@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Mathematics, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Rose, Simon Charles Florian"@en ; dcterms:issued "2012-04-19T17:07:04Z"@en, "2012"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description "In this thesis we produce a generating function for the number of hyperelliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of P. A. MacMahon's generalized sum-of-divisors functions, and prove that they are quasi-modular forms."@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/42091?expand=metadata"@en ; skos:note "Counting Hyperelliptic Curves in Abelian Surfaces with Quasi-modular Forms by Simon Charles Florian Rose B.Sc. Honours, The University of Alberta, 2005 M.Sc., The University of British Columbia, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2012 c© Simon Charles Florian Rose 2012 Abstract In this thesis we produce a generating function for the number of hyper- elliptic curves (up to translation) on a polarized Abelian surface using the crepant resolution conjecture and the Yau-Zaslow formula. We present a formula to compute these in terms of P. A. MacMahon’s generalized sum- of-divisors functions, and prove that they are quasi-modular forms. ii Preface The material from Chapter 6 is taken from [3], which is to appear in Journal für die Reine und Angewandte Mathematik (Crelle’s Journal). iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Abelian surfaces and Kummer surfaces . . . . . . . . . . . . 7 2.2 Gromov-Witten theory of hyperelliptic curves . . . . . . . . 12 2.3 The case A ∼= S × F . . . . . . . . . . . . . . . . . . . . . . . 22 3 Main Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Computation of the invariants on Km(An) . . . . . . . . . . 24 iv 3.2 Application of the crepant resolution conjecture . . . . . . . 32 3.3 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . 39 3.4 A simple consequence . . . . . . . . . . . . . . . . . . . . . . 42 4 Proofs for Low Genera . . . . . . . . . . . . . . . . . . . . . . 45 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Genus 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3 Genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.4 Proof of Göttsche’s genus 2 formula . . . . . . . . . . . . . . 52 5 Structure of the Moduli Space . . . . . . . . . . . . . . . . . 55 6 MacMahon’s Generalized Sum-of-Divisors Functions . . . 72 6.1 Statement of results . . . . . . . . . . . . . . . . . . . . . . . 72 6.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.2 Further research . . . . . . . . . . . . . . . . . . . . . . . . . 82 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 v List of Tables 1.1 Coefficients for genus 3 curves . . . . . . . . . . . . . . . . . . 5 vi List of Figures 2.1 A map in a component of M ([A/± 1];k;n) . . . . . . . . . . 20 4.1 A collapsing component joining two non-collapsing ones . . . 46 vii Acknowledgements I would like to thank my advisor Jim Bryan for introducing me to orb- ifolds, Gromov-Witten theory, and for helping bring me back to a proper appreciation of geometry. I would like to thank my family, friends, and everyone who has been there to help me with, or to distract me from what I should be doing. viii Dedication To my grandfather, Dr. J. C. F. MacDonald. ix Chapter 1 Introduction Let (Ah−1, Lh−1) be a polarized abelian surface with polarization of type (1, h − 1). Up to translation in Ah−1, there is an (h − 2)-dimensional fam- ily of curves of arithmetic genus h in the homology class c1(Lh−1)∨. The codimension of the hyperelliptic locus in Mh,0 is h− 2, and so the following natural question arises: Question 1.1. Given a polarized abelian surface (Ah−1, Lh−1), how many curves (up to translation in Ah−1) of geometric genus g in the class c1(Lh−1)∨ are hyperelliptic? We will often write Ah−1 or Lh−1 simply as A or L if there is no possibility of confusion, and when the degree of the polarization is not important. We will also for convenience make the substitution n = h− 1 which works to make most of the formulæ cleaner. Furthermore, throughout this thesis, whenever we say “the number of curves in A...” we will always be referring to the number of curves in the class c1(L)∨ up to translation in A. Let Ng,h denote the number of hyperelliptic curves of geometric genus g 1 and arithmetic genus h in a fixed (A,L). Let Fg(u) = ∞∑ h=g Ng,hu h−1 be the generating function for these numbers. We will give an explicit for- mula for Fg(u) in terms of quasi-modular forms. Remark 1.2. Note that in the case g = 2, all curves are hyperelliptic. In [11], it is shown that F2(u) = ∞∑ d=0 σ1(d)ud = E2(u) + 1 24 where σ1(d) = ∑ k|d k, and E2 is the Eisenstein series of weight 2. Thus we see that F2 is in the ring of quasi-modular forms. The goal of this thesis is to transform this natural enumerative problem into the language of orbifold Gromov-Witten theory, and to use the crepant resolution conjecture [5] and the Yau-Zaslow formula [22] to compute the generating functions Fg for all g. We should remark that this number Ng,h is not necessarily well defined— that is, independent of the choice of A—nor necessarily finite. In Section 2.2 we will interpret it in terms of Gromov-Witten invariants which will be defined for all polarized A with c1(L)∨ primitive; in the case that A is sufficiently generic, we expect that this will coincide with the honest count of hyperelliptic curves of geometric and arithmetic genera g and h, respectively. In fact, we provide a refinement of this count. Let A[2] denote the collection of 2-torsion points in A. As we will see in Section 2.2, we can 2 translate a hyperelliptic curve so that all of its Weirstrass points all lie on points of A[2]. We can then use the number of Weirstrass points lying over each v ∈ A[2] to refine our count, as follows. Let k : A[2] → Z≥0 be a function, denote by |k| = ∑ v∈A[2] k(v), and let g be such that 2g + 2 = |k|. Let Nk,h denote the number of curves C ⊂ A of geometric genus g and arithmetic genus h whose normalizations C are hyperelliptic and with k(v) Weirstrass points lying over v for each v ∈ A[2] (See Section 2.2, equation (2.4) for a precise definition). Let P be the collection of v ∈ A[2] so that k(v) is odd. The main theorem of this thesis is the following. Theorem 1.3. Assume the Gromov-Witten crepant resolution conjecture for the resolution Km(A)→ A/± 1 (See Section 3.2). Then the generating function Fg,k(u) = ∑∞ h=gNk,hu h−1 is given by Fg,k(u) = E(u) 1 2 |S|−2 ∏ v∈S Ak(v)−1 2 (u4) ∏ v/∈S Ck(v) 2 (u2) (1.1) when P satisfies an easily-verified condition (see Remark 3.2), and is zero otherwise. The functions E(q), Ai(q), and Ci(q) are given by E(q) = ∞∑ k=0 σ1(2k + 1)q2k+1 Ai(q) = ∑ 0 s citi for i ≤ s It should be noted that the crepant resolution conjecture, as stated, only applies to ordinary Gromov-Witten invariants and not to reduced Gromov- Witten invariants as we use in our case. The short explanation is simply that it appears to