Related papers: Modular Forms and Certain ${}_2F_1(1)$ Hypergeomet…

Under mild hypotheses on the residual representation, we prove the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras using a novel combination of the methods of…

We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a…

A description is given of all primitive differential series mod p of order 1 which are eigenvectors of all the Hecke operators and which are differential Fourier expansions of differential modular forms of arbitrary order and given weight;…

Two integral structures on the Q-vector space of modular forms of weight two on X_0(N) are compared at primes p exactly dividing N. When p=2 and N is divisible by a prime that is 3 mod 4, this comparison leads to an algorithm for computing…

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight $2-k\in 2\Z$ for the full modular group $\SL_2(\Z)$. The space has a distinguished…

Generalizing a result of \cite{Z1991, CPZ} about elliptic modular forms, we give a closed formula for the sum of all Hilbert Hecke eigenforms over a totally real number field with strict class number $1$, multiplied by their period…

Given a Hecke eigenform $f$ of weight $2$ and square-free level $N$, by the work of Kohnen, there is a unique weight $3/2$ modular form of level $4N$ mapping to $f$ under the Shimura correspondence. Furthermore, by the work of Waldspurger…

Hypergeometric motives are family of motives associated to hypergeometric local systems. Their special features, in particular their rigidity, makes them more tractable than general motives. In the present article we prove most of the…

The family of circular Jacobi $\beta$ ensembles has a singularity of a type associated with Fisher and Hartwig in the theory of Toeplitz determinants. Our interest is in the Fourier transform of the corresponding bulk scaled spectral…

We show that signs of Fourier coefficients, on certain sub-families, determine the half-integral weight cuspidal eigenform uniquely, up to a positive constant. We also study sign change results for the product of the Fourier coefficients of…

We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of…

We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$…

Let $\rho: SL(2,\mathbb{Z})\to GL(2,\mathbb{C})$ be an irreducible representation of the modular group such that $\rho(T)$ has finite order $N$. We study holomorphic vector-valued modular forms $F(\tau)$ of integral weight associated to…

We prove that Hecke eigenvalues for any Hilbert and Siegel modular forms are algebraic integers. Our method does not rely on cohomologicality nor Galois representations. We apply the integrality of Hecke eigenvalues for Hilbert modular…

We compute the Fourier coefficients of analogues of Kohnen and Zagier's modular forms $f_{k,D}$ of weight $2$ and negative discriminant. These functions can also be written as twisted traces of certain weight $2$ Poincar\'e series with…

We state and verify up to weight 172 a conjecture on the existence of a certain generating set for spaces of classical Siegel modular forms. This conjecture is particularly useful for calculations involving Fourier expansions. Using this…

This article describes results of joint work with Michael Rapoport and Tonghai Yang. First, we construct an modular form \phi(\tau) of weight 3/2 valued in the arithmetic Chow group of the arithmetic surface M attached toa Shimura curve…

We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent $q$-hypergeometric function of Andrews as…

We discover a non-trivial relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers. This relation yields a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$, where $N > 1$ is…

Hecke expected that an explicit set of theta series obtained from maximal orders of the definite quaternion algebra over Q which is ramified at a prime N will be a basis of the space of holomorphic modular forms of weight 2 and level N.…