Abstract: Suppose that E is an elliptic curve defined over Q, and that p is a prime where E has good ordinary reduction. Under the assumption that E is modular, one can define nonnegative integers λ E , μ alg E , λ anal E , and μ anal E . The “algebraic” Iwasawa invariants λ alg E and μ E are defined in terms of the structure of the p-primary subgroup SelE(Q∞)p of the Selmer group for E over the cyclotomic Zp-extension Q∞ of Q. The definition of the “analytic” invariants μ E and λ anal E is in terms of the p-adic L-function for E constructed by Mazur and Swinnerton-Dyer [MSD74]. We will recall these definitions below. Our purpose in this article is to prove in certain cases that μ E = μ anal E = 0 and that λ alg E = λ anal E . These equalities, together with a deep theorem of Kato, imply the main conjecture for E over Q∞. In this introduction, we will discuss the nature of our results and give an outline of the proofs for the case of modular elliptic curves. We want to point out, however, that the theorems proven in the text apply to modular forms, as well as to elliptic curves with multiplicative reduction at p. We will not attempt to state the most general versions here. If K is any algebraic extension of Q, then the Selmer group for E over K is a certain subgroup of H(GK , E(Q)tors), where GK = Gal (Q/K). The Selmer group fits into an exact sequence 0 → E(K)⊗Q/Z → SelE(K) → XE(K)

Abstract: Preface 1 Overview of modular forms 2 Representations of a group 3 Representations and modular forms 4 Galois cohomology 5 Modular L-values and Selmer groups Bibliography Subject index List of statements List of symbols

Abstract: On a problem of injectivity for the Radon transform on a paraboloid by M. Agranovsky Anti-self-dual symplectic forms and integral geometry by J. C. Alvarez Holomorphic extendibility of functions via nonlinear Fourier transforms by T. T. Banh Division-interpolation methods and Nullstellensatze by C. A. Berenstein and A. Yger Uniqueness and non-uniqueness for microanalytic continuation of ultradistributions by J. Boman A rational Landen transformation. The case of degree six by G. Boros and V. H. Moll A perturbation result for linear differential operators admitting a global right inverse on $\mathcal{D}^\prime$ by R. W. Braun, R. Meise, and B. A. Taylor Weyl's law with error estimate by S. Catto, J. Huntley, N. Moh, and D. Tepper Dirac equation in the octonionic algebra by F. Colombo, I. Sabadini, and D. C. Struppa A note on Siegel's proof of Hamburger's theorem by W. Culp-Ressler and W. d. A. Pribitkin Sheaves and $\mathcal{D}$-modules in integral geometry by A. D'Agnolo Positivity conditions and squared norms of holomorphic polynomial mappings by J. P. D'Angelo The exponential x-ray transform and Fritz John's equation. I. Range description by L. Ehrenpreis, P. Kuchment, and A. Panchenko Two lemmas in local analytic geometry by C. L. Epstein and G. M. Henkin Partitions and theta constant identities by H. M. Farkas and I. Kra The Radon transform and spectral rigidity of the Grassmannians by J. Gasqui and H. Goldschmidt Complex powers of convolution operators on the Heisenberg group by D. Geller Estimates for generalized Radon transforms in three and four dimensions by A. Greenleaf, A. Seeger, and S. Wainger Irregular sampling and the Radon transform by E. Grinberg and I. Pesenson Involution of $\Lambda$-adic analytic spaces and the $U_p$-operator for half-integral weight modular forms by P. Guerzhoy Maximal sequences of compact Riemann surfaces by R. C. Gunning On BMO singularities of solutions of analytic complex vector fields by J. Hounie and J. Tavares The initial value problem for a fifth order shallow water equation on the real line by A. A. Himonas and G. Misiolek Mappings between degenerate real analytic hypersurfaces in $C^n$ by X. Huang, J. Merker, and F. Meylan Analysis of artifacts in local tomography with nonsmooth attenuation by A. Katsevich The Hecke convergence factor and modular forms of weight zero by M. I. Knopp and W. d. A. Pribitkin Hypoellipticity at points of infinite type by J. J. Kohn Semi-global analytic regularity for ${{\overline{\partial}}_b}$ on CR submanifolds of $\mathbb{C}^2$ by M. Derridj and D. S. Tartakoff Cauchy transform and Hardy spaces for rough planar domains by L. Lanzani On real analytic planar vector fields near the character set by A. Meziani On longest increasing subsequences in random permutations by A. M. Odlyzko and E. M. Rains Convergence of Poincare series with two complex coweights by P. C. Pasles Eisenstein series and Eichler integrals by W. d. A. Pribitkin Real hypersurfaces with no infinitesimal CR automorphisms by N. K. Stanton Extension of cohomology classes by F. Treves Two-radius support theorems for spherical Radon transforms on manifolds by Y. Zhou and E. T. Quinto.