work. The longer explanation is that our situation (dealing with fibre-wise Gromov-Witten invariants of a non-Kähler 3-fold) is similar enough to the CY3 case that it should work. Moreover, the local picture around the singular points is the same as that of the resolution T ∗P1 → [C2/ ± 1], a case where the equivariant version of the crepant resolution conjecture has been proven [5]. The only issue is whether or not we can extant this globally to the orbifold [A/± 1], which appears to be the case. As is usually the case, the change-of-variables is forced upon us by know- ing a few of the invariants and choosing data to match those. In our case this comes from two sources. As stated above, we have a good understanding of the local picture around a singular point, so we can use that knowledge. Moreover, we already know the number of genus 2 curves in an Abelian surface due to Göttsche [11]. These two facts allow us to derive the full change-of-variables for the restricted potential function, which is given as 33 follows. The function L : H∗ ( Km(A) ) → H∗orb([A/ ± 1]) used in the crepant resolution conjectures is the following. Let yv denote the formal cohomo- logical variable corresponding to the dual of an exceptional divisor Ev, and let zv denote the variable corresponding to the class in the twisted sector of [A/± 1]. Similarly, let yS and yF denote the formal variables corresponding to the classes S and F (with zS and zF downstairs), respectively. Then the map L on the formal variables is given by L(yv) = izv L(yS) = zS L(yF ) = zF while the roots of unity are given by applying, for w = ∑ v∈A[2] av 2 Ev qw 7→ (−1)w := (−1) P v∈A[2] av 2 and the substitutions qS/2 7→ tS/2 qF/2 7→ −tF/2 As in the statement of Theorem 3.6, we will continue with the equivalent substitution of u = ( exp(zF )tF )1/2 . which yields that u 7→ −u under this substitution. All of this together yields the following. Proposition 3.8. Assume the crepant resolution conjecture holds for the 34 resolutions Km(An) → [An/ ± 1]. Then the restricted genus 0 orbifold Gromov-Witten potential function for [A/ ± 1], when summed over all po- larizations, is given by F [A/±1] = ( exp(zS)tS )1/2 u2 ∆(u2) ·( ∏ v∈ε0 e−izv/2 ∑ w∈K u−〈w,w−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w + ∏ v∈ε1 e−izv/2 ∑ w∈K u−〈w,w−2bε1〉+1 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w ) . where (−1)w := (−1) P v∈A[2] av 2 for w = ∑ v∈A[2] av 2 Ev. Proof. If we combine all of the transformations given above, we find that the remaining composite terms in the potential transform as λ0 7→ ( exp(zS)tS )1/2 ∏ v∈ε0 eizv/2 λ1 7→ − ( exp(zS)tS )1/2 ∏ v∈ε1 eizv/2 This yields the result as claimed. Remark 3.9. We should remark that for simplicity, we will omit the term( exp(zS)tS )1/2 occurring at the beginning of the expression, which con- tributes no further enumerative information. To simplify this, we recall that from corollary 2.5 we can write any element w̄ ∈ K as w̄ = w + η̂ for w ∈ Λ and η ∈ Π3. This allows us to replace the summation over K by a double summation over Λ and over Π3. More precisely, let η ∈ Π3, and (noting that (−1)bη = 1 by the definition 35 above) define F [A/±1]η = u2 ∆(u2) ( ∏ v∈ε0 e−izv/2 ∑ w∈Λ u−〈w+bη,w+bη−2bε0〉 · ∏ v∈A[2] exp ( i( ∫ w+bη γv)zv)(−1)w + ∏ v∈ε1 e−izv/2 ∑ w∈Λ u−〈w+bη,w+bη−2bε1〉+1 · ∏ v∈A[2] exp ( i( ∫ w+bη γv)zv)(−1)w). Then we have that F [A/±1] = ∑ η∈Π3 F [A/±1] η . Since the intersection form restricted to Λ is diagonal, the functions F [A/±1]η can now be computed. For simplicity, the superscript [A/± 1] will now be omitted. Lemma 3.10. The function Fη can be written as Fη = u2 ∆(u2) ( u 1 2 |η+ε0|−2 ∏ v∈η+ε0 h(zv, u) ∏ v/∈η+ε0 g(zv, u) + u 1 2 |η+ε1|−2 ∏ v∈η+ε1 h(zv, u) ∏ v/∈η+ε1 g(zv, u) ) where h(z, u) = 2 ∞∑ k=0 (−1)k sin ((2k + 1)z 2 ) u2k 2+2k g(z, u) = 1 + 2 ∞∑ k=1 (−1)k cos(kz)u2k2 . Proof. Throughout this proof we will only work with the terms involving ε0; 36 all of the proofs for ε1 are nearly identical. We begin by noting that 〈w + η̂, w + η̂ − 2ε̂0〉 = 〈w,w + 2η̂ − 2ε̂0〉+ 〈η̂, η̂〉 − 2〈η̂, ε̂0〉. Since |ε0| = 4, and for any two subsets η1, η2 ∈ Π0, we have 〈η̂1, η̂2〉 = −12 |η1 ∩ η2| (see remark 2.3), it follows that 〈η̂, η̂〉 − 2〈η̂, ε̂0〉 = 〈η̂, η̂〉 − 2〈η̂, ε̂0〉+ 〈ε̂0, ε̂0〉+ 2 = −1 2 (|η| − 2|η ∩ ε0|+ |ε0|)+ 2 = −1 2 |η + ε0|+ 2 where we recall that the summation is done in Π0 (and so is the symmetric difference of the two sets). We work now to simplify the expression ∏ v∈ε0 e−izv/2 ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w+bη γv)zv)(−1)w. To begin with, we note that exp ( i( ∫bη γv)zv) = exp(izv/2) if v ∈ η (and is 1, otherwise), and so this term is ∏ v∈ε0\\η e−izv/2 ∏ v∈η\\ε0 eizv/2 ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w Because of the fact that the intersection form restricted to Λ is diagonal, we 37 can write the latter sum as a product ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w = ∏ v∈η\\ε0 ∞∑ k=−∞ (−1)ku2k2+2keikzv · ∏ v∈ε0\\η ∞∑ k=−∞ (−1)ku2k2−2keikzv · ∏ v/∈η+ε0 ∞∑ k=−∞ (−1)ku2k2eikzv . By rearranging the summation index on the middle product it can be written as ∏ v∈ε0\\η ∞∑ k=−∞ (−1)ku2k2−2keikzv = ∏ ε0\\η (− eizv) ∞∑ k=−∞ (−1)ku2k2+2keikzv and so the whole term becomes ∏ v∈ε0\\η e−izv/2 ∏ v∈η\\ε0 eizv/2 ∑ w∈Λ u−〈w,w+2bη−2bε0〉 ∏ v∈A[2] exp ( i( ∫ w γv)zv ) (−1)w = ∏ v∈η+ε0 eizv/2 ∞∑ k=−∞ (−1)ku2k2+2keikzv ∏ v/∈η+ε0 ∞∑ k=−∞ (−1)ku2k2eikzv Since we have that ∞∑ k=−∞ (−1)ku2k2eikz = 1 + 2 ∞∑ k=1 (−1)k cos(kz)u2k2 and also that eiz/2 ∞∑ k=−∞ (−1)ku2k2+2keikz = 2 ∞∑ k=0 sin ( (2k + 1)z/2 ) u2k 2+2k 38 the conclusion follows. 3.3 Proof of Theorem 1.3 To prove Theorem 1.3 it remains to study the functions h and g given above. As they are written in terms of trigonometric functions, we begin by writing them in terms of Chebyshev polynomials. Recall that Chebyshev polynomials Tn(x) are given by the relationship Tn(cos θ) = cos(nθ). They can equivalently be defined recursively via T0(x) = 1 T1(x) = x Tn(x) = 2xTn−1(x)− Tn−2(x). (3.2) They also satisfy the relationships T2n+1(sin θ) = (−1)n sin ( (2n+ 1)θ ) and Tn(1− 2x2) = (−1)nT2n(x). With these last two in mind, we have the following fact. 39 Proposition 3.11. The functions h(x, u) and g(x, u) can be written as h(z, u) = 2 ∞∑ n=0 T2n+1 ( sin(z/2) ) u2n 2+2n g(z, u) = 1 + 2 ∞∑ n=1 T2n ( sin(z/2) ) u2n 2 Proposition 3.11 and Lemma 3.10 give us an explicit description for the (relevant portion of the) potential function of the orbifold [A/± 1]. We now perform the substitution xv = 2 sin(zv/2) (See Theorem 2.13), and use the following result from Chapter 6. Theorem 3.12 (Theorem 6.1). Let H(x, u) = 2 ∑∞ n=0 T2n+1(x/2)q n2+n and G(x, u) = 1 + 2 ∑∞ n=1 T2n(x/2)q n2. Then H(x, u) = (q2; q2)3∞ ∞∑ k=0 Ak(q2)x2k+1 G(x, u) = (q; q)∞ (−q; q)∞ ∞∑ k=0 Ck(q)x2k where (a; q)∞ = ∏∞ k=0(1− aqk) is a q-Pockhammer symbol, and Ak, Ck are MacMahon’s generalized sum-of-divisors functions (See [16]). We are now in a position to prove theorem 1.3. Proof of theorem 1.3. From Lemma 3.10, Proposition 3.11, and Theorem 