Abstract: We attach p-adic L-functions to critical modular forms and study them. We prove that those L-functions fit in a two-variables p-adic L-function defined locally everywhere on the eigencurve.

Abstract: We introduce a class of deformations of the values of the Goss zeta function. We prove, with the use of the theory of deformations of vectorial modular forms as well as with other techniques, a formula for their value at 1, and some arithmetic properties of values at other positive integers. Our formulas involve Anderson and Thakur’s function !. We discuss how our formulas may be used to investigate the existence of a kind of functional equation for the Goss zeta function.

Abstract: The congruence subgroup property is established for the modular representations associated to any modular tensor category. This result is used to prove that the kernel of the representation of the modular group on the conformal blocks of any rational, C_2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup. The Galois symmetry of the modular representations is obtained and the order of the anomaly for those modular categories satisfying some integrality conditions is determined.

Abstract: We study sums over primes of trace functions of $\ell$-adic sheaves. Using an extension of our earlier results on algebraic twist of modular forms to the case of Eisenstein series and bounds for Type II sums based on similar applications of the Riemann Hypothesis over finite fields, we prove general estimates with power-saving for such sums. We then derive various concrete applications.

Abstract: By developing a connection between partial theta functions and Appell-Lerch sums, we find and prove a formula which expresses Hecke-type double sums in terms of Appell-Lerch sums and theta functions. Not only does our formula prove classical Hecke-type double sum identities such as those found in work Kac and Peterson on affine Lie Algebras and Hecke modular forms, but once we have the Hecke-type forms for Ramanujan's mock theta functions our formula gives straightforward proofs of many of the classical mock theta function identities. In particular, we obtain a new proof of the mock theta conjectures. Our formula also applies to positive-level string functions associated with admissable representations of the affine Lie Algebra $A_1^{(1)}$ as introduced by Kac and Wakimoto.

Abstract: We show that the rank generating function U(t; q) for strongly unimodal sequences lies at the interface of quantum modular forms and mock modular forms. We use U(-1; q) to obtain a quantum modular form which is “dual” to the quantum form Zagier constructed from Kontsevich’s “strange” function F(q). As a result, we obtain a new representation for a certain generating function for L-values. The series U(i; q) = U(-i; q) is a mock modular form, and we use this fact to obtain new congruences for certain enumerative functions.

Abstract: Recently, we introduced a new procedure for computing a class of one-loop BPS-saturated amplitudes in String Theory, which expresses them as a sum of one-loop contributions of all perturbative BPS states in a manifestly T-duality invariant fashion. In this paper, we extend this procedure to all BPS-saturated amplitudes of the form \int_F \Gamma_{d+k,d} {\Phi}, with {\Phi} being a weak (almost) holomorphic modular form of weight -k/2. We use the fact that any such {\Phi} can be expressed as a linear combination of certain absolutely convergent Poincare series, against which the fundamental domain F can be unfolded. The resulting BPS-state sum neatly exhibits the singularities of the amplitude at points of gauge symmetry enhancement, in a chamber-independent fashion. We illustrate our method with concrete examples of interest in heterotic string compactifications.

Abstract: Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.

Abstract: Given a weight two modular form f with associated p-adic Galois representation V_f, for certain quadratic imaginary fields K one can construct canonical classes in the Galois cohomology of V_f by taking the Kummer images of Heegner points on the modular abelian variety attached to f. We show that these classes can be interpolated as f varies in a Hida family, and construct an Euler system of big Heegner points for Hida's universal ordinary deformation of V_f. We show that the specialization of this big Euler system to any form in the Hida family is nontrivial, extending results of Cornut and Vatsal from modular forms of weight two and trivial character to all ordinary modular forms, and propose a horizontal nonvanishing conjecture for these cohomology classes. The horizontal nonvanishing conjecture implies, via the theory of Euler systems, a conjecture of Greenberg on the generic ranks of Selmer groups in Hida families.

Abstract: In this article we refine the method of Bertolini and Darmon and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

Abstract: Let bl(n) denote the number of l-regular partitions of n. Recently Andrews, Hirschhorn, and Sellers proved that b4(n) satisfies two infinite families of congruences modulo 3, and Webb established an analogous result for b13(n). In this paper we prove similar families of congruences for bl(n) for other values of l.

Abstract: We construct regular integral canonical models for Shimura varieties attached to Spin groups at (possibly ramified) odd primes. We exhibit these models as schemes of 'relative PEL type' over integral canonical models of larger Spin Shimura varieties with good reduction. Work of Vasiu-Zink then shows that the classical Kuga-Satake construction extends over the integral model and that the integral models we construct are canonical in a very precise sense. We also construct good compactifications for our integral models. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla's program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.

Abstract: We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for $\Gamma_1(N)$: an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.

