Modular Forms, Hecke Operators, and Modular Abelian Varieties
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Canonical Heights on Varieties with Morphisms Compositio Mathematica, Tome 89, No 2 (1993), P
COMPOSITIO MATHEMATICA GREGORY S. CALL JOSEPH H. SILVERMAN Canonical heights on varieties with morphisms Compositio Mathematica, tome 89, no 2 (1993), p. 163-205 <http://www.numdam.org/item?id=CM_1993__89_2_163_0> © Foundation Compositio Mathematica, 1993, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 89: 163-205,163 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands. Canonical heights on varieties with morphisms GREGORY S. CALL* Mathematics Department, Amherst College, Amherst, MA 01002, USA and JOSEPH H. SILVERMAN** Mathematics Department, Brown University, Providence, RI 02912, USA Received 13 May 1992; accepted in final form 16 October 1992 Let A be an abelian variety defined over a number field K and let D be a symmetric divisor on A. Néron and Tate have proven the existence of a canonical height hA,D on A(k) characterized by the properties that hA,D is a Weil height for the divisor D and satisfies A,D([m]P) = m2hA,D(P) for all P ~ A(K). Similarly, Silverman [19] proved that on certain K3 surfaces S with a non-trivial automorphism ~: S ~ S there are two canonical height functions hs characterized by the properties that they are Weil heights for certain divisors E ± and satisfy ±S(~P) = (7 + 43)±1±S(P) for all P E S(K) . -
Identities Between Hecke Eigenforms Have Been Studied by Many Authors
Identities between Hecke Eigenforms D. Bao February 20, 2018 Abstract In this paper, we study solutions to h = af 2 + bfg + g2, where f,g,h are Hecke newforms with respect to Γ1(N) of weight k > 2 and a, b = 0. We show that the number of solutions is finite for 6 all N. Assuming Maeda’s conjecture, we prove that the Petersson inner product f 2, g is nonzero, where f and g are any nonzero cusp h i eigenforms for SL2(Z) of weight k and 2k, respectively. As a corollary, we obtain that, assuming Maeda’s conjecture, identities between cusp 2 n eigenforms for SL2(Z) of the form X + i=1 αiYi = 0 all are forced by dimension considerations. We also give a proof using polynomial P identities between eigenforms that the j-function is algebraic on zeros of Eisenstein series of weight 12k. 1 Introduction arXiv:1701.03189v1 [math.NT] 11 Jan 2017 Fourier coefficients of Hecke eigenforms often encode important arithmetic information. Given an identity between eigenforms, one obtains nontrivial relations between their Fourier coefficients and may in further obtain solu- tions to certain related problems in number theory. For instance, let τ(n) be the nth Fourier coefficient of the weight 12 cusp form ∆ for SL2(Z) given by ∞ ∞ ∆= q (1 qn)24 = τ(n)qn − n=1 n=1 Y X 1 11 and define the weight 11 divisor sum function σ11(n) = d n d . Then the Ramanujan congruence | P τ(n) σ (n) mod691 ≡ 11 can be deduced easily from the identity 1008 756 E E2 = × ∆, 12 − 6 691 where E6 (respectively E12) is the Eisenstein series of weight 6 (respectively 12) for SL2(Z). -
Abelian Varieties and Theta Functions Associated to Compact Riemannian Manifolds; Constructions Inspired by Superstring Theory
ABELIAN VARIETIES AND THETA FUNCTIONS ASSOCIATED TO COMPACT RIEMANNIAN MANIFOLDS; CONSTRUCTIONS INSPIRED BY SUPERSTRING THEORY. S. MULLER-STACH,¨ C. PETERS AND V. SRINIVAS MATH. INST., JOHANNES GUTENBERG UNIVERSITAT¨ MAINZ, INSTITUT FOURIER, UNIVERSITE´ GRENOBLE I ST.-MARTIN D'HERES,` FRANCE AND TIFR, MUMBAI, INDIA Resum´ e.´ On d´etailleune construction d^ue Witten et Moore-Witten (qui date d'environ 2000) d'une vari´et´eab´elienneprincipalement pola- ris´eeassoci´ee`aune vari´et´ede spin. Le th´eor`emed'indice pour l'op´erateur de Dirac (associ´e`ala structure de spin) implique qu'un accouplement naturel sur le K-groupe topologique prend des valeurs enti`eres.Cet ac- couplement sert commme polarization principale sur le t^oreassoci´e. On place la construction dans un c^adreg´en´eralce qui la relie `ala ja- cobienne de Weil mais qui sugg`ereaussi la construction d'une jacobienne associ´ee`an'importe quelle structure de Hodge polaris´eeet de poids pair. Cette derni`ereconstruction est ensuite expliqu´eeen termes de groupes alg´ebriques,utile pour le point de vue des cat´egoriesTannakiennes. Notre construction depend de param`etres,beaucoup comme dans la th´eoriede Teichm¨uller,mais en g´en´erall'application de p´eriodes n'est que de nature analytique r´eelle. Abstract. We first investigate a construction of principally polarized abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000. The index theorem for the Dirac operator associated to the spin structure implies integrality of a natural skew pairing on the topological K-group. The latter serves as a principal polarization. -