40 3.12, we can write Fη = u2 ∆(u2) ( u 1 2 |η+ε0|−2 ∏ v∈η+ε0 (u4;u4)3∞ ∞∑ k=0 Ak(u4)x2k+1v · ∏ v/∈η+ε0 (u2;u2)∞ (−u2;u2)∞ ∞∑ k=0 Ck(u2)x2kv + u 1 2 |η+ε1|−2 ∏ v∈η+ε1 (u4;u4)3∞ ∞∑ k=0 Ak(u4)x2k+1v · ∏ v/∈η+ε1 (u2;u2)∞ (−u2;u2)∞ ∞∑ k=0 Ck(u2)x2kv ) Collecting all the Pockhammer symbols (and the extra powers of u) we find (for the ε0 term) u 1 2 |η+ε0|−2 u 2 ∆(u2) ( (u4;u4)3∞ )|η+ε0|( (u2;u2)∞ (−u2;u2)∞ )16−|η+ε0| . Using the fact that u 2 ∆(u2) = (u2;u2)−24∞ and that (q; q)∞(−q; q)∞ = (q2; q2)∞ this simplifies to ( u ( (u2;u2)∞(−u2;u2)2∞ )4)12 |η+ε0|−2 Using a theorem of Legendre [15] which states that ( (q; q)∞(−q; q)2∞ )4 =∑∞ k=0 σ1(2k + 1)q k, it follows that this term is in fact E(u) 1 2 |η+ε0|−2 where E(u) = ∞∑ k=0 σ1(2k + 1)u2k+1 = 1 16 ϑ2(u)4 and where ϑ2 is the Jacobi theta function ϑ2(q) = ∑∞ k=0 q (k+1/2)2 . An 41 identical computation yields that the terms in front of the ε1 term simplify to E(u) 1 2 |η+ε1|−2. We can now rewrite Fη as Fη = E(u) 1 2 |η+ε0|−2 ∏ v∈η+ε0 ∞∑ k=0 Ak(u4)x2k+1v ∏ v/∈η+ε0 ∞∑ k=0 Ck(u2)x2kv + E(u) 1 2 |η+ε1|−2 ∏ v∈η+ε1 ∞∑ k=0 Ak(u4)x2k+1v ∏ v/∈η+ε1 ∞∑ k=0 Ck(u2)x2kv . This is the generating function for counting hyperelliptic curves f : Σ→ An with the property that f(w) ∈ A[2] for all Weirstrass points w where we include the data of how many Weirstrass points lie on a given 2-torsion point. Specifically, the coefficient of the monomial ∏ v∈A[2] x k(v) v gives the number of such curves with k(v) Weirstrass points having as image the point v ∈ A[2]. From this description, the conclusion follows. Corollary 3.13. The coefficient of a monomial ∏ v∈A[2] x k(v) v in the gener- ating function F lies in the ring of quasimodular forms. Proof. This follows directly from Theorem 1.3 and from Corollary 6.4. 3.4 A simple consequence One consequence of this formula is a prediction for the lower bound on the arithmetic genus of a hyperelliptic curve in an Abelian surface A, which we prove. Note that the lowest degree terms of Ak(q) and Ck(q), respectively, are 42 ( k+1 2 ) and k2. Thus the lowest degree term of E(u) 1 2 |S|−2 ∏ v∈S Ak(v)−1 2 (u4) ∏ v/∈S Ck(v) 2 (u2) (see equation (1.1)) is given by −2 + ∑ v∈S 1 2 + ∑ v∈S 4 (1 2(k(v)− 1) + 1 2 ) + ∑ v/∈S 2 (k(v) 2 )2 = −2 + ∑ v∈A[2] 1 2k(v) 2. It follows that this formula predicts that the minimal arithmetic genus of a hyperelliptic curves in A (with discrete data k) is −1 + ∑ v∈A[2] 1 2k(v) 2 by the definition of Fg,k(u). We now show without assuming the crepant resolution conjecture that this is true. Theorem 3.14. Fix k, and let g be such that 2g+2 = |k|, and suppose that A has Picard number 1. Then the minimal arithmetic genus of a hyperelliptic curve in A with discrete data k is ∑ v∈A[2] 1 2k(v) 2 − 1. Proof. Given such discrete data, the geometric genus is given by g = 12 |k|−1. For each v ∈ A[2] with k(v) > 1, we note that any curve which produces such data must be nodal, since more than one Weirstrass point will have the same image. More specifically, we introduce at least ( k(v) 2 ) nodes for each 43 k(v) > 1. As such, the total arithmetic genus is given by 1 2 |k| − 1 + ∑ v∈A[2] ( k(v) 2 ) = 12 ∑ v∈A[2] k(v)− 1 + ∑ v∈A[2] 1 2 ( k(v)2 − k(v)) = −1 + ∑ v∈A[2] 1 2k(v) 2 as claimed. Corollary 3.15. There are no smooth hyperelliptic curves in A of genus greater than 5. Proof. We first note that if any k(v) > 1, then there must be at least one node in the image curve, as we have two Weirstrass points with the same image, v. Next, note that due to the requirement that P , the set of those v with k(v) odd must be congruent to εi mod K (or more accurately, mod Π3) yields that no more than twelve of the sixteen 2-torsion points can have k(v) = 1 (this maximal case is when P is the complement of ε0). In this case, we have that 2g + 2 = 12, or g = 5. 44 Chapter 4 Proofs for Low Genera 4.1 Introduction We will prove the formula (1.1) for the case of genus 1 and genus 2 curves. Recall that a hyperelliptic curve of genus g yields a P1 with 2g + 2 stacky points, and so to enumerate genus g curves we need only consider those monomials of total degree 2g + 2. For genus 1, this is degree 4 (we will explain below what a genus one curve in A, which should generically have no elliptic curves, means), and for genus 2, this is of degree 6. We assume as in Section 2.3 that the Abelian surface is a product A ∼= S × F of non-isogenous elliptic curves. Recall that there is then an elliptic fibration A/± 1→ S/± 1 with general fibre F , and with four special fibres F/ ± 1 over the points v ∈ S[2]. We will further (by abuse of notation) denote by S and by F the classes in A/ ± 1 of a section S/ ± 1 and fibre F/± 1, respectively. In the case that the Picard number of A is 1 and that the curve class is primitive, the source curves are combs with collapsing teeth. This allows us to perform the substitution given in in Theorem 2.13. When the Picard number of A is greater than 1—as is the case when A ∼= S × F—or when the class is not primitive, it is possible that there are collapsing components 45 0 × × × × n1 n2 Figure 4.1: A collapsing component joining two non-collapsing ones 46 which join two components mapped into A with non-zero degree (see figure 4.1). These do not contribute to the Gromov-Witten invariant (see Proposi- tion 5.9). From these two considerations, we can consider only those maps which do not collapse any components. Let f : Σ → [A/ ± 1] be a rational curve. As we saw, the source is a tree of rational curves Σ = Σ0 ∪ Σ1 ∪ · · · ∪ Σk, and such that f∗[Σ0] = S and f∗[Σi] = niF for i ≥ 1 with ∑ ni = n (and ni > 0, as discussed above). Moreover, the leaves of the tree of curves must have at least 3 stacky points on them (plus a node which connects them to the tree). For convenience, we now need to label the 2-torsion points of A. We label them as 2-torson from F // E3?> =< 89 :; E7 E11 E15 E2 E6 E10 E14 Translates of Soo vvmmm mmm mmm hhQQQQQQQQQ E1 E5 E9 E13 E076 5401 23E4 E8 E12 2-torsion from Soo (4.1) with monomials labeled similarly. With this labelling, we have the explicit descriptions ε0 = 12(E0+E4+E8+E12) and ε1 = 1 2(E1+E2+E3)+ 1 2(E4+ E8 + E12). 47 4.2 Genus 1 The genus 1 case is somewhat of an aberation; there are of course no elliptic curves in a generic Abelian surface. It turns out that the genus 1 case corresponds to a ‘polarization’ of type (1, 0). By Poincaré’s Reducibility Theorem [14, Chapter 5, Theorem 3.5], such an A is isogenous to a product of elliptic curves. Since the source curve is genus 1, on the orbifold side we have exactly 4 marked stacky points. From the description above, if the source curve were to consist of more than one component, it would have at least 6 marked points, and so the source curve must be irreducible with f mapping it iso- morphically onto a section S/±1. There is only one such map (as S, F were chosen to be generic). The only term with monomials of degree 4 from all of the functions Fη are x0x4x8x12, x1x5x9x13, x2x6x10x14, and x3x7x11x15. If we look at the formula (1.1), the prediction for each of these is also 1, and so the conjecture is verified. It should be noted that in terms of computing the actual number of such curves, these latter three monomials yield curves that are simply translations of the first, and so the number of genus 1 curves in A in the class S is 1, as we would expect. 