operator). Let k ≥ 0 and p ∈ PN have uniformizer Suppose φ : Γ0 (N)\X → C. Then we define (T (p )φ)(z) := Npk , i.e. p = ( ). : X → C by k 1 k T (pk )φ k−j φ j=0 ρ∈O/pj 0 ρ j ·z where the inner sum over ρ is over a set of inequivalent representatives of O/pj . The operator T (pk ) is called the pk -Hecke operator. Remark. This definition is independent of the choice of representatives for O/pj and uniformizer as Γ = Γ0 (N) contains all elements of the form θ 0 1 , where 41 ∈ O× , θ ∈ O. First, we note that the Hecke operator is in fact an action on Γ0 (N)-invariant functions, and further on the space of Maaß cusp forms. Proposition 3.13 (§14.6 of [IK04]). The Hecke operator T (p) for p ∈ PN acts on the space of Γ0 (N)-invariant functions, i.e. {φ : Γ0 (N)\X → C}. Corollary 3.14 (§8.5 of [Iwa02]). The Hecke operator T (n) for n ∈ NN is an L2 -bounded linear operator acting on the space of Maaß cusp forms of Γ0 (N). Now, we describe the multiplicative relations of Hecke operators on the space of Maaß cusp forms of Γ0 (N). We shall first discuss the collection of {T (pk )}∞ k=0 for a fixed prime ideal p ∈ PN . Note that the normalization factor (Npk )−1/2 affects 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 ∈ PN be given. Then for k ≥ 2, T (pk ) = T (pk−1 )T (p) − T (pk−2 ) on the space of Maaß cusp forms of Γ0 (N). The second relation demonstrates that Hecke operators for distinct prime ideals p, q ∈ PN commute. Proposition 3.16 (Proposition 14.9 of [IK04]). Let p, q ∈ PN be distinct prime ideals. Then T (p)T (q) = T (q)T (p) on the space of Maaß cusp forms of Γ0 (N). Thus, we may extend our definition of Hecke operators to all ideals of NN multiplicatively. Definition 3.17 (n-Hecke operators). For m, n ∈ NN are relatively prime, i.e. (m, n) = (1) = O, define T (mn) := T (m)T (n). With this definition, we may collect our results, and summarize them in the following theorem. Theorem 3.18 (Proposition 14.9 of [IK04], or §8.5 of [Iwa02]). On the space of Maaß cusp forms of Γ0 (N), the Hecke operators {T (n) : n ∈ NN } satisfy the following multiplicative relations: 42 (i) T (O) = id. T (ii) T (m)T (n) = d|(m,n) 3.4 mn d2 for m, n ∈ NN . Hecke-Maaß Cusp Forms and their Whittaker Coefficients We are now in a position to define the key object of interest: Hecke-Maaß cusp forms. In order to possess well-defined Hecke operators, recall that we have assumed Γ = Γ0 (N) for some ideal N. Definition 3.19 (Hecke-Maaß cusp form). A function φ : X → C is a Hecke-Maaß cusp form (with respect to Γ0 (N)) provided • φ is a Maaß cusp form • φ is an eigenfunction of every Hecke operator T (n) for all ideals n. More explicitly, φ satisfies all of the following: (i) φ is Γ0 (N)-invariant, i.e. φ(γ · z) = φ(z) for all γ ∈ Γ0 (N) and z ∈ X . (ii) φ is an eigenfunction of ∆v for every v | ∞, i.e. ∆v φ(z) = λv φ(z) where λv = sv (1 − sv ) ∈ C. Denote λ = (λv )v and s = (sv )v . (iii) φ ∈ L2 (Γ0 (N)\X), i.e. ||φ||2L2 = |φ(z)|2 dvol(z) < ∞ F where F is a fundamental domain for Γ0 (N)\X . (iv) The zeroth Fourier coefficient of φ at every cusp σ of Γ0 (N) vanishes, i.e. φσ (y; 0) = 0 for all cusps σ of Γ0 (N). (v) φ is an eigenfunction of every Hecke operator T (n) for all ideals n, i.e. T (n)φ(z) = λφ (n)φ(z) where λφ (n) ∈ C is the n-Hecke eigenvalue of φ. 43 Remark. In the previous section, Hecke operators T (n) were defined for certain ideals n, namely n ∈ NN , albeit one can define them for all ideals n. In the literature, Hecke-Maaß cusp forms are simultaneous eigenforms of all Hecke operators, but as already noted, we shall only require T (n) for n ∈ NN . The Hecke eigenvalues λφ (n) for n ∈ NN naturally inherit the multiplicative properties of their Hecke operators as described in the previous section. Proposition 3.20. Let φ be a Hecke-Maaß cusp form with respect to Γ0 (N). Then (i) λφ (O) = 1. (ii) λφ (pk ) = λφ (pk−1 )λφ (p) − λφ (pk−2 ) for p ∈ PN and k ≥ 2. (iii) λφ (m)λφ (n) = λφ d|(m,n) mn for m, n ∈ NN . d2 Proof. This is immediate from Proposition 3.15 and Theorem 3.18. Note (ii) is a specific case of (iii). For a Maaß cusp form φ, recall that we have the following Whittaker expansion from Corollary 3.11. cσ (φ; α)|Nα|−1/2 W (s; α · z) φσ (z) = α∈(DN)−1 \{0} where σ is a cusp of Γ0 (N). If we additionally assume that φ is a Hecke-Maaß cusp form, then the Whittaker coefficients cσ (φ; α) are closely related to the Hecke eigenvalues of φ. Proposition 3.21. Let φ be a Hecke-Mass cusp form of Γ0 (N) and σ be a cusp. For p ∈ PN , let be a uniformizer for p. If α ∈ (DN)−1 is a unit modulo p, then cσ (φ; α k ) = cσ (φ; α) · λφ (pk ) for k ≥ 1. Proof. We will follow the argument structure of [IK04, §14.6]. Applying T (pk ) and then gσ to φ(z), we utilize the Whittaker expansion from Corollary 3.11 to deduce (N p)k/2 (gσ · T (pk ))φ(z) k = φσ k−j z +ρ j (3.8) j=0 ρ∈O/pj = β∈(DN)−1 \{0} cσ (φ; β) |Nβ|1/2 k W s; β · j=0 ρ∈O/pj 44 k−j z j +ρ We prove that the RHS of (3.8) is actually the Whittaker expansion of T (pk )φσ (z). To do this, we determine the non-zero terms of (3.8) by considering the inner sum over ρ for a fixed non-zero β ∈ (DN)−1 and 0 ≤ j ≤ k. We remark that, in the following arguments, the rearranging of terms is valid, as the Whittaker expansion is absolutely convergent due to the infinite differentiability of φσ . Notice by Definition 3.9 and Definition 3.10, we have W s; β · k−j z +ρ j ρ∈O/pj W s; |β = k−2j k−j x | · y e Tr β · +ρ (3.9) j ρ∈O/pj = W s; |β k−2j k−2j |y e Tr β e Tr x βρ j ρ∈O/pj For this last sum, we claim e Tr βρ j ρ∈O/pj (Np)j = 0 νp (β) ≥ j (3.10) else where νp : 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 ρ belong to N. Observe that the map O → C× 1 ρ → e(Tr( βρj )) is a character of O/pj . If νp (β) ≥ j, then β −j ∈ (DN)−1 implying Tr(β −j ρ) ∈ Z for all ρ ∈ N by definition of the absolute different. Thus, the character is trivial for ρ ∈ N and hence for all the representatives ρ in the sum. Further noting #O/pj = (Np)j gives the result in this case. If νp (β) < j, then since p ∈ PN is unramified and p N, the character is non-trivial (evaluate at any ρ ∈ p), so by orthgonality of characters, the sum is zero. Thus, from (3.10), the terms involving e(Tr(β (DN)−1 . k−2j x)) in (3.9) are non-zero only if β As a result, the RHS of (3.8) is indeed the Whittaker expansion of T (pk )φ k−2j ∈ σ (z). Now, our goal is to collect terms in (3.9) and determine the coefficient of e(Tr(αx)) for a given α ∈ (DN)−1 which is a unit modulo p. For α ∈ (DN)−1 , define Aα = {(β, j) : α = β k−2j 45 , β ∈ (DN)−1 pj }. Then by (3.8), (3.9) and orthogonality of Fourier coefficients, we see that (Np)k/2 (gσ T (pk )φ)(y; α) = W s; |α|y (β,j)∈Aα cσ (φ; β) · Npj , |Nβ|1/2 (3.11) so it remains to determine the pairs (β, j) ∈ Aα . Evidently, for a given α ∈ (DN)−1 , the value of j determines β ∈ (DN)−1 by the formula β := α 2j−k . Since α is a p-adic unit by assumption, we see that νp (β) = 2j − k. On the other hand, from our previous observations, νp (β) ≥ j, and so 2j − k ≥ j implying k , k). j = k. Thus, Aα consists of exact one element (α Substituting this result into (3.11), we see that (gσ T (pk )φ)(y; α) = cσ (φ; α k )|Nα|−1/2 W (s; |α|y). On the other hand, gσ T (pk )φ = λφ (pk )φσ , and so we also have (gσ T (pk )φ)(y; α) = λφ (pk )cσ (φ; α)|Nα|−1/2 W (s; |α|y). Comparing the previous two equations, we have the desired result. Corollary 3.22. Let φ be a Hecke-Mass cusp form of Γ0 (N) and σ be a cusp. For n ∈ NN , choose η ∈ n such that n = (η). If α ∈ (DN)−1 is a unit modulo n, then cσ (φ; αη) = cσ (φ; α) · λφ (n). Proof. This follows immediately from Proposition 3.21 and Proposition 3.20. Corollary 3.22 provides significant information about the Whittaker coefficients. In the classical case of F = Q for [Sou10], it is the statement of Weak Multiplicity One; in other words, all Whittaker coefficients correspond directly to Hecke eigenvalues, up to a constant depending only on φ. However, for a general number field F , the Whittaker coefficients cσ (φ; α) behave differently for Hecke operators of ramified ideals or ideals dividing N, so the coefficients cannot be exactly identified with Hecke eigenvalues. Nonetheless, the relation in Corollary 3.22 between Whittaker coefficients and Hecke eigenvalues will fundamentally drive the material in the following chapter and the proof of the main result. In order to provide natural definitions for the following chapter, we rearrange the Whittaker expansion in a more practical form via the following lemma. 