Automorphisms of a Symmetric Product of a Curve (With an Appendix by Najmuddin Fakhruddin)
Documenta Math. 1181 Automorphisms of a Symmetric Product of a Curve (with an Appendix by Najmuddin Fakhruddin) Indranil Biswas and Tomas´ L. Gomez´ Received: February 27, 2016 Revised: April 26, 2017 Communicated by Ulf Rehmann Abstract. Let X be an irreducible smooth projective curve of genus g > 2 defined over an algebraically closed field of characteristic dif- ferent from two. We prove that the natural homomorphism from the automorphisms of X to the automorphisms of the symmetric product Symd(X) is an isomorphism if d > 2g − 2. In an appendix, Fakhrud- din proves that the isomorphism class of the symmetric product of a curve determines the isomorphism class of the curve. 2000 Mathematics Subject Classification: 14H40, 14J50 Keywords and Phrases: Symmetric product; automorphism; Torelli theorem. 1. Introduction Automorphisms of varieties is currently a very active topic in algebraic geom- etry; see [Og], [HT], [Zh] and references therein. Hurwitz’s automorphisms theorem, [Hu], says that the order of the automorphism group Aut(X) of a compact Riemann surface X of genus g ≥ 2 is bounded by 84(g − 1). The group of automorphisms of the Jacobian J(X) preserving the theta polariza- tion is generated by Aut(X), translations and inversion [We], [La]. There is a universal constant c such that the order of the group of all automorphisms of any smooth minimal complex projective surface S of general type is bounded 2 above by c · KS [Xi]. Let X be a smooth projective curve of genus g, with g > 2, over an al- gebraically closed field of characteristic different from two. -
Associating Abelian Varieties to Weight-2 Modular Forms: the Eichler-Shimura Construction
Ecole´ Polytechnique Fed´ erale´ de Lausanne Master’s Thesis in Mathematics Associating abelian varieties to weight-2 modular forms: the Eichler-Shimura construction Author: Corentin Perret-Gentil Supervisors: Prof. Akshay Venkatesh Stanford University Prof. Philippe Michel EPF Lausanne Spring 2014 Abstract This document is the final report for the author’s Master’s project, whose goal was to study the Eichler-Shimura construction associating abelian va- rieties to weight-2 modular forms for Γ0(N). The starting points and main resources were the survey article by Fred Diamond and John Im [DI95], the book by Goro Shimura [Shi71], and the book by Fred Diamond and Jerry Shurman [DS06]. The latter is a very good first reference about this sub- ject, but interesting points are sometimes eluded. In particular, although most statements are given in the general setting, the book mainly deals with the particular case of elliptic curves (i.e. with forms having rational Fourier coefficients), with little details about abelian varieties. On the other hand, Chapter 7 of Shimura’s book is difficult, according to the author himself, and the article by Diamond and Im skims rapidly through the subject, be- ing a survey. The goal of this document is therefore to give an account of the theory with intermediate difficulty, accessible to someone having read a first text on modular forms – such as [Zag08] – and with basic knowledge in the theory of compact Riemann surfaces (see e.g. [Mir95]) and algebraic geometry (see e.g. [Har77]). This report begins with an account of the theory of abelian varieties needed for what follows. -
Mirror Symmetry of Abelian Variety and Multi Theta Functions