4.3 Genus 2 The component which maps isomorphically onto a section curve S/±1 must have three of the six marked points, and so our source curve must consist 48 of exactly two components, Σ = Σ0 ∪ Σ1, and where f∗[Σ1] = nF . Thus the component Σ1 is an n-fold cover of F/± 1 (with certain data about the images of the stacky points in Σ1), and so we are reduced to computing the number of such covers. Analogous to the genus 1 case, since we only care up to translation in A, we need only focus on certain monomials. These are x1x2x3x4x8x12 x 3 0x4x8x12 x0x4x8x12x 2 1 x0x4x8x12x 2 2 x0x4x8x12x 2 3 and the corresponding predictions are E(u) A1(u4) C1(u2) C1(u2) C1(u2). The first comes simply from the term ε1 and the others all come from ε0. In each case, we are counting maps Σ1 → [F/ ± 1] of a curve with four Z/2-points to the orbifold [F/± 1]. In this case, we can lift to the cover E //  F  Σ1 // F/± 1 and so we can compute this by counting covers of F by elliptic curves E satisfying certain conditions based on the image of the 2-torsion of E. These are the following. 1. The first monomial corresponds to those maps E → F such that the 2-torsion of E surjects onto the 2-torsion of F . 49 2. The second monomial corresponds to those maps such that all 2- torsion of E maps to 0 ∈ F . 3. The last three monomials all correspond to the three possible cases where E[2] surjects onto a subgroup 〈ei〉 for ei a non-zero 2-torsion point in F . We first need a classically known fact. Proposition 4.1. Let E be an elliptic curve. Then the number of degree n isogenies F → E is given by σ1(n) = ∑ d|n d. Proof. As maps of Abelian varieties are determined by their lattices, count- ing degree n isogenies F → E is the same as counting index n sublattices of a fixed rank 2 lattice. Up to change of basis of the respective lattices, this is the same as counting matrices of the form a b 0 d  with ad = n and 0 ≤ b < d. This is clearly equal to σ1(n) as claimed. All that is now required are the following lemmata. We will provide a proof of Lemma 4.4, the rest of them having a similar flavour. Lemma 4.2. Suppose that g : E → F is a map of elliptic curves. Then the degree of g is odd if and only if the 2-torsion surjects. In this case the generating function is simply the odd-degree part of A1(u), i.e. E(u). 50 Lemma 4.3. Suppose that g : E → F is a map of elliptic curves such that g(E[2]) = 0. Then g factors through the degree 4 map E g // ## F F/F [2] ;;xxxxxxxxx In this case, we are actually counting all covers of the curve F/F [2]—but this is of course the same as counting covers of degree n/4 of an arbitrary curve. That is, we obtain the function A1(u4) as desired. Lemma 4.4. Suppose that g : E → F is a map of elliptic curves such that g(E[2]) surjects onto the subgroup 〈v〉 for some non-zero v ∈ F [2]. Then g factors through the degree 2 map E g // \"\" F F/〈v〉 < 0 be congruent to 2 (mod 4). Then σ1(n) = 3σ1(n/2). Lemma 4.6. Let n > 0 be congruent to 0 (mod 4). Then σ1(n) = 3σ1(n/2)− 2σ1(n/4). Proof. Write n = 2km with 2 - m and with k ≥ 2. Then as σ1 is a multi- plicative function, we have that σ1(n) = σ1(2k)σ1(m) = (2k+1 − 1)σ1(m) = ( 3(2k − 1)− 2(2k−1 − 1))σ1(m) = 3σ1(n/2)− 2σ1(n/4) as claimed. In [11], the following is proven. Theorem 4.7 (Göttsche, Theorem 3.2). Let (A,L) be a polarized Abelian surface with polarization of type (1, n). Then the generating function for the number of genus 2 curves in the linear system |L|, summed over all polarization types, is given by F̃2(u) = ∞∑ n=1 n2σ1(n)un = D2A1(u) 53 To see that these are equivalent, we note that the difference between the two counts—that is, curves in a fixed linear system vs. curves up to translation—comes from translating by elements in the kernel of the isogeny A → Â = Pic0(A) given by a 7→ L ⊗ t∗aL−1. If the polarization is of type (1, n), then the map on lattices H1(A)→ H1(Â) can be represented by the matrix (2.2) (see Section 2.1). The map is thus of degree n2, and so the kernel consists of exactly n2 elements. This yields the claim. 54 Chapter 5 Structure of the Moduli Space In this Chapter we gather a few facts about the structure of the moduli space M ([A/± 1];k;n), and in particular about its reduced virtual fundamental class. All throughout we assume that A is a generic Abelian surface and has Picard number 1. As stated before, the Gromov-Witten theory of an Abelian or K3 surface, strictly speaking is trivial, as any of these can be deformed into a non- algebraic surface. To account for that, we look at a reduced theory for these surfaces. For more detail, see [7, 6, 17]. To construct the reduced class on M ([A/± 1]; 2g + 2;n), we use the following approach. Let A be a fixed polarized Abelian surface with polar- ization of type (1, n), and let B be the family of Kähler metrics arising from the hyperkähler structure. Note that B ∼= S2, the real 2-sphere. Let A pi−→ B be the family of Abelian surfaces over B given by this family of metrics. That is, Ab = pi−1(b) is A with the Kähler structure given by b. We can take the fiber-wise quotient to obtain the family [A / ± 1] pi−→ B which we use to construct our reduced class. 55 Remark 5.1. The family A is not an algebraic family. In fact, to work with this we must leave the algebraic setting and move into the complex analytic category. However, while it is not algebraic, it is fiber-wise Kähler, and so we are still able to work with Gromov-Witten invariants of this family. It is worth noting that the construction of the reduced class for families of K3 surfaces has been done in [17] purely in the algebraic category. It seems likely that their methods would work similarly to obtain an algebraic reduced class for the moduli space of maps into an Abelian surface, and that we should similarly be able to do this for the orbifold [A/ ± 1]. We do not however pursue this approach in this work. In the end we use the notion of the Twistor family as it is a well- understood and concrete approach. This concrete approach suits us well, as it permits us to define our invariants with as little pain as possible. Note that we have an inclusion ι : [A/ ± 1] → [A / ± 1] as one of the fibres. For brevity’s sake, define M =M ([A/± 1]; 2g + 2;β) and M ′ =M ([A /± 1]; 2g + 2; ι∗β) . We have the following lemma. Lemma 5.2. Let [A/ ± 1] be as above, and suppose that C ⊂ [A/ ± 1] is a holomorphic curve. Then the only Kähler structure in B that has a holomorphic curve in the class [C] is the original Kähler structure for which C is holomorphic. Proof. Suppose that there are two complex structures b, b′ for which there are curves in the class [C] which are holomorphic. We can then lift these 56 