46 Lemma 3.23. Let φ be a Hecke-Mass form of Γ0 (N) and σ be a cusp. Then cσ (φ; α) = cσ (φ; α) for ∈ V and α ∈ (DN)−1 , where V ⊆ O× is as in Proposition 2.6. Proof. Since 0 0 1 ∈ Λ ⊆ Γ0 (N)(σ) for ∈ V , we have φσ ( ·z) = φσ (z). Applying the Whittaker expansion, and sending α → α −1 , we see φ( · z) = cσ (φ; α)W (s; α · z) = α∈(DN)−1 \{0} cσ (φ; α −1 )W (s; α · z) α∈(DN)−1 \{0} Comparing with φ(z), we deduce cσ (φ; α) = cσ (φ; α −1 ). Fix a finite set of representatives ξ for O× /V and a single generator α ∈ (DN)−1 for every principal fractional ideal n of (DN)−1 . Then, by Lemma 3.23 we may define the n-Whittaker ideal coefficient (of φ at cusp σ for ξ) c(ξ) σ (φ; n) := cσ (φ; ξα) yielding the following corollary. Corollary 3.24. Let φ be a Hecke-Mass form of Γ0 (N) and σ be a cusp. Then c(ξ) σ (φ; n) φσ (z) = ξ∈O× /V n⊆(DN)−1 n=(α)=(0) W (s; ξα · z) ∈V where the infinite sums are absolutely and uniformly convergent. Further, if n ∈ NN and m is a fractional ideal of (DN)−1 such that m and n are coprime, then (ξ) c(ξ) σ (φ; mn) = cσ (φ; m)λφ (n) for ξ ∈ O× /V. Proof. The Whittaker ideal expansion is immediate from Lemma 3.23. The absolute and uniform convergence is inherited from Corollary 3.11. The last property is a restatement of Corollary 3.22. 47 Chapter 4 Mock P-Hecke Multiplicative Functions The purpose of this chapter is to abstract functions satisfying the properties of Whittaker coefficients 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 fixed subset of unramified prime ideals of F not dividing N. Denote the set of P-friable ideals by N = N (P) := {n ⊆ O : p | n =⇒ p ∈ P}. Note the ideal (1) = O ∈ N vacuously and (0) ∈ N . We begin with a definitions motivated by the properties from Proposition 3.20. Definition 4.1 (P-Hecke Multiplicative). For a subset P of unramified 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) = fP d|(m,n) mn d2 48 Remark. If P = PN = {p ⊆ O : p unramified principal prime ideals, p N}, then for a HeckeMaaß cusp form of Γ0 (N), the Hecke eigenvalues λφ : 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 ∈ 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 definition is motivied by the property of Whittaker coefficients in Corollary 3.24, and the fact that Hecke eigenvalues of a Maaß form are P-Hecke multiplicative for an appropriate choice of P. Definition 4.2 (Mock P-Hecke Multiplicative). Let P be a set of unramified 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 ∈ N and m a fractional ideal of (DN)−1 with m and n coprime. Remark. Let φ be a Hecke-Maaß cusp form of Γ0 (N), σ be a cusp, and ξ ∈ O× /V . If we define P = PN = {p ⊆ O : p unramified principal prime ideal, p N} as before, and set −1 c(ξ) σ (φ; n) n principal fractional ideal of (DN) f (n) := 0 else then, by Corollary 3.24, f is mock P-Hecke mutliplicative where fP (n) = λφ (n) for n ∈ NN . For a fixed mock P-Hecke multiplicative function f of level N, our aim is to understand the decay of |f (n)|2 Nn≤y/Y where 1 ≤ Y ≤ y. We shall consider Y to vary, and y to be fixed. We present the main technical theorem. 49 Theorem 4.3. Let P be a set of unramified 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 |f (n)|2 P Nn≤y/Y 1 + log Y √ Y |f (n)|2 . Nn≤y 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 fixed value henceforth. For Y ≥ 1, define F(Y ) = F(Y ; y) := Nn≤y/Y Nn≤y |f (n)|2 |f (n)|2 Theorem 4.3 then equivalently asserts that F(Y ) P 1 + log Y √ 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 defining 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 ∈ P and n is a fractional ideal of (DN)−1 . Then f (np) f (n)fP (p) = f (np) + f (np−1 ) p n (4.3) p|n so in particular |f (n)fP (p)| ≤ |f (np)| + |f (np−1 )|, 50 (4.4) and |f (np)| ≤ |f (n)fP (p)| + |f (np−1 )|. (4.5) Similarly, f (np2 ) f (n)fP (p2 ) = f (np2 ) + f (n) f (np2 ) + f (n) + f (np−2 ) p n p n (4.6) p2 | n so in particular |f (n)fP (p2 )| ≤ |f (np2 )| + |f (n)| + |f (np−2 )|. (4.7) |f (np2 )| ≤ |f (n)fP (p2 )| + |f (n)| + |f (np−2 )|. (4.8) and 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 = pk m 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 (p) P = f (m) · f (pk+1 ) + f (pk−1 ) P P f (np) k=0 = f (np) + f (np−1 ) k ≥ 1 k=0 k≥1 as required. Similarly, we have f (n)fP (p2 ) = f (m)fP (pk )fP (p2 ) f (p2 ) P = f (m) · fP (p3 ) + fP (p) f (pk+1 ) + f (pk−1 ) P P f (np2 ) k=0 = f (np2 ) + f (n) k=1 f (np) + f (np−1 ) k ≥ 2 as required. 