Abstract: We show that the n-fold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model, as well as various other n-fold integrals of the "Ising class", or n-fold integrals from enumerative combinatorics, like lattice Green functions, are actually diagonals of rational functions. As a consequence, the power series expansions of these solutions of linear differential equations "Derived From Geometry" are globally bounded, which means that, after just one rescaling of the expansion variable, they can be cast into series expansions with integer coefficients. Besides, in a more enumerative combinatorics context, we show that generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions. We give a large set of results illustrating the fact that the unique analytical solution of Calabi-Yau ODEs, and more generally of MUM ODEs, is, almost always, diagonal of rational functions. We revisit Christol's conjecture that globally bounded series of G-operators are necessarily diagonals of rational functions. We provide a large set of examples of globally bounded series, or series with integer coefficients, associated with modular forms, or Hadamard product of modular forms, or associated with Calabi-Yau ODEs, underlying the concept of modularity. We finally address the question of the relations between the notion of integrality (series with integer coefficients, or, more generally, globally bounded series) and the modularity (in particular integrality of the Taylor coefficients of mirror map), introducing new representations of Yukawa couplings.

Abstract: Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to the corresponding Hilbert modular form at the places above $p$ where $E$ has split multiplicative reduction. Under some mild restrictions on $p$ and the conductor of $E$ we deduce the exceptional zero conjecture in the strong form (i.e.\ where the automorphic $p$-adic periods are replaced by the $\cL$-invariants of $E$ defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the $p$-adic $L$-function of $E$ in terms of local data.

Abstract: We review the relationship between the largest Mathieu group and various modular objects, including recent progress on the relation to mock modular forms. We also review the connections between these mathematical structures and string theory on K3 surfaces.

Abstract: Quasi-Historical Introduction. Remarks on Unique Factorization. Elementary Methods. Kummer's Arguments. Why Do We Believe Wiles? More Quasi-History. Diophantus and Fermat. A Child's Introduction to Elliptic Functions. Local and Global. Curves. Modular Forms. The Modularity Conjecture. The Functional Equation. Zeta Functions and L -Series. The ABC-Conjecture. Heights. Class Number of Imaginary Quadratic Number Fields. Wiles' Proof. Appendices. Index.

Abstract: In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomolo- gies. In this way dierential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of ellip- tic curves. For the full modular group such a dierential equation is calculated and it turns out to be the Ramanujan dierential equation between Eisenstein se- ries. We also explain the notion of period map constructed from elliptic integrals. This turns out to be the bridge between the algebraic notion of a quasi-modular form and the one as a holomorphic function on the upper half plane. In this way we also get another interpretation, essentially due to Halphen, of the Ramanu- jan dierential equation in terms of hypergeometric functions. The interpretation of quasi-modular forms as sections of jet bundles and some related enumerative problems are also presented.

Abstract: The goal of this paper is to study certain p-adic differential operators on automorphic forms on U(n,n). These operators are a generalization to the higher-dimensional, vector-valued situation of the p-adic differential operators constructed for Hilbert modular forms by N. Katz. They are a generalization to the p-adic case of the C^{\infty}-differential operators first studied by H. Maass and later studied extensively by M. Harris and G. Shimura. The operators should be useful in the construction of certain p-adic L-functions attached to p-adic families of automorphic forms on the unitary groups U(n) x U(n).

Abstract: Let p be a prime. We construct and study integral and torsion invariants, such as integral and torsion Weil-Deligne representations, associated to potentially semi-stable representations and tor- sion potentially semi-stable representations respectively. As applications, we prove the compatibility between local Langlands correspondence and Fontaine's construction for Galois representations attached to Hilbert modular forms, and Neron-Ogg-Shafarevich criterion of finite level for potentially semi-stable representations.

Abstract: -Preface (K. Alladi and F. Garvan).- 1. MacMahon's dream (G. E. Andrews and P. Paule).- 2. Ramanujan's elementary method in partition congruences (B. Berndt, C. Gugg, and S. Kim).- 3. Coefficients of harmonic Maass forms (K. Bringmann and K. Ono).- 4. On the growth of restricted partition functions (E. R. Canfield and H. Wilf).- 5. On applications of roots of unity to product identities (Z. Cao).- 6. Lecture hall sequences, q-series, and asymmetric partition identities (S. Corteel, C. Savage and A. Sills).- 7. Generalizations of Hutchinson's curve and the Thomae formula (H. Farkas).- 8. On the parity of the Rogers-Ramanujan coefficients (B. Gordon).- 9. A survey of the classical mock theta functions (B. Gordon and R. McIntosh).- 10. An application of the Cauchy-Sylvester theorem on compound determinants to a BC_n Jackson integral (M. Ito and S. Okada).- 11. Multiple generalizations of q-series identities found in Ramanujan's Lost Notebook (Y. Kajihara).- 12. Non-terminating q-Whipple transformations for basic hypergeometric series in U(n) (S. C. Milne and J. W. Newcomb).