to the cover A → [A/ ± 1] to obtain two differing complex structures on A which support curves in a fixed homology class; this contradicts [6, Lemma 3.4]. From this we obtain the following. Proposition 5.3. The moduli spaces M and M ′ are isomorphic in the cat- egory of complex analytic stacks. Proof. There is an obvious mapM →M ′ induced by the inclusion ι. Specif- ically, for a family C̃ //  A  C //  [A/± 1] T si GG we can compose with the inclusion ι to obtain C̃ //  A  // A  C //  [A/± 1] ι // [A /± 1] T si GG For the reverse direction, note that all holomorphic curves in [A /± 1] land 57 in a fixed fibre [A/± 1]b. This given a diagram C̃ //  A  C f //  [A /± 1] T si GG we note that the map f factors through this fixed fibre [A/±1]b. This yields the inverse map j :M ′ →M . We compute the virtual dimension of M ′ to be ∫ ι∗β c1 ( [A /± 1])+ (1− g)(dim[A /± 1]︸ ︷︷ ︸ =3 −3) + (2g + 2)− 2g+2∑ i=1 age(pi)︸ ︷︷ ︸ =1 = 0 and so we have a virtual fundamental class [M ′]vir in degree zero. Definition 5.4 We define the reduced virtual fundamental class on M to be [M ]red = j∗[M ′]vir. We next investigate the structure of the spaceMk,n =M ([A/± 1];k;n). Recall that, for k : A[2]→ Z≥0, this is the moduli space of genus 0 twisted stable maps into [A/± 1] such that k(v) stacky points have as image k(v). Denote by λ ` k a ‘multipartition’ of k. That is, a collection of partitions λv ` k(v) with parts (λv1, . . . , λvrv), indexed by v ∈ A[2]. We say that a 58 twisted stable map has partition type λ if we can write the source curve as Σ = Σ0 ∪ ⋃ v∈A[2] rv⋃ i=1 Σvi with each Σvi a (potentially nodal) genus 0 curve with λ v i marked Z/2-points, and where Σvi is attached to the curve Σ0 at some point, which is will be stacky depending on the parity of λvi . The map then collapses each Σ v i to the stacky point in [A/± 1] corresponding to v ∈ A[2]. Lastly, denote byMλ,n the closed substack consisting of those maps with partition type λ. The main result is the following. Proposition 5.5. If the Picard number of A is 1, then Mk,n = ∐ λ`k Mλ,n. Proof. As stated before, each Mλ,n is a closed substack, and so we must show that they are also open in Mk,n. We claim that any deformation of the nodes connected a tooth to the handle cannot be smoothed. We first note that the collection of rational curves (excluding collapsing components) in [A/ ± 1] is 0-dimensional. Indeed, if it were not then by looking at the proper transform we would obtain a positive dimensional family of rational curves in Km(A), which cannot exist (See, e.g., [8]). Now, since the Picard number is 1 and the curve class is primitive, the source curve must be the normalization of its image. We will assume for simplicity that the number of teeth on the curve is 1, and that this tooth is itself irreducible. Thus if we were to smooth the node joining this tooth, 59 the resulting source curve must be irreducible. Consider now a flat family of twisted stable maps into [A/ ± 1] over a punctured base T ′ = T \\ {p}. C //  [A/± 1] T ′ Since each fibre is the normalization of the image curve in [A/ ± 1], this family must be constant. As the moduli spaces Mk,n are separated and proper, there is a unique way to fill in the central fibre, which in this case must also be a constant family. In particular, the resulting central fibre is also the normalization of the image, and so must have no teeth. It follows that no nodes joining the teeth to the handle can be smoothed. The main result of this decomposition is the following. Recall (See Proposition 2.11) that all curves in M ([A/± 1]; 2g + 2;n) are comb curves. We claimed that the components corresponding to even parts contribute zero to the Gromov-Witten invariant—this is equivalent to saying that if a comb curve has a tooth which is joined to the handle at a non-orbifold point, then it contributes zero to the total invariant. Lemma 5.6. Let v ∈ A[2] be fixed, and let M0;2g+2|2k,v denote the compo- nent consisting of those twisted stable maps into [A/ ± 1] which collapse a component to v with 2k marked stacky points, and with 2g+2 marked stacky 60 points elsewhere. Then ∫ [M0;2g+2|2k,v ]red 1 = 0 Proof. We will assume without loss of generality (by induction on the num- ber of collapsing components) that the source curve consists of two compo- nents, Σ1 and Σ2 joined at a non-stacky point P , and with Σ1 being the handle. We begin with a little notation. Define M1 =M◦ ([A/± 1]; 2g + 2, 1;n) to be the moduli space of twisted stable maps into [A/±1] with no collapsing components and with one ordinary (i.e. non-stacky) marked point. This has (reduced) virtual dimension 1. Similarly, define M2 =M (BZ/2; 2k, 1; 0) to be the moduli space of twisted stable maps into BZ/2 = [C2/± 1], which we think of as the local model for one of the stacky points in [A/± 1]. This also has virtual dimension 1. Now, unlike the case where the node is stacky, we do not have an iso- morphism M0;2g+2|2k,v ∼=M1 ×M2 due to the fact that the curves are joined at a non-stacky point. What we 61 do have, however, is a morphism ι :M0;2g+2|2k,v →M1 ×M2 which fits into the gluing diagram M0;2g+2|2k,v  ι //M1 ×M2 ev×ev  I[A/± 1] ∆ // I[A/± 1]× I[A/± 1] where the evaluation maps on the right are from the non-stacky points. From this we obtain (see [2, Proposition 5.3.1]) that [M0;2g+2|2k,v]red = ∆! ( [M1]red × [M2]red ) . Now, since the evaluations are at non-stacky points, they in fact lie in the non-twisted sector, which is [A/±1]. Since [A/±1] satisfies Poincaré duality (rationally, at least), we can choose a basis (γi) of the cohomology of A/±1 so that this is given by ∆! ( [M1]red × [M2]red ) = ∑ i ∫ ev∗[M1]red γi ∫ ev∗[M2]red γi. Since [M1]red and [M2]red are classes in H2, we see that the only cohomol- ogy classes which may contribute are those in dimension 2. However, since the map ev : M2 → [A/ ± 1] is constant (recall that this is a collapsing component), it doesn’t intersect any classes in H2, and so each of the in- tegrals ∫ ev∗[M2]red γ i are zero. It follows then that [M0;2g+2|2k,v]red = 0 as claimed. We will next provide the proofs of several facts concerning the non- 62 enumerative nature of the Gromov-Witten invariants stated in section 2.2. First, recall that we have the decomposition of moduli spaces Uλ,n =M◦ ([A/± 1];kλ;n)× ∏ v∈A[2] rv∏ i=1 M (BZ/2;λvi + 1; 0) together with a projection map pλ : Uλ,n →M◦ ([A/± 1];kλ;n). Theorem 5.7 (Theorem 2.12). Let λ = (λv)v∈A[2] be a collection of parti- tions of k, all of which consist of odd parts. Then the virtual degree of pλ is(− 14) 12 (|k|−|kλ|). That is, (pλ)∗[Uλ,n]red = ( − 1 4 ) 1 2 (|k|−|kλ|)[ M◦ ([A/± 1];kλ;n) ]red . The proof of this is virtually identical to that in [20, Section 3.6]. Let pi : C → Uλ,n denote the universal curve, and let Σ = Σ0 ∪ Σ1 ∪ · · · ∪ Σk denote a comb curve. Since all deformations of the nodes which join the teeth to the handle are obstructed, we have the exact sequence 0→ ⊕ Pi pi∗(TPiΣ0 ⊗ TPiΣi)→ Obs(f)→ Obs(Σ, f)→ 0. (5.1) We will compute