51 k=0 k=1 k≥2 The next lemma gives an easy bound of the magnitude of fP in relation to F at prime ideals p ∈ 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 ∈ P. If Np ≤ y, then 2 F(Np)1/2 |fP (p)| ≤ and, if Np ≤ √ y, then |fP (p)| ≤ 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 first bound, observe that (4.4) of Lemma 4.4 implies |fP (p)f (n)|2 ≤ 2(|f (np)|2 + |f (np−1 )|2 ) by Cauchy-Schwarz. Applying this inequality, we find |fP (p)|2 (|f (np)|2 + |f (np−1 )|2 ). |f (n)|2 ≤ 2 Nn≤y/q Nn≤y/q Since Np = q, notice Nnp ≤ y and Nnp−1 ≤ y provided Nn ≤ y/q, so the sum on the RHS is |f (n)|2 ≤4 Nn≤y as required. Similarly, for the second bound, equation (4.4) of Lemma 4.4 implies |fP (p)f (n)|2 ≤ 3(|f (np2 )2 + |f (n)|2 + |f (np−2 )|2 ) by Cauchy-Schwarz. Applying this inequality, we find |fP (p2 )|2 (|f (np2 )|2 + |f (n)|2 + |f (np−2 )|2 ). |f (n)|2 ≤ 3 Nn≤y/q 2 Nn≤y/q 2 Since Np = q, notice the norms of np2 , n and np−2 are ≤ y provided Nn ≤ y/q 2 , so the sum on the RHS is |f (n)|2 . ≤9 Nn≤y Combining the two above inequalities, we conclude |fP (p2 )|2 ≤ 52 9 F(Np2 ) Then by the Hecke relations given in (4.1), 3 +1 F(Np2 )1/2 |fP (p)|2 ≤ |fP (p2 )| + 1 ≤ Since F(Np2 ) ≤ 1, we have 1 ≤ F(Np2 )−1/2 , yielding the result. Proposition 4.6. Suppose a ∈ N is a square-free integral ideal. Then (1 + |fP (p)|2 )F(Y · Na) |f (n)|2 ≤ τ (a) p|a Nn≤y/Y a|n |f (n)|2 Nn≤y where τ (a) denotes the number of integral ideals dividing a. Furthermore, |f (n)|2 ≤ τ3 (a) (2 + |fP (p)|2 )F(Y · Na2 ) p|a Nn≤y/Y a2 |n |f (n)|2 Nn≤y where τ3 (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 find by induction that |fP (s)||f (mt−1 )| |f (ma)| ≤ st=a 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 |f (s)|2 |f (mt−1 )|2 |f (ma)|2 ≤ τ (a) st=a where τ (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 |f (n)|2 = |f (ma)|2 Nm≤y/(Y ·Na) Nn≤y/Y a|n ≤ |fP (s)|2 |f (mt−1 )|2 τ (a) Nm≤y/(Y ·Na) st=a |f (mt−1 )|2 . |fP (s)|2 = τ (a) st=a Nm≤y/(Y ·Na) 53 Recall that f (mt−1 ) = 0 for t m by convention. Thus, for t | a we have Nt ≥ 1 so we may bound the inner sum as follows: |f (mt−1 )|2 = |f (mt−1 )|2 Nm≤y/(Y ·Na) t|m Nm≤y/(Y ·Na) |f (m )|2 = Nm ≤y/(Y ·Nat) |f (m)|2 ≤ Nm≤y/(Y ·Na) |f (n)|2 . = F(Y · Na) Nn≤y which is independent of t. Using this bound in the previous inequality, we find |f (n)|2 ≤ τ (a) n≤y/Y a|n |fP (s)|2 F(Y · Na) s|a |f (n)|2 . Nn≤y For the sum over s | a, we use the Hecke multiplicativity of fP to deduce |fP (s)|2 = s|a (1 + fP (p))2 p|a because a is square-free. Substituting this into the former inequality, we have the desired first bound. Similarly, for the second bound, let m be a fractional ideal of (DN)−1 . Using (4.8) from Lemma 4.4, we may find by induction that |fP (r2 )||f (mt−2 )| |f (ma2 )| ≤ rst=a 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 |fP (r2 )|2 |f (mt−2 )|2 |f (ma2 )|2 ≤ τ3 (a) rst=a where τ3 (a) denotes the number of integral ideal triples (r, s, t) such that rst = a. Summing 54 this inequality over all m ≤ y/(Y · Na2 ) and commuting the sums, we have |f (ma2 )|2 |f (n)|2 = Nm≤y/(Y ·Na2 ) Nn≤y/Y a2 |n ≤ |fP (r2 )|2 |f (mt−2 )|2 τ3 (a) rst=a Nm≤y/(Y ·Na2 ) rst=a Recall that f (mt−2 ) = 0 for t2 |f (mt−2 )|2 . |fP (r2 )|2 = τ3 (a) Nm≤y/(Y ·Na2 ) m by convention. Thus, for t | a we have Nt ≥ 1 so we may bound the inner sum as follows: |f (mt−2 )|2 = Nm≤y/(Y ·Na2 ) |f (mt−2 )|2 Nm≤y/(Y ·Na2 ) t2 |m |f (m )|2 = Nm ≤y/(Y ·N(at)2 ) |f (m)|2 ≤ Nm≤y/(Y ·Na2 ) = F(Y · Na2 ) |f (n)|2 . Nn≤y which is independent of t. Using this bound in the previous inequality, we find |f (n)|2 ≤ τ3 (a) |fP (r2 )|2 F(Y · Na2 ) rst=a n≤y/Y a2 |n |f (n)|2 . (4.9) Nn≤y For the sum over triples rst = a, we see that |fP (r2 )|2 τ (ar−1 ) |fP (r2 )|2 = rst=a r|a Since fP is Hecke-multiplicative and a is square-free, we may write each term in the above sum as |fP (r2 )|2 τ (ar−1 ) = |fP (p2 )|2 p|r 55 2. p|ar−1 Using this identity and the former equation. we deduce (2 + |fP (p2 )|2 ). |fP (r2 )|2 = rst=a p|a 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 . Specifically, we assume that the following limit δ = δ(P) := lim t→∞ #{p ∈ P : Np ≤ t} #{p prime ideal : Np ≤ t} exists and is positive, i.e. δ > 0. It is well known [Lan94, p. 315] that this is equivalent to #{p ∈ P : Np ≤ t} ∼ δt . log t In Chapter 5, we will require the natural density of PN . Proposition 4.7. The set of ideals PN = {p ⊆ O : p unramified principal prime ideal with p N} 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 differs from PN by only finitely many prime ideals. Let L be the Hilbert class field 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 ∈ 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 #{1} #Gal(L/F ) = 1 hF . Now, we return to our generic set of prime ideals P with positive natural density. Y ≥ 1, define √ √ P(Y ) := {p ∈ P : Np ∈ [ Y /2, Y ]}. 