1 Mirror symmetry of Abelian variety and Multi Theta functions by Kenji FUKAYA (深谷賢治) Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto Japan Table of contents § 0 Introduction. § 1 Moduli spaces of Lagrangian submanifolds and construction of a mirror torus. § 2 Construction of a sheaf from an affine Lagrangian submanifold. § 3 Sheaf cohomology and Floer cohomology 1 (Construction of a homomorphism). § 4 Isogeny. § 5 Sheaf cohomology and Floer cohomology 2 (Proof of isomorphism). § 6 Extension and Floer cohomology 1 (0 th cohomology). § 7 Moduli space of holomorphic vector bundles on a mirror torus. § 8 Nontransversal or disconnected Lagrangian submanifolds. ∞ § 9 Multi Theta series 1 (Definition and A formulae.) § 10 Multi Theta series 2 (Calculation of the coefficients.) § 11 Extension and Floer cohomology 2 (Higher cohomology). § 12 Resolution and Lagrangian surgery. 2 § 0 Introduction In this paper, we study mirror symmetry of complex and symplectic tori as an example of homological mirror symmetry conjecture of Kontsevich [24], [25] between symplectic and complex manifolds. We discussed mirror symmetry of tori in [12] emphasizing its “noncom- mutative” generalization. In this paper, we concentrate on the case of a commutative (usual) torus. Our result is a generalization of one by Polishchuk and Zaslow [42], [41], who studied the case of elliptic curve. The main results of this paper establish a dictionary of mirror symmetry between symplectic geometry and complex geometry in the case of tori of arbitrary dimension. We wrote this dictionary in the introduction of [12]. We present the argument in a way so that it suggests a possibility of its generalization. -
Modular Forms, the Ramanujan Conjecture and the Jacquet-Langlands Correspondence
Appendix: Modular forms, the Ramanujan conjecture and the Jacquet-Langlands correspondence Jonathan D. Rogawski1) The theory developed in Chapter 7 relies on a fundamental result (Theorem 7 .1.1) asserting that the space L2(f\50(3) x PGLz(Op)) decomposes as a direct sum of tempered, irreducible representations (see definition below). Here 50(3) is the compact Lie group of 3 x 3 orthogonal matrices of determinant one, and r is a discrete group defined by a definite quaternion algebra D over 0 which is split at p. The embedding of r in 50(3) X PGLz(Op) is defined by identifying 50(3) and PGLz(Op) with the groups of real and p-adic points of the projective group D*/0*. Although this temperedness result can be viewed as a combinatorial state ment about the action of the Heckeoperators on the Bruhat-Tits tree associated to PGLz(Op). it is not possible at present to prove it directly. Instead, it is deduced as a corollary of two other results. The first is the Ramanujan-Petersson conjec ture for holomorphic modular forms, proved by P. Deligne [D]. The second is the Jacquet-Langlands correspondence for cuspidal representations of GL(2) and multiplicative groups of quaternion algebras [JL]. The proofs of these two results involve essentially disjoint sets of techniques. Deligne's theorem is proved using the Riemann hypothesis for varieties over finite fields (also proved by Deligne) and thus relies on characteristic p algebraic geometry. By contrast, the Jacquet Langlands Theorem is analytic in nature. The main tool in its proof is the Seiberg trace formula. -
AN EXPLORATION of COMPLEX JACOBIAN VARIETIES Contents 1
AN EXPLORATION OF COMPLEX JACOBIAN VARIETIES MATTHEW WOOLF Abstract. In this paper, I will describe my thought process as I read in [1] about Abelian varieties in general, and the Jacobian variety associated to any compact Riemann surface in particular. I will also describe the way I currently think about the material, and any additional questions I have. I will not include material I personally knew before the beginning of the summer, which included the basics of algebraic and differential topology, real analysis, one complex variable, and some elementary material about complex algebraic curves. Contents 1. Abelian Sums (I) 1 2. Complex Manifolds 2 3. Hodge Theory and the Hodge Decomposition 3 4. Abelian Sums (II) 6 5. Jacobian Varieties 6 6. Line Bundles 8 7. Abelian Varieties 11 8. Intermediate Jacobians 15 9. Curves and their Jacobians 16 References 17 1. Abelian Sums (I) The first thing I read about were Abelian sums, which are sums of the form 3 X Z pi (1.1) (L) = !; i=1 p0 where ! is the meromorphic one-form dx=y, L is a line in P2 (the complex projective 2 3 2 plane), p0 is a fixed point of the cubic curve C = (y = x + ax + bx + c), and p1, p2, and p3 are the three intersections of the line L with C. The fact that C and L intersect in three points counting multiplicity is just Bezout's theorem. Since C is not simply connected, the integrals in the Abelian sum depend on the path, but there's no natural choice of path from p0 to pi, so this function depends on an arbitrary choice of path, and will not necessarily be holomorphic, or even Date: August 22, 2008. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