Obs(f), and use this exact sequence to compute the ob- struction bundle Obs(Σ, f). Lemma 5.8. Over a point [f : Σ→ A /± 1], the bundle Obs(f) is isomor- phic to Obs(f) ∼= H1(Σ0, f∗T [A /± 1]|Σ0)⊕ k⊕ i=1 H1(Σi, ρ1 ⊕ ρ1) 63 where ρ1 is the non-trivial representation of Z/2. Proof. As above, let f : Σ → [A / ± 1] be a comb curve with teeth Σi for 1 ≤ i ≤ k, and let T = f∗T [A /± 1]. Recall that over such a point that Obs(f) ∼= H1(Σ, T ). To compute this, we look at the normalization sequence H0(Σ0, T )⊕ k⊕ i=1 H0(Σi, T |Σi)→ k⊕ i=1 H0(Pi, T |Pi)→ → H1(Σ, T )→ k⊕ i=0 H1(Σi, T |Σi)→ 0 where Pi is the node joining Σi to Σ0. As f is representable, the image of Pi must lie in the twisted sector, and so we see that T |Σi ∼= T |Pi ∼= ρ1⊕ρ1⊕ρ0 where ρ1 and ρ0 denote the non-trivial and trivial representations of Z/2, respectively. Since Σi has stacky points, and since Pi ∼= BZ/2, we have that H0(Σ0, ρ1) ∼= 0 (and similarly for Pi). Moreover, it is clear that H0(Σi, ρ0)→ H0(Pi, ρ0) is surjective. It follows then that Obs(f) ∼= H1(Σ, T ) ∼= H1(Σ0, T |Σ0)⊕ k⊕ i=1 H1(Σi, T |Σi). Since T |Σi ∼= ρ1 ⊕ ρ1 ⊕ ρ0 and ρ0, being the trivial representation, has no higher cohomology, the lemma follows. Proof of Theorem 2.12. Since M◦0;kλ is zero-dimensional, on components of Uλ,n the summand of Obs(f) coming from H1 ( Σ0, f∗T [A / ± 1]|Σ0 ) is a 64 fixed vector space, and so it corresponds to a trivial summand. Moreover, as discussed in [21], the terms H1(Σi, ρ1⊕ρ1) contributed a dual Hodge bundle summand; that is, Obs(f) ∼= O⊕d⊕⊕ki=1 E∨i ⊕E∨i . Our exact sequence (5.1) thus reads 0→ ⊕ Pi pi∗(TPiΣ0 ⊗ TPiΣi)→ O⊕d ⊕ k⊕ i=1 E∨i ⊕ E∨i → Obs(Σ, f)→ 0. from which we compute that the total Chern class of Obs(Σ, f) over Uλ,n is given by c(O⊕d)︸ ︷︷ ︸ =1 k∏ i=1 c(E∨i )2 c ( pi∗(TPiΣ0 ⊗ TPiΣi) ) . We need to integrate this obstruction class over the fibres of the projection map pλ : Uλ,n → M◦ ([A/± 1];kλ;n). Specifically, we need to compute the integrals (using the notation of [10]) ∫ M(BZ/2;2gi+2;0) c(Ei)2 1− 12ψ1 (5.2) which are computed in [21, 10] to be (−14)gi . Since the fibre is the product ∏ v∈A[2] rv∏ i=1 M (BZ/2;λvi + 1; 0) 65 it follows that the degree of the pushforward is ∏ v∈A[2] rv∏ i=1 ( − 1 4 )1 2 (λ v i−1) = ∏ v∈A[2] ( − 1 4 )1 2k(v)− 1 2 rv = ( − 1 4 )1 2 (|k|−|kλ|) as claimed. Consider now case that A ∼= S ×F . The same reasoning as above yields that on any component of M ([A/± 1]; 2g + 2;n) which collapses a compo- nent between two non-collapsing components (see figure 4.1), the virtual class is obtained by computing the integral ∫ M(BZ/2;2g+2;0) c(E)2 (1− 12ψ1)(1− 12ψ2) which arises due to the two nodes whose smoothings are obstructed. Proposition 5.9. Let g > 0. Then the integral ∫ M(BZ/2;2g+2;0) c(E)2 (1− 12ψ1)(1− 12ψ2) is zero. Proof. We follow a method similar to the one given in [10] to prove that (5.2) is equal to (−14)g. More specifically, we assemble the Gromov-Witten invariants into a generating function, which we will see must be equal to zero. Let X = [C2/ ± 1]. Let 1 and v, respectively, denote the generators 66 of the untwisted and twisted sectors of H∗(X ), and let 〈· · · 〉 denote the integral ∫ M · · · where the integral is over the appropriate moduli space of genus 0 twisted stable maps into X . Let g > 1, and let a, b be non-negative integers. The topological recursion relations in this case yield 〈v2g−1, τa+1v, τbv, v〉 = 2 g−1∑ i=1 ( 2g − 1 2i ) 〈v2i, τav, v〉〈v2(g−i)−1, τbv, v, v〉 + 2 g∑ i=1 ( 2g − 1 2i− 1 ) 〈v2i−1, τav, 1〉〈v2(g−i), τbv, v, 1〉 = 2 g−1∑ i=1 ( 2g − 1 2i ) 〈v2i+1, τav〉〈v2(g−i)+1, τbv〉 + 2 g∑ i=1 ( 2g − 1 2i− 1 ) 〈v2i−1, τa−1v〉〈v2(g−i)+1, τb−1v〉 where the second equality is given by the string equation. For g = 1, this reads 〈v2, τa+1v, τbv〉 = 2〈v, τav, 1〉〈v, τbv, 1〉 due to the requirement that each component of the curve have an even number of stacky points. Multiplying both sides of this by 2−a−b−1 and summing a and b from 0 67 to ∞ yields 〈 v2g, v 1− 12ψ1 , v 1− 12ψ2 〉 − 〈 v2g+1, v 1− 12ψ2 〉 = g−1∑ i=1 ( 2g − 1 2i )〈 v2i+1, v 1− 12ψ1 〉〈 v2(g−i)+1, v 1− 12ψ2 〉 + 1 4 g∑ i=1 ( 2g − 1 2i− 1 )〈 v2i−1, v 1− 12ψ1 〉〈 v2(g−i)+1, v 1− 12ψ2 〉 (5.3) We now assemble these into a generating series. Let H(q) and h(q) be given by H(q) = ∞∑ g=1 〈 v2g, v 1− 12ψ1 , v 1− 12ψ2 〉 q2g−1 (2g − 1)! h(q) = q + ∞∑ g=1 〈 v2g+1, v 1− 12ψ 〉 q2g+1 (2g + 1)! From [10], we see that h(q) = 2 sin(q/2). Moreover, if we multiply (5.3) by q2g−1 (2g−1)! and sum from g = 1 to ∞, we obtain H(q)− h′′(q) = ∞∑ g=1 g−1∑ i=1 〈 v2i+1, v 1− 12ψ1 〉 q2i (2i)! 〈 v2(g−i)+1, v 1− 12ψ2 〉 q2(g−i)−1 (2g − 2i− 1)! + 1 4 ∞∑ g=1 g∑ i=1 〈 v2i−1, v 1− 12ψ1 〉 q2i−1 (2i− 1)! 〈 v2(g−i)+1, v 1− 12ψ2 〉 q2(g−i) (2g − 2i)! (5.4) A somewhat tedious computation yields that the right-hand side of (5.4) is 68 equal to ( h′(q)− 1)h′′(q) + 1 4 h(q)h′(q) = h′(q)h′′(q)− h′′(q) + 1 4 h(q)h′(q). Since h′′(q) = −14h(q), this is simply equal to −h′′(q). It follows then that H(q) = 0. Since 〈 v2g, v 1− 12ψ1 , v 1− 12ψ2 〉 = ∫ M(BZ/2;2g+2;0) c(E)2 (1− 12ψ1)(1− 12ψ2) the claim follows. Next, recall that we defined the generating functions Fn(zv) = ∑ k:A[2]→Z≥0 GWk,n ∏ v∈A[2] z k(v) v k(v)! F ◦n(xv) = ∑ k:A[2]→Z≥0 GW ◦k,n ∏ v∈A[2] x k(v) v k(v)! . We prove now the following. Theorem 5.10 (Theorem 2.13). The two generating functions Fn and F ◦n are equal after the substitution xv = 2 sin(zv/2). Proof. We prove this by computing F ◦n ( 2 sin(zv/2) ) , and showing that this is equal to Fn(zv). For simplicity we use the notation A∨ = Map(A[2],Z≥0). Note that 2 sin(zv/2) = ∞∑ `=0 ( − 1 4 )` z2`+1v (2`+ 1)! . 69 Substituting this into the definition for F ◦n we obtain F ◦n ( 2 sin(zv/2) ) = ∑ k∈A∨ GW ◦k,n ∏ v∈A[2] 1 k(v)! ( ∞∑ `=0 ( − 1 4 )` z2`+1v (2`+ 1)! )k(v) = ∑ k∈A∨ GW ◦k,n ∏ v∈A[2] ∞∑ `=0 ( − 1 4 )` z2`+k(v)v (2`+ k)(v)! s ( 2`+ k(v),k(v) ) where s(k, `) = 1 `! ∑ a1+···+a`=k ai odd ( k a1, . . . , a` ) . Exchanging the order of the summation over ` and the product over v in this expression we find that F ◦n ( 2 sin(zv/2) ) = ∑ k∈A∨ GW ◦k,n ∑ `∈A∨ ( − 1 4 )|`| ∏ v∈A z (2`+k)(v) v (2`+ k)(v)! s ( (2`+ k)(v),k(v) ) = ∑ k,`∈A∨ GW ◦k,n ( − 1 4 )|`| ∏ v∈A z (2`+k)(v) v (2`+ k)(v)! s ( (2`+ k)(v),k(v) ) If we then re-index the summation by letting k′ = 2`+k (and for simplicity of notation omitting the ′), we find F ◦n ( 2 sin(zv/2) ) = ∑ k,`∈A∨ GW ◦k−2`,n ( − 1 4 )|`| ∏ v∈A z k(v) v k(v)! s ( k(v), (k− 2`)(v)) The claim then that these two generating functions are equal is equivalent then to the claim that GWk,n = ∑ `∈A∨ GW ◦k−2`,n ( − 1 4 )|`| ∏ v∈A s ( k(v), (k− 2`)(v)) 70 This follows from theorem 2.12 and from the following interpretation of the numbers s(k, `). The number s(k, `) gives the count of all possible ways of partitioning k marked points into ` (unordered) odd-sized collections of points. In our case, by summing over all possible functions ` : A[2] → Z≥0, the numbers s ( k(v), (k − 2`)(v)) yield the count of all possible ways of partitioning the k(v) points mapping to a given 2-torsion point v by bubbling off collapsing components (all of which must have odd numbers of marked points). Order does not matter as they all map to the same point. For each such possibility, the Gromov-Witten invariant is then GW ◦k−2` (the invariant coming from the non-collapsing component) times (−14)|`| (the virtual degree of the map which forgets the collapsing components), as discussed above. This proves the theorem. 