Abstract: Keywords: modular curves ; modular forms ; $L$-functions ; nonvanishing Reference TAN-ARTICLE-2000-002 Record created on 2008-11-14, modified on 2017-05-12

Abstract: In this paper, we set up Shimura and Shintani correspondences between Jacobi forms and modular forms of integral weight for arbitrary level and character, and generalize the Eichler-Zagier isomorphism between Jacobi forms and modular forms of half-integral weight to higher levels. Using this together with the known results, we get a strong multiplicity 1 theorem in certain cases for both Jacobi cusp newforms and half-integral weight cusp newforms. As a consequence, we get, among other results, the explicit Waldspurger theorem.

Abstract: We suggest to compactify the universal covering of the moduli space of complex structures by non-commutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of abelian varieties this approach gives quantum tori as a non-commutative boundary of the moduli space. Relations to mirror symmetry, modular forms and deformation theory are discussed.

Abstract: We study graded traces of vectors in free bosonic vertex operator algebras and lattice vertex operator algebras. We show in particular that trace functions in these two theories always have the shape f(q)/\eta(q)^d where f(q) is quasi-modular in the case of d free bosons, and modular (i.e., a sum of holomorphic modular forms of various weights) in the case of theories based on a lattice L of rank d. We also show how spherical harmonic polynomials with respect to L are related to primary fields in lattice theories.

Abstract: We determine the space of 1-point correlation functions associated with the Moonshine module: they are precisely those modular forms of non-negative integral weight which are holomorphic in the upper half plane, have a pole of order at most 1 at infinity, and whose Fourier expansion has constant 0. There are Monster-equivariant analogues in which one naturally associates to each element in the Monster a modular form of fixed weight k, the case k=0 corresponding to the original ``Moonshine'' of Conway and Norton.

Abstract: It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that the partial differential equations they satisfy are governed by Gauss--Manin connections, whose monodromy groups are well-known. Modular theta functions provide a concrete interpretation of our result, and we study their differential properties in detail in the case of degree 2.

Abstract: Fix a prime number p and choose, once and for all, an embedding of the algebraic closure of Q into Qp. Let k and N be integers, and suppose N is not divisible by p. If f is a modular form of weight k, level N, and trivial character which is an eigenform for the p-th Hecke operator Tp, we define the slope of f to be the p-adic valuation of the eigenvalue of Tp. This paper reports on computations that suggest that there is quite a lot of structure to the set of slopes for eigenforms of varying weight k. In particular, we find that the slopes are often smaller than expected, that they are almost always integers, that there is evidence of a connection between fractional slopes and slopes which are "bigger than usual", and that there are some hints of a connection to the theory of theta-cycles.

Abstract: In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R. Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. MotivatedbyRussell'stheorem,westateandproveitscubicanaloguewhichallowsustoconstructRussell-type modular equations in the theory of signature 3.

Abstract: This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered product of regular trees. We construct explicit arithmetic examples using quaternion algebras over totally real fields. Here we reduce the Ramanujan property to special cases of the Ramanujan-Petersson conjecture for holomorphic Hilbert modular forms, many of which are known.

Abstract: On pages 51–53 of his lost notebook, S. Ramanujan expressed several integrals of products of Dedekind eta-functions in terms of incomplete elliptic integrals of the first kind. In this paper, we prove these identities using only results found in Ramanujan’s notebooks. We then construct several new elliptic integrals of this type using modular identities associated with certain “Hauptmoduls.”

Abstract: Ramanujan recorded several hundred modular equations in his three notebooks [7]; no other mathematician has ever discovered nearly so many. Complete proofs for all the modular equations in Ramanujan’s three notebooks can be found in Berndt’s books [1]–[3]. In particular, Chapters 19–21 in Ramanujan’s second notebook are almost exclusively devoted to modular equations. Ramanujan used modular equations to evaluate class invariants, certain q-continued fractions including the Rogers-Ramanujan continued fraction, theta-functions, and certain other quotients and products of theta-functions and eta-functions [3].