56 For Since P has positive density δ, we have √ δ 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 ≤ Y , |fP (p)| ≤ 2 . F(Y )1/4 Therefore, setting J := 1 log(1/F(Y )) + 3 4 log 2 we see that 0 ≤ |fP (p)| ≤ 2J−1 for all p ∈ P(Y ). (4.11) We may thus partition P(Y ) into sets P0 , P1 , . . . , PJ such that P0 = P0 (Y ) = {p ∈ P(Y ) : |fP (p)| ≤ 2−1 } and for 1 ≤ j ≤ J, Pj = Pj (Y ) = {p ∈ P(Y ) : 2j−2 < |fP (p)| ≤ 2j−1 }. For k ≥ 1, define 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, define 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-defined as P consists of unramified prime ideals. Proposition 4.8. Keep the notations above. For 2 ≤ k ≤ |P0 |/4, we have |f (n)|2 ≤ Nn≤y/Y n∈N0 (k) 4k |P0 | |f (n)|2 . Nn≤y Further, if 1 ≤ j ≤ J, and 1 ≤ k ≤ |Pj |/4 − 1, we have |f (n)|2 ≤ Nn≤y/Y n∈Nj (k) 212 k 2 24j |Pj |2 57 |f (n)|2 . Nn≤y Proof. Compare with Proposition 3.3 of [Sou10]. Since |fP (p)| ≤ 1/2 for p ∈ P0 , we have by the Hecke relations |fP (p2 )| = |fP (p2 )−1| ≥ 3/4. Thus, |fP (p2 )|2 ≥ |f (n)|2 p∈P0 p2 n Nn≤y/Y n∈N0 (k) |f (n)|2 p∈P0 p2 n Nn≤y/Y n∈N0 (k) ≥ 9 (|P0 | − k) 16 9 16 |f (n)|2 Nn≤y/Y n∈N0 (k) ≥ 27 |P0 | 64 (4.12) |f (n)|2 . Nn≤y/Y n∈N0 (k) In the last step, we have noted k ≤ |P0 |/4. To achieve an upper bound on the LHS of (4.12), we first claim that for p ∈ P0 (n) and p2 n, we have |f (n)fP (p2 )| ≤ |f (np2 )|. For p n, we have equality by (4.6) of Lemma 4.4. For p n, again by (4.6), we see f (n)(fP (p2 ) − 1) = f (np2 ). Since |fP (p2 )| ≤ 1/2, it follows that |f (n)fP (p2 )| ≤ |f (n)(fP (p2 ) − 1)| = |f (np2 )|. This proves the claim. Thus, the LHS of (4.12) is |f (np2 )|2 . ≤ Nn≤y/Y p∈P0 n∈N0 (k) p2 n In the above sum, p ∈ P0 , we have Np2 ≤ Y , and so Nnp2 ≤ y. Also, n ∈ 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 |f (m)|2 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 |f (m)|2 = |f (m)|2 . 1 ≤ (k + 1) m=np2 Nm≤y (4.13) Nm≤y Nn≤y/Y,n∈N0 (k) p∈P0 ,p2 n For the second assertion, we argue similarly. Since |fP (p)| ≥ 2j−2 for p ∈ Pj , by Hecke multiplictativity, we have |fP (p1 p2 )| = |fP (p1 )fP (p2 )| ≥ 22j−4 for distinct p1 , p2 ∈ Pj . Therefore, |f (n)|2 Nn≤y/Y n∈Nj (k) 1 2 |fP (p1 p2 )|2 ≥ p1 ,p2 ∈Pj p1 =p2 p1 n, p2 n |f (n)|2 Nn≤y/Y n∈Nj (k) 58 24j−9 p1 ,p2 ∈Pj p1 =p2 p1 n, p2 n (4.14) The inner sum on the RHS counts the number of ordered pairs of distinct prime ideals in Pj not dividing n. By definition, n ∈ Nj (k) is divisible by at most k distinct prime ideals of Pj , and so there are at least |Pj | − k prime ideals of Pj not dividing n. Thus, there are 2 |Pj |−k 2 terms in the inner sum, so the RHS ≥ 22j−9 · 2 |Pj | − k 2 |f (n)|2 Nn≤y/Y n∈Nj (k) = 22j−9 (|Pj | − k)(|Pj | − k − 1) |f (n)|2 (4.15) Nn≤y/Y n∈Nj (k) ≥ 22j−9 · 9 |Pj |2 16 |f (n)|2 Nn≤y/Y n∈Nj (k) since 2 ≤ k ≤ |Pj |/4 − 1. As for an upper bound, notice that since f is mock P-Hecke multiplicative, the LHS of (4.14) = 1 2 |f (np1 p2 )|2 . Nn≤y/Y p1 ,p2 ∈Pj n∈Nj (k) p1 =p2 p1 n, p2 n In the above sum, since p1 , p2 ∈ Pj , we have Np ≤ √ Y for = 1, 2, and so Nnp1 p2 ≤ y. Also, n ∈ Nj (k) implies that the product np1 p2 is divisible by at most k + 2 distinct prime ideals of Pj . Thus, in the sum, the terms are of the form |f (m)|2 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 |f (m)|2 ≤ 3k 2 Nm≤y |f (m)|2 . Nm≤y Combining this result with (4.15), we have |f (n)|2 ≤ Nn≤y/Y n∈Nj (k) 213 k 2 3 · 22j |Pj |2 |f (m)|2 ≤ Nm≤y 212 k 2 22j |Pj |2 |f (m)|2 Nm≤y as desired. We have thus established the necessary facts to prove Theorem 4.3, the key technical result. 59 4.4 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 fixed value y ≥ 1, recall that we defined Nn≤y/Y F(Y ) = F(Y ; y) := Nn≤y |f (n)|2 |f (n)|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 suffices to show 1 + log Y √ Y F(Y ) ≤ C for Y ≥ y0 , where C and y0 are positive constants depending only on P. Replacing C by √ max{C, y0 }, 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 δ2 (4.16) where δ = δP ∈ (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 1 + log Y √ Y F(Y ) > C . (4.17) Since F (Y ) = 0 for Y > y, we may choose Y ∈ [1, y] maximal with respect to the property that Y satisfies the above inequality and no value larger than Y +1 does. Since Y ≥ y0 ≥ cP , we have √ #P(Y ) ≥ δ 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 defined in (4.11), so P(Y ) = Jj=0 Pj . It follows by the Pigeonhole √ √ Principle, either #P0 ≥ δ Y /(4 log Y ) or #Pj ≥ δ Y /(4J log Y ) for some 1 ≤ j ≤ J. The arguments below are divided by these two distinguished cases. √ Case 1: #P0 ≥ δ Y /(4 log Y ). Set K := [(#P0 )F(Y )/8]. Then since Y satisfies (4.17), we have √ δ Y K≥ ·C 4 log Y 1 + log Y √ Y · 1 δC −1≥ − 1 ≥ 218 8 32 since C ≥ 224 . On the other hand, we can easily see that K ≤ (#P0 )/4 since F(Y ) ≤ 1. Thus, 60 we may apply Proposition 4.8 using K to conclude 1 |f (n)|2 ≤ F(Y ) 2 |f (n)|2 Nn≤y Nn≤y/Y n∈N0 (K) by noting 4K/(#P0 ) ≤ 12 F(Y ) from the definition of K. From this inequality, it follows that 1 |f (n)|2 ≥ F(Y ) 2 |f (n)|2 . (4.18) Nn≤y Nn≤y/Y n∈N0 (K) If n ∈ N0 (K), then n must be divisible by at least K + 1 squares of prime ideals in P0 . There #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 ( Y /2)K+1 are since P0 ⊆ P(Y ). A fractional ideal n ∈ N0 (K) must be divisible by the square of one of these ideals, say a ∈ P0 (K + 1). To summarize, a2 | n and a is a square-free integral ideal composed entirely of prime ideals in P0 ⊆ P. Thus, by Proposition 4.6, we have |f (n)|2 ≤ τ3 (a) (2 + |fP (p)|2 )F(Y · Na2 ) p|a Nn≤y/Y a2 |n |f (n)|2 Nn≤y ≤ 3K+1 · 3K+1 · F(Y · (Y /4)K+1 ) |f (n)|2 . Nn≤y In the above, we noted (i) τ3 (a) = 3K+1 as a is a product of exactly K + 1 distinct prime ideals, (ii) |fP (p)|2 ≤ 1/2 ≤ 1 by definition of p ∈ P0 , and (iii) F is a decreasing function and Na2 ≥ (Y /4)K+1 . Summing this inequality over all a ∈ P0 (K +1), by our previous observations, we deduce |f (n)|2 ≤ |f (n)|2 a∈P0 (K+1) Nn≤y/Y a2 |n Nn≤y/Y n∈N0 (K) ≤ (4.19) #P0 3K+1 · 3K+1 F(Y · (Y /4)K+1 ) K +1 |f (n)|2 . Nn≤y 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 61 K+1 < 24 F(Y ) K+1 . Further, by the maximality of Y and (4.17), we know 1 + log(Y K+2 · 4−(K+1) ) Y (K+2)/2 · 4−(K+1)/2 F(Y · (Y /4)K+1 ) ≤ C ≤ 2K+1 C 1 + (K + 2) log Y Y (K+2)/2 ≤ 2K+1 C 1 + log Y Y 1/2 ≤ 2K+1 C F(Y ) C K+2 K+2 . Combining the last two inequalities into (4.19), we deduce that |f (n)|2 ≤ 24 F(Y ) K+1 3K+1 · 3K+1 · 2K+1 C F(Y ) C K+2 |f (n)|2 Nn≤y Nn≤y/Y n∈N0 (K) ≤ 432 C K+1 |f (n)|2 F(Y ) Nn≤y 1 < F(Y ) 2 |f (n)|2 Nn≤y since C ≥ 224 and K ≥ 218 , hence contradicting (4.18). √ Case 2: #Pj ≥ δ 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+ log(Y 1/2 /C) log Y log F(Y )−1 ≤3+ ≤ . 4 log 2 4 log 2 4 With this bound on J and (4.17), we have √ 22j−9 δ Y K≥ · C 1/2 4J log Y 1 + log Y √ Y 1/2 −1≥ 2−7 δC 1/2 Y 1/4 − 1. (log Y )3/2 by also noting j ≥ 1. If we additionally observe that C = 224 · δ −2 and Y ≥ 2, we see that the RHS is ≥ 2−7 · 212 · 1 − 1 ≥ 24 , 1 so K ≥ 24 . 62 On the other hand, for p ∈ Pj , we have 22j−4 ≤ |fP (p)|2 ≤ 4/F(Y )1/2 The left inequality follows by definition 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 1 |f (n)|2 ≤ F(Y ) 2 Nn≤y Nn≤y/Y n∈Nj (K) In the above, we have noted 212 K 2 24j (#Pj )2 it follows that |f (n)|2 . ≤ 21 F(Y ) from the definition of K. From this inequality, 1 |f (n)|2 ≥ F(Y ) 2 |f (n)|2 . (4.20) Nn≤y Nn≤y/Y n∈Nj (K) If n ∈ 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 ( Y /2)K+1 since Pj ⊆ P(Y ). A fractional ideal n ∈ Nj (K) must be divisible by one of these ideals, say a. To summarize, a | n and a ∈ Pj (K + 1) is a square-free integral ideal composed entirely of prime ideals in Pj ⊆ P. Thus, by Proposition 4.6, we have |f (n)|2 ≤ τ (a) Nn≤y/Y a|n (1 + |fP (p)|2 )F(Y · N a) p|a ≤ 2K+1 · (22j−1 )K+1 F(Y · (Y /4)(K+1)/2 ) |f (n)|2 . Nn≤y In the above, we noted (i) τ (a) = 2K+1 as a is a product of exactly K + 1 distinct prime ideals, (ii) |fP (p)|2 ≤ 22j−2 by definition of p ∈ Pj , and (iii) F is a decreasing function and Na ≥ (Y /4)2 . Summing this inequality over all a ∈ Pj (K + 1), by our previous observations, 63 we deduce |f (n)|2 ≤ |f (n)|2 a∈Pj (K+1) Nn≤y/Y a|n Nn≤y/Y n∈Nj (K) ≤ (4.21) #P0 22j(K+1) F(Y · (Y /4)(K+1)/2 ) K +1 2 |f (n)| . Nn≤y 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 2j−9 2 F(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)/2 C 1 + 21 (K + 3) log Y Y (K+3)/4 ≤ 2(K+1)/2 C 1 + log Y Y 1/2 ≤ 2(K+1)/2 C F(Y ) C (K+3)/2 (K+3)/2 . Combining the last two inequalities into (4.21), we deduce that |f (n)|2 ≤ K+1 3 · 2−2j+9 F(Y )1/2 22j(K+1) · 2(K+1)/2 C 212 C (K+3)/2 |f (n)|2 Nn≤y Nn≤y/Y n∈Nj (K) ≤ F(Y ) C K+1 |f (n)|2 F(Y ) Nn≤y 1 < F(Y ) 2 |f (n)|2 Nn≤y since C ≥ 224 and K ≥ 24 , hence contradicting (4.20). This completes the proof. 64 Remark. If one wishes to determine a constant C0 = C0 (P) > 0 such that F(Y ) ≤ C0 · 1 + log Y √ Y for all Y ≥ 1 (instead of just for sufficiently large Y as per the above proof), then one may √ simply take C0 := max{ y0 , C}, where y0 and C are as in (4.16), because F(Y ) ≤ 1 for all Y ≥ 1. 