71 Chapter 6 MacMahon’s Generalized Sum-of-Divisors Functions 6.1 Statement of results In this chapter we prove some relations satisfied by the generalized sum-of- divisors functions introduced by P. A. MacMahon in [16, pp. 303, 309]. In particular, as stated in Theorem 1.3, we prove that they are in the ring of quasi-modular forms. Recall that the sum-of-divisors function σk(n) is defined to be σk(n) = ∑ d|n dk. For k = 1, this has as a generating function A1(q) = ∞∑ k=1 σ1(n)qn = ∞∑ k=1 qk (1− qk)2 . 72 Generalizing this notion, MacMahon introduces the generating functions Ak = ∑ 0 0. Note that for k = 1 this is nothing but σ1(n), the usual sum-of-divisors function. It can then be shown that we have Ak(q) = ∞∑ n=1 an,kq n. Similarly, we define cn,k to be cn,k = ∑ s1 · · · sk where the sum is over all partitions of n into n = s1(2m1 − 1) + · · ·+ sk(2mk − 1) with, as before 0 < m1 < · · · < mk. For k = 1 this is the sum over all 73 divisors whose conjugate is an odd number. As for the case of an,k, we have Ck(q) = ∞∑ n=1 cn,kq n. We recall also that Chebyshev polynomials are defined via the relation Tn(cos θ) = cos(nθ). With these we form the following generating functions. F (x, q) := 2 ∞∑ n=0 T2n+1(12x)q n2+n G(x, q) := 1 + 2 ∞∑ n=1 T2n(12x)q n2 . The results of this Chapter are the following. Theorem 6.1. We have the following equalities: F (x, q) = (q2; q2)3∞ ∞∑ k=0 Ak(q2)x2k+1 G(x, q) = (q; q)∞ (−q; q)∞ ∞∑ k=0 Ck(q)x2k where (a; q)∞ = ∏∞ k=0(1− aqk). 74 Corollary 6.2. The functions Ak(q) and Ck(q) can be written as Ak(q) = (−1)k (2k + 1)!(q; q)3∞ ∞∑ n=0 (−1)n(2n+ 1)(n+ k)! (n− k)!q 1 2n(n+1) Ck(q) = (−1)k(−q; q)∞ (2k)!(q; q)∞ ∞∑ n=1 (−1)n2n(n+ k − 1)! (n− k)! q n2 . Corollary 6.3. The functions Ak and Ck satisfy the recurrence relations Ak(q) = 1 (2k + 1)2k (( 6A1(q) + k(k − 1) ) Ak−1(q)− 2q d dq Ak−1(q) ) Ck(q) = 1 2k(2k − 1) (( 2C1(q) + (k − 1)2 ) Ck−1(q)− q d dq Ck−1(q) ) . The final result of this chapter settles a long-standing conjecture of P. A. MacMahon. In his paper [16, p. 328] he makes the claim The function Ak = ∑ an,kq n has apparently the property that the coefficient an,k is expressible as a linear function of the sum of the uneven powers of the divisors of n. I have not succeeded in reaching the general theory... What we prove is the following. Corollary 6.4. The functions Ak(q) are in the ring of quasi-modular forms. 75 6.2 Proofs Proof of theorem 6.1. Beginning with the series F (x, q), and letting x = 2 cos θ we find F (x, q) = 2 ∞∑ n=0 T2n+1(cos θ)qn 2+n = 2 ∞∑ n=0 cos ( (2n+ 1)θ ) qn 2+n = ∞∑ n=0 ( ei(2n+1)θ + e−i(2n+1)θ ) qn 2+n = ∞∑ n=0 ei(2n+1)θqn 2+n + ∞∑ n=0 e−i(2n+1)θqn 2+n where in the latter sum, letting n 7→ −n− 1 we obtain F (x, q) = eiθ ∞∑ n=−∞ e2niθqn 2+n. Using the Jacobi triple product we see that this is equal to eiθ ∞∑ n=−∞ e2niθqn 2+n = eiθ(−e−2iθ; q2)∞(−q2e2iθ; q2)∞(q2; q2)∞ = (eiθ + e−iθ)︸ ︷︷ ︸ x (q2; q2)∞ ∞∏ m=1 ( 1 + 2 cos(2θ)︸ ︷︷ ︸ x2−2 q2m + q4m ) = x(q2; q2)∞ ∞∏ m=1 ( (1− q2m)2 + x2q2m) = x(q2; q2)3∞ ∞∏ m=1 ( 1 + x2 q2m (1− q2m)2 ) = (q2; q2)3∞ ∞∑ k=0 Ak(q2)x2k+1 76 and thus comparing coefficients of x2k+1 yeilds the result. We ply a similar trick for G(x, k). In that case we have G(x, q) = 1 + 2 ∑ n>0 T2n(cos θ)qn 2 = 1 + 2 ∑ n>0 cos(2nθ)qn 2 = ∞∑ n=−∞ e2niθqn 2 which, again, by the Jacobi triple product yields ∞∑ n=−∞ e2niθqn 2 = (q2; q2)∞(−qe2iθ; q2)∞(−qe−2iθ; q2)∞ = (q2; q2)∞ ∞∏ m=1 ( 1 + 2 cos(2θ)︸ ︷︷ ︸ x2−2 q2m−1 + q4m−2 ) = (q2; q2)∞ ∞∏ m=1 ( (1− q2m−1)2 + x2q2m−1) = (q2; q2)∞(q; q2)2∞︸ ︷︷ ︸ (q;q)∞ (−q;q)∞ ∞∏ m=1 ( 1 + x2 q2m−1 (1− q2m−1)2 ) = (q; q)∞ (−q; q)∞ ∞∑ k=0 Ck(q)x2k which completes the theorem. To deduce Corollary 6.2, we begin by expanding the series F (x, q) (and 77 similarly, G(x, q)) in powers of x, i.e. F (x, q) = xf0(q) + x3f1(q) + x5f2(q) + · · ·+ x2k+1fk(q) + · · · G(x, q) = g0(q) + x2g1(q) + x4g2(q) + · · ·+ x2kgk(q) + · · · . Now, the coefficients of x2k in 2T2n(12x) and of x 2k+1 in 2T2n+1(12x) are respectively given by 2n(−1)n−k (n+ k − 1)! (n− k)!(2k)! (−1) n−k(2n+ 1) (n+ k)! (n− k)!(2k + 1)! and thus we have fk(q) = (−1)k (2k + 1)! ∞∑ n=0 (−1)n(2n+ 1)(n+ k)! (n− k)!q n2+n gk(q) = (−1)k (2k)! 2 ∞∑ n=1 (−1)nn(n+ k − 1)! (n− k)! q n2 . with g0 defined to be 1 + 2 ∑∞ n=1(−1)nqn 2 . As Theorem 6.1 implies that fk(q) = (q2; q2)3∞Ak(q2) and gk(q) = (q;q)∞ (−q;q)∞Ck(q), we see that Corollary 6.2 follows. Next, letting f0(q) = ∞∑ n=0 (−1)n(2n+ 1)qn2+n = (q2; q2)3∞ g0(q) = 1 + 2 ∞∑ n=1 (−1)nqn2 = (q; q)∞ (−q; q)∞ and defining the operators D` = q ddq − `(`− 1) and D′` = q ddq − (`− 1)2, we 78 then have that fk(q) = (−1)k (2k + 1)! Dk · · ·D1f0(q) gk(q) = (−1)k (2k)! D′k · · ·D′1g0(q). From these formulae we note that the functions fk, gk satisfy the recur- sion relations fk(q) = −1 (2k + 1)2k ( q d dq − k(k − 1) ) fk−1(q) gk(q) = −1 2k(2k − 1) ( q d dq − (k − 1)2 ) gk−1(q). Noting again that fk(q) = (q2; q2)3∞Ak(q2) and gk(q) = (q;q)∞ (−q;q)∞Ck(q), together with the relations q d dq (q2; q2)3∞ = −6(q2; q2)3∞A1(q2) q d dq (q; q)∞ (−q; q)∞ = −2 (q; q)∞ (−q; q)∞C1(q), we now obtain the recurrence relation of Corollary 6.3 between the functions Ak(q) and Ck(q). Our final result requires a bit of explanation. It is well known that the ring of modular forms for the full modular group Γ = PSL2(Z) is the polynomial ring in the generators E4, E6, where E2k(q) = 1 2 ζ(1− 2k) + ∞∑ n=1 σ2k−1(n)qn 79 are the Eisenstein series of weights 2k. There are no modular forms of weight 2 for Γ, but E2 = − 124 + ∑∞ n=1 σ1(n)q n is a quasi-modular form (See [13]). The relevant fact for this paper is that the ring of all such objects (which contains the ring of modular forms as a subring) is the ring generated either by E2, E4, and E6, or by q ddq and by E2. Noting then thatA1(q) = E2(q)+ 1 24 , the recurrence relation from Corollary 6.3 implies that each Ak(q) lies in this ring, and hence the conclusion follows. 80 Chapter 7 Conclusion 7.1 Summary The main result of this thesis is the complete solution (assuming the crepant resolution conjecture for the resolution Km(A) → A/ ± 1) of the problem stated in Question 1.1 regarding the number of hyperelliptic curves in a generic polarized Abelian surface. In face we provide a stronger result, in that we refine this count based on the discrete data of the image of the Weirstrass points in A[2]. Moreover, we show that there are strong restrictions on what possible patterns of these images are allowed, which is not a priori obvious. The formula obtained from this thesis is also of interest. It has a certain elegance to it, and it provides further connections between geometry, number theory, and combinatorics. For example, it is by no means obvious that counting certain restricted partitions of h − 1 should provide information about the number of hyperelliptic curves of arithmetic genus h in a generic abelian surface A. This also proves once more the importance of the study of modular forms in geometry. Lastly, this work offered as a nice side-effect the proof of a more-than- 90-year old conjecture of P. A. MacMahon’s [16] regarding the patterns that 81 arise when studying the functions Ak(q) and Ck(q). While MacMahon did not have available the language of (quasi-) modular forms to describe the generating functions he constructed to count partitions, it is in some ways exactly the result of Corollary 6.4 that he predicted. 