Abstract: We give the exact expressions of the partial susceptibilities ?(3)d and ?(4)d for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi?Yau ODEs, and more specifically, 3F2([1/3, 2/3, 3/2],?[1, 1];?z) and 4F3([1/2, 1/2, 1/2, 1/2],?[1, 1, 1];?z) hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for ?(3)d and ?(4)d. We also give new results for ?(5)d. We see, in particular, the emergence of a remarkable order-6 operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the n-fold integrals of the Ising model are not only ?derived from geometry? (globally nilpotent), but actually correspond to ?special geometry? (homomorphic to their formal adjoint). This raises the question of seeing if these ?special geometry? Ising operators are ?special? ones, reducing, in fact systematically, to (selected, k-balanced, ...) q + 1Fq hypergeometric functions, or correspond to the more general solutions of Calabi?Yau equations.

Abstract: Let V be a strongly regular vertex operator algebra. For a state h ∈ V1 satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions TrM qL(0)-c/24ζh(0) (M a V-module) is a vector-valued weak Jacobi form of weight 0 and a certain index 〈h, h〉/2. We discuss refinements and applications of this result when V is holomorphic, in particular we prove that if g = eh(0) is a finite-order automorphism then TrV qL(0)-c/24g is a modular function of weight 0 on a congruence subgroup of SL2(ℤ).

Abstract: Chapter 1. Introduction.- Chapter 2. Review of Chains and Cochains.- Chapter 3. Review of Intersection Homology and Cohomology.- Chapter 4. Review of Arithmetic Quotients.- Chapter 5. Generalities on Hilbert Modular Forms and Varieties.- Chapter 6. Automorphic vector bundles and local systems.- Chapter 7. The automorphic description of intersection cohomology.- Chapter 8. Hilbert Modular Forms with Coefficients in a Hecke Module.- Chapter 9. Explicit construction of cycles.- Chapter 10. The full version of Theorem 1.3.- Chapter 11. Eisenstein Series with Coefficients in Intersection Homology.- Appendix A. Proof of Proposition 2.4.- Appendix B. Recollections on Orbifolds.- Appendix C. Basic adelic facts.- Appendix D. Fourier expansions of Hilbert modular forms.- Appendix E. Review of Prime Degree Base Change for GL2.- Bibliography.

Abstract: The fake monster Lie algebra is determined by the Borcherds function Phi_{12} which is the reflective modular form of the minimal possible weight with respect to O(II_{2,26}). We prove that the first non-zero Fourier-Jacobi coefficient of Phi_{12} in any of 23 Niemeier cusps is equal to the Weyl-Kac denominator function of the affine Lie algebra of the root system of the corresponding Niemeier lattice. This is an automorphic answer (in the case of the fake monster Lie algebra) on the old question of I. Frenkel and A. Feingold (1983) about possible relations between hyperbolic Kac-Moody algebras, Siegel modular forms and affine Lie algebras.

Abstract: Recently, Kac and Wakimoto established specialized character formulas for irreducible highest weight s (m, 1)∧ modules and established a main exponential term in their asymptotic expansions. By different methods, we improve upon the Kac-Wakimoto asymptotics for these characters, obtaining an asymptotic expansion with an arbitrarily large number of terms beyond the main term. More specifically, it is well known that in the case of holomorphic modular forms, asymptotic information may be obtained using modular transformation properties. However, here this is not the case due to the analytic nature of the Kac-Wakimoto series as discovered recently by the first author and Ono. We first “complete” these series by adding to them certain integrals, obtaining functions that exhibit suitable modular transformation laws, at the expense of the completed objects being nonholomorphic. We then exploit this mock-modular behavior of the Kac-Wakimoto series to obtain our asymptotic expansion. In particular, we show that beyond the main term, the asymptotic behavior is dictated by the nonholomorphic part of the completed Kac-Wakimoto characters, which is a priori invisible. Euler numbers (equivalently, zeta-values) appear as coefficients.

Abstract: We describe a computational approach to the verification of Maeda's conjecture for the Hecke operator T2 on the space of cusp forms of level one. We provide experimental evidence for all weights less than 12000, as well as some applications of these results. The algorithm was implemented using the mathematical software Sage, and the code and resulting data were made freely available.

Abstract: Using deformed or twisted Eisenstein Series, we construct a Jacobi-Serre derivative on even-weight Jacobi forms that generalizes the classical Serre derivative on modular forms. As an application, we obtain Ramanujan equations for the index $1$ Eisenstein series $E_{4,1}, E_{6,1}$ and a newly defined $E_{2,1}$. Finally, we relate the deformed Eisenstein Series directly to the classical first Jacobi theta function.