65 Chapter 5 Elimination of Escape of Mass We may now return to the motivating problem: elimination of escape of mass. Let F be a number field and let Γ = Γ0 (N). As before, X is the product of hyperbolic 2- and 3-spaces associated to F , and our interest lies with the quotient space Γ\X . Suppose {φj }∞ j=1 are a sequence of Hecke-Maaß cusp forms of Γ, with associated probability measures dµφj = |φj (z)|2 dvol(z) ||φj ||2L2 on Γ\X . If the probability measures µφj weak-∗ converge to a measure µ, then: Is µ still a probability measure? In other words, is µ(Γ\X ) = 1? In this chapter, we will retain this setup and answer this question in the affirmative. 5.1 Decay High in the Cusp From Theorem 2.13, we have gained a very clear understanding of the structure of Γ\X . Using a single parameter y ≥ 1, we may divide a fundamental domain F of Γ\X into one compact set S(y), and a finite collection of non-compact cusps {gσ D(y) : σ ∈ Γ\P1 (F )} of the form {compact} × R>0 . This decomposition is best characterized by Figure 2.2. If the measure µ supposedly lost mass, then the mass must have “escaped” into a cusp. In other words, one should expect that, for some cusp σ, the quantity µφj (gσ D(y)) does not go to zero as y → ∞, for all sufficiently large j. Thus, in order to eliminate escape of mass, it suffices to show that, for fixed j, the measure µφj possesses some uniform decay in every cusp gσ D(y) as y → ∞. The following proposition addresses this key issue by applying Theorem 4.3 to the Fourier coefficients of a Hecke-Maaß cusp form. 66 Proposition 5.1. Let F be a number field, φ be a Hecke-Maaß cusp form for Γ = Γ0 (N), and σ be a cusp of Γ\X . Then for Y ≥ 1, we have 1 ||φ||2L2 |φ(z)|2 dvol(z) N gσ D(Y ) 1 + log Y √ . Y Proof. Compare with Proposition 2 of [Sou10]. Keep notation as in Chapters 2 and 3. We may assume that Y is sufficiently 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 φσ (z) = φ(gσ z) using Corollary 3.11, yielding |φσ (z)|2 dvol(z) |φ(z)|2 dvol(z) = D(Y ) gσ D(Y ) |cσ (φ; α)|2 = D(Y ) α∈(DN)−1 \{0} |W (s; αz)|2 dvol(z). |Nα| (5.1) The integral on the LHS converges since Y ≥ Y0 , and by Theorem 2.13, D(Y ) injects into a fundamental domain of Γ(σ) \X on which φσ 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 coefficients by non-zero principal fractional ideals n = (α) of (DN)−1 , and a finite set of representatives ξ of O× /V . Thus, the RHS of (5.1) 2 |c(ξ) σ (φ; n)| = D(Y ) ξ∈O× /V n⊆(DN)−1 n=(α)=(0) W(s; αξ · z) dz Nn (5.2) where |W (s; z)|2 . W(s; z) = ∈V Note |W (s; z)| = |W (s; y)| by Definition 3.10, so it follows that W(s; z) = W(s; y). Now, recall D(Y ) = {(x, y) ∈ F∞ × Rm >0 : x ∈ U, y ∈ T (Y )} where T (Y ) = {y ∈ Rm >0 : y ∈ V, Ny ∈ (Y, ∞)}, so we may write (5.2) as an iterated integral dvol(z) = dV1 (x)dV2 (y) as in (2.1). Using this parametrization, we find that (5.2) is 2 |c(ξ) σ (φ; n)| = U T (Y ) ξ∈O× /V n⊆(DN)−1 n=(α)=(0) 67 W(s; |αξ|y) dV2 (y) dV1 (x). Nn By 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 → (yv /|αv ξv |) = y/|αξ|, then T (Y ) maps bijectively to αξ · T (Y ), and dV2 (y) → Nn · dV2 (y) similar to (2.2). Thus, the above expression 2 |c(ξ) σ (φ; n)| = U ξ∈O× /V W(s; y) dV2 (y) dV1 (x). (5.3) αξ·T (Y ) n⊆(DN)−1 n=(α)=(0) Observe that the quantity αξ is determined up to multiplication by a unit in V , but since W(s; · y) = W(s; y) for ∈ 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 α ∈ (DN)−1 \ {0} and ξ ∈ O× /V . By definition of W, we have |W (s; y)|2 dy. W(s; y)dy = αξ·T (Y ) ∈V (5.4) αξ·T (Y ) Substituting y = (yv )v → (yv /| v |), we have αξ · T (Y ) maps to αξ · T (Y ) and dV2 (y) → |N |dV2 (y) = dV2 (y). Thus, the RHS |W (s; y)|2 dy. = ∈V Now notice, for (5.5) αξ ·T (Y ) ∈V, αξ · T (Y ) = {y ∈ Rm >0 : y ∈ αξ · V, Ny ∈ (Y · Nn, ∞)} since ξ = ξ . From the proof of Theorem 2.7, the collection of sets { · V : ∈ V } are disjoint, and their union is the set Y = {y ∈ Rm >0 : Ny = 1}. As αξ has norm 1, and Y is a group under multiplication, it follows that the sets {αξ · V : ∈ V } are also disjoint, and their union is Y. Thus, we have shown αξ · T (Y ) = {y ∈ Rm >0 : Ny ∈ (Y · Nn, ∞)} = ∈V · T (Y · Nn). ∈V 68 As a result, the expression in (5.5) and hence the LHS of (5.4) |W (s; y)|2 dV2 (y) |W (s; y)|2 dV2 (y) = = ·T (Y ·Nn) ∈V ∈V T (Y ·Nn) W(s; y)dV2 (y), = T (Y ·Nn) by doing the substitution y → y and swapping the sum and integral again. With this observation, we see that (5.3) 2 |c(ξ) σ (φ; n)| = U ξ∈O× /V n⊆(DN)−1 W(s; y) dV2 (y) dV1 (x). T (Y ·Nn) n=(α)=(0) Swapping the sum over ideals and inner integral back, notice W(s; y) has