7.2 Further research The most obvious direction to guide further research is to remove the reliance of this proof on the crepant resolution conjecture. That is, we would like to prove that the generation functions given hold using only the methods of orbifold Gromov-Witten theory. The first stages and the general method are outlined in Chapter 4 where the case of genus 1 and genus 2 are proven, but for higher genera more care is required. This is first seen in the genus 3 case. As stated in Section 2.3, the moduli space M ([A/± 1]; 2g + 2;n) in the case that A ∼= S × F will be a product of moduli spaces of Hurwitz covers of the rational curve F/ ± 1. In the case of genus 3, it can be shown that this space is one dimensional, although its virtual dimension is zero. Thus we need to take care with virtual fundamental classes and obstruction bundles to obtain our desired results, as well as a careful study of certain moduli spaces of Hurwitz covers. The use of Gromov-Witten theory to study Hurwitz covers has proven to be a fruitful method of study (see [19]), but as yet is limited to some strong restrictions on allowable monodromy. As the monodromy which results from computing the number of higher genus hyperelliptic curves in A ∼= S × F falls out of this class, it would be worthwhile to work to extend these known 82 results. A further approach that is of interest is to prove Theorem 1.3 is to follow along the ideas for proving the genus 2 case given in [11]. In that paper, the count of genus 2 curves is given by a purely lattice-theoretic approach. The idea is that a map C → A for an Abelian variety A factors as C // ''OO OOO OOO OOO OOO J(C)  A where J(C) is the Jacobian variety of the curve C. Since J(C) is itself an Abelian variety, we can study the map C → A up to translation in A by instead looking at the map H1 ( J(C) ) ∼= H1(C)→ H1(A) using this method. As the Jacobian of hyperelliptic curves has some special structure (see [18]), it follows that we should be able to derive these results simply by studying the maps of lattices. Finally, one other direction of interest is inspired by Section 4.4 and Equation (4.2): A1(u) = E(u) + 3A1(u2)− 2A1(u4). We see in this case that a certain linear combination of quasi-modular forms of higher level combine to produce one of lower level. An analysis of the genus 3 case yields a similar phenomena for the generation functions A2(u) and for C2(u), and so the question arises under which conditions this can occur. This leads into the Galois theory of modular functions, in an attempt 83 to understand the particulars of how linear combinations of non-invariant objects become invariant under the larger group PSL2(Z). 84 Bibliography [1] Dan Abramovich, Tom Graber, and Angelo Vistoli. Algebraic orbifold quantum products. In Orbifolds in mathematics and physics (Madison, WI, 2001), volume 310 of Contemp. Math., pages 1–24. Amer. Math. Soc., Providence, RI, 2002. [2] Dan Abramovich, Tom Graber, and Angelo Vistoli. Gromov-Witten theory of Deligne-Mumford stacks. Amer. J. Math., 130(5):1337–1398, 2008. [3] George E. Andrews and Simon C.F. Rose. Macmahon’s sum-of-divisors functions, Chebyshev polynomials, and quasi-modular forms. J. Reine Angew. Math., 2010. To appear. [4] W. Barth, C. Peters, and A. Van de Ven. Compact Complex Surfaces, chapter VIII. Springer-Verlag, 1984. [5] Jim Bryan and Tom Graber. The crepant resolution conjecture. In Algebraic geometry—Seattle 2005. Part 1, volume 80 of Proc. Sympos. Pure Math., pages 23–42. Amer. Math. Soc., Providence, RI, 2009. 85 [6] Jim Bryan and Naichung Conan Leung. Generating functions for the number of curves on abelian surfaces. Duke Math. J., 99(2):311–328, 1999. [7] Jim Bryan and Naichung Conan Leung. The enumerative geometry of K3 surfaces and modular forms. J. Amer. Math. Soc., 13(2):371–410 (electronic), 2000. [8] X. Chen. Rational curves on K3 surfaces. J. Algebraic Geom., 8:245– 278, 1999. [9] William Gillam. Hyperelliptic Gromov-Witten Theory. PhD thesis, Columbia University, 2008. [10] William Gillam. Letter to J. wise, 2008. [11] Lothar Göttsche. A conjectural generating function for numbers of curves on surfaces. Comm. Math. Phys., 196(3):523–533, 1998. [12] Tom Graber. Enumerative geometry of hyperelliptic plane curves. J. Algebraic Geom., 10(4):725–755, 2001. [13] Masanobu Kaneko and Don Zagier. A generalized Jacobi theta function and quasimodular forms. In The moduli space of curves (Texel Island, 1994), volume 129 of Progr. Math., pages 165–172. Birkhäuser Boston, Boston, MA, 1995. [14] H. Lange and Ch. Birkenhake. Complex Abelian Varieties, chapter 5. Springer-Verlag, 1992. 86 [15] A. M. Legendre. Traité des Fonctions Elliptiques, pages 133–134. Im- primerie de Huzard-Courcier, 1828. [16] P. A. MacMahon. Divisors of numbers and their continuations in the theory of partitions. In G. Andrews, editor, Reprinted: Percy A. MacMahon Collected Papers, pages 305–341. MIT Press, Cambridge, 1986. [17] D. Maulik, R. Pandharipande, and R. P. Thomas. Curves on K3 sur- faces and modular forms. J. Topol., 3(4):937–996, 2010. With an ap- pendix by A. Pixton. [18] D. Mumford. Tata Lectures on Theta: Jacobian theta functions and differential equations. Progress in mathematics. Birkhäuser, 1984. [19] A. Okounkov and R. Pandharipande. Gromov-witten theory, hurwitz theory, and completed cycles. Anal. Math., 163(2):517–560, 2006. [20] Jonathan Wise. The genus zero Gromov-Witten invariants of [Sym2P2] and the enumerative geometry of hyperelliptic curves in P2. ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Brown University. [21] Jonathan Wise. A hyperelliptic hodge integral, 2008. [22] Shing-Tung Yau and Eric Zaslow. BPS states, string duality, and nodal curves on K3. Nuclear Phys. B, 471(3):503–512, 1996. 87"@en ; edm:hasType "Thesis/Dissertation"@en ; vivo:dateIssued "2012-05"@en ; edm:isShownAt "10.14288/1.0080648"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Mathematics"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "Attribution-NonCommercial-NoDerivatives 4.0 International"@en ; ns0:rightsURI "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Counting hyperelliptic curves in Abelian surfaces with quasi-modular forms"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/42091"@en .