Torsion Subgroups of Abelian Varieties
Torsion Subgroups of Abelian Varieties Raoul Wols April 19, 2016 In this talk I will explain what abelian varieties are and introduce torsion subgroups on abelian varieties. k is always a field, and Vark always denotes the category of varieties over k. That is to say, geometrically integral sepa- rated schemes of finite type with a morphism to k. Recall that the product of two A; B 2 Vark is just the fibre product over k: A ×k B. Definition 1. Let C be a category with finite products and a terminal object 1 2 C. An object G 2 C is called a group object if G comes equipped with three morphism; namely a \unit" map e : 1 ! G; a \multiplication" map m : G × G ! G and an \inverse" map i : G ! G such that id ×m G × G × G G G × G m×idG m G × G m G commutes, which tells us that m is associative, and such that (e;id ) G G G × G idG (idG;e) m G × G m G commutes, which tells us that e is indeed the neutral \element", and such that (id ;i)◦∆ G G G × G (i;idG)◦∆ m e0 G × G m G 1 commutes, which tells us that i is indeed the map that sends \elements" to inverses. Here we use ∆ : G ! G × G to denote the diagonal map coming from the universal property of the product G × G. The map e0 is the composition G ! 1 −!e G. id Now specialize to C = Vark. The terminal object is then 1 = (Spec(k) −! Spec(k)), and giving a unit map e : 1 ! G for some scheme G over k is equivalent to giving an element e 2 G(k). -
Singularities of Integrable Systems and Nodal Curves
Singularities of integrable systems and nodal curves Anton Izosimov∗ Abstract The relation between integrable systems and algebraic geometry is known since the XIXth century. The modern approach is to represent an integrable system as a Lax equation with spectral parameter. In this approach, the integrals of the system turn out to be the coefficients of the characteristic polynomial χ of the Lax matrix, and the solutions are expressed in terms of theta functions related to the curve χ = 0. The aim of the present paper is to show that the possibility to write an integrable system in the Lax form, as well as the algebro-geometric technique related to this possibility, may also be applied to study qualitative features of the system, in particular its singularities. Introduction It is well known that the majority of finite dimensional integrable systems can be written in the form d L(λ) = [L(λ),A(λ)] (1) dt where L and A are matrices depending on the time t and additional parameter λ. The parameter λ is called a spectral parameter, and equation (1) is called a Lax equation with spectral parameter1. The possibility to write a system in the Lax form allows us to solve it explicitly by means of algebro-geometric technique. The algebro-geometric scheme of solving Lax equations can be briefly described as follows. Let us assume that the dependence on λ is polynomial. Then, with each matrix polynomial L, there is an associated algebraic curve C(L)= {(λ, µ) ∈ C2 | det(L(λ) − µE) = 0} (2) called the spectral curve. -
Deligne Pairings and Families of Rank One Local Systems on Algebraic Curves
DELIGNE PAIRINGS AND FAMILIES OF RANK ONE LOCAL SYSTEMS ON ALGEBRAIC CURVES GERARD FREIXAS I MONTPLET AND RICHARD A. WENTWORTH Abstract. For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface. Contents 1. Introduction2 2. Relative Connections and Deligne Pairings5 2.1. Preliminary definitions5 2.2. Gauss-Manin invariant7 2.3. Deligne pairings, norm and trace9 2.4. Metrics and connections 10 3. Trace Connections and Intersection Connections 12 3.1. Weil reciprocity and trace connections 12 3.2. Reformulation in terms of Poincaré bundles 17 3.3. Intersection connections 20 4. Proof of the Main Theorems 23 4.1. The canonical extension: local description and properties 23 4.2. The canonical extension: uniqueness 29 4.3. Variant in the absence of rigidification 31 4.4. Relation between trace and intersection connections 32 arXiv:1507.02920v1 [math.DG] 10 Jul 2015 4.5. Curvatures 34 5. Examples and Applications 37 5.1. Reciprocity for trivial fibrations 37 5.2. Holomorphic extension of analytic torsion 40 5.3.