a contribution in the (ξ) amount |cσ (φ; n)|2 if and only if y ∈ T (Y · Nn). This occurs equivalently when y ∈ T (Y0 ) and Ny ≥ Y · Nn. Thus, the above equation 2 |c(ξ) σ (φ; n)| dV2 (y) dV1 (x). W(s; y) = U ξ∈O× /V T (Y0 ) (5.6) Nn≤Ny/Y n=(α)=(0) For each ξ ∈ O× /V , define f and P = PN as in the remark following Definition 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 find that (5.6) is log(eY /Y0 ) N Y /Y0 2 |c(ξ) σ (φ; n)| dV2 (y)dV1 (x). W(s; y) U ξ∈O× /V T (Y0 ) Nn≤Ny/Y0 n=(α)=(0) 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 ) √ Y |φσ (z)|2 dvol(z). D(Y0 ) Since Y0 ≥ 1 depends only on F , it may be absorbed into the implicit constant. Finally, by Theorem 2.13, the set gσ D(Y0 ) injects into a fundamental domain for Γ0 (N)\X, so the above integral is bounded by the L2 -norm of φ, as required. 69 5.2 No Escape of Mass As per the discussion in the previous section, with a uniform decay in the cusps, we may finally eliminate the possibility of escape of mass for probability measures of Hecke-Maaß 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 field, and N be an integral ideal. Suppose {φj }∞ j=1 are Hecke-Maaß cusp forms on Γ0 (N)\X with probability measures µφj . Suppose µφj → µ weak-∗, that is to say, 1 ||φj ||2L2 |φj (z)|2 dvol(z) → dµ. A A for every compact set A contained in a fundamental domain of Γ0 (N)\X. Then µ 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 sufficiently large (depending on F ) such that F = S(Y ) gσ D(Y ) σ∈Ω where F is a fundamental domain for Γ0 (N)\X. In other words, F may be written as a compact centre S(Y ) and a finite union of cusps gσ D(Y ). We shall analyze the convergence of probability measures on each of these sets. On the compact set S(Y ), by definition of weak-∗ convergence, we have that S(Y ) dµφj = dµ + oj (1) S(Y ) as j → ∞. On the other hand, from Proposition 5.1, we have gσ D(Y ) dµφj N log(eY ) √ . Y Notice that this bound is independent of φj . Combining these two observations and noting µφj are probability measures, we deduce 1= F dµφj = S(Y ) = dµφj + σ∈Ω gσ D(Y ) dµ + oj (1) + S(Y ) ON σ∈Ω 70 dµφj log(eY ) √ . Y where Ω ⊆ P1 (F ) is an inequivalent set of representatives of cusps of Γ0 (N). Taking j → ∞, and Y → ∞, we see that 1 = lim Y →∞ S(Y ) dµ dµ = F since by Theorem 2.13(v), S(Y ) → F as Y → ∞. This completes the proof. Remark. In applying Proposition 5.1, for a fixed φj , we only require some decay in the cusp as Y → ∞, as long as it is uniform with respect to φj . 71 Conclusion We have eliminated the possibility of escape of mass occurring for Hecke-Maaß 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. Sup2 pose {φj }∞ j=1 ⊆ L (M ) is a sequence of Hecke-Maaß cusp forms with Laplace eigenvalues (j) wk-∗ (j) λ(j) = (λv )v|∞ such that λv → ∞ for some v | ∞. If µφj −→ µ, then µ = c · vol for some c ∈ [0, 1]. Corollary. Assume Conjecture 2 holds. Then by Theorem 5.2, AQUE holds for Hecke-Maaß 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 technical result, Theorem 4.3, on mock P-Hecke multiplicative functions. This mysterious argument about Whittaker coefficients 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 72 with the Whittaker expansion, and so further study is necessary. 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Escape of mass on Hilbert modular varieties Zaman, Asif Ali 2012
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Title | Escape of mass on Hilbert modular varieties |
Creator |
Zaman, Asif Ali |
Publisher | University of British Columbia |
Date Issued | 2012 |
Description | Let F be a number field, G = PGL(2,F_∞), and K be a maximal compact subgroup of G. We eliminate the possibility of escape of mass for measures associated to Hecke-Maaß 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)\H², and follows his approach closely. First, we define M, a congruence locally symmetric space covered by G/K, and articulate the details of its structure. Then we define Hecke-Maass cusp forms and provide their Whittaker expansion along with identities regarding the Whittaker coefficients. 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-Maass cusp forms on M. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2012-08-30 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0073094 |
URI | http://hdl.handle.net/2429/43097 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2012-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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