Abstract: A well-known theorem of Ken Ribet asserts that, under certain assumptions, a modular form (mod )o nΓ0(N) can be "lifted" to yield a newform on Γ0(N ) with the same modular Galois representation. For further progress in the modular theory of automorphic forms one will need to understand this phenomenon for automorphic forms on reductive groups other than GL(2). In this paper we prove such a result for the unitary group of rank 3, under suitable assumptions. The proof relies on the modular representation theory of p-adic reductive groups. 1. Introduction. Suppose N 1 is an integer and Γ0(N) SL (2, )i s the usual Hecke subgroup; S2(Γ0(N)) denotes the space of cusp forms of weight 2. Let f S2(Γ0(N)) be a new form (with trivial Nebentypus) that is an eigen- form for all the Hecke operators T( p), with Fourier expansion

Abstract: We provide a new proof of Rademacher's celebrated exact formula for the partition function. Along the way we present a simple treatment of an integral which is ubiquitous in the theory of nonanalytic automorphic forms.

Abstract: The purpose of this short paper is to state the weight-monodromy conjecture for l-adic representation associated to modular forms, which is implicitly proved in the paper [10], and to give a correct argument on an ambiguous point in the proof in it. The monodromy-weight conjecture is mentioned in [8, (3.2)]. We will prove the corresponding property for Hilbert modular forms in a forthcoming paper [11].

Abstract: Our aim is to study the cohomology groups of some coherent sheaves on a Griffiths–Schmid variety associated with an anisotropic Q-from of the unitary group SU(2,1). We define some transforms relating this cohomology to the coherent cohomology groups of some sheaves defined on certain threefolds, which are fibered in projective lines over Picard modular surfaces. In particular, we give a complete and explicit description, in terms of classical Picard modular forms, of the holomorphic (resp. anti-holomorphic) part of the 1-cohomology of the Griffiths-Schmid variety. From this, it results an explicit generating system for the part of the 2–cohomology which correspond to those automorphic representations whose archimedean component is a degenerate limit of discrete series.

Abstract: Invited Talks.- The Complexity of Some Lattice Problems.- Rational Points Near Curves and Small Nonzero | x 3 ? y 2| via Lattice Reduction.- Coverings of Curves of Genus 2.- Lattice Reduction in Cryptology: An Update.- Contributed Papers.- Construction of Secure C ab Curves Using Modular Curves.- Curves over Finite Fields with Many Rational Points Obtained by Ray Class Field Extensions.- New Results on Lattice Basis Reduction in Practice.- Baby-Step Giant-Step Algorithms for Non-uniform Distributions.- On Powers as Sums of Two Cubes.- Factoring Polynomials over ?-Adic Fields.- Strategies in Filtering in the Number Field Sieve.- Factoring Polynomials over Finite Fields and Stable Colorings of Tournaments.- Computing Special Values of Partial Zeta Functions.- Construction of Tables of Quartic Number Fields.- Counting Discriminants of Number Fields of Degree up to Four.- On Reconstruction of Algebraic Numbers.- Dissecting a Sieve to Cut Its Need for Space.- Counting Points on Hyperelliptic Curves over Finite Fields.- Modular Forms for GL(3) and Galois Representations.- Modular Symbols and Hecke Operators.- Fast Jacobian Group Arithmetic on C ab Curves.- Lifting Elliptic Curves and Solving the Elliptic Curve Discrete Logarithm Problem.- A One Round Protocol for Tripartite Diffie-Hellman.- On Exponential Sums and Group Generators for Elliptic Curves over Finite Fields.- Component Groups of Quotients of J 0(N).- Fast Computation of Relative Class Numbers of CM-Fields.- On Probable Prime Testing and the Computation of Square Roots mod n.- Improving Group Law Algorithms for Jacobians of Hyperelliptic Curves.- Central Values of Artin L-Functions for Quaternion Fields.- The Pseudoprimes up to 1013.- Computing the Number of Goldbach Partitions up to 5 108.- Numerical Verification of the Brumer-Stark Conjecture.- Explicit Models of Genus 2 Curves with Split CM.- Reduction in Purely Cubic Function Fields of Unit Rank One.- Factorization in the Composition Algebras.- A Fast Algorithm for Approximately Counting Smooth Numbers.- Computing All Integer Solutions of a General Elliptic Equation.- A Note on Shanks's Chains of Primes.- Asymptotically Fast Discrete Logarithms in Quadratic Number Fields.- Asymptotically Fast GCD Computation in ?[i].

Abstract: In this paper the author will give new proofs of the Borweins' cubic theta function identity and a related identity relying on the properties of elliptic functions and the technique of comparing constant terms.

Abstract: We prove three modular equations of Ramanujan using theta-function identities. Proofs via methods known to Ramanujan were not available hitherto. One had previously been proved by classical methods, and two had been proved using the theory of modular forms.