Related papers: On the kernel of the theta operator mod p

Let $p$ be a prime number and let $n$ be a positive integer prime to $p$. By an Ihara result we mean existence of an injection with torsion-free cokernel from a full lattice in the space of $p$-old modular forms, into a full lattice in the…

In a letter to Tate (published in Israel J. Math. in 1996), J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions on an adelic double coset space…

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;…

We define Hecke operators on vector valued modular forms transforming with the Weil representation associated to a discriminant form. We describe the properties of the corresponding algebra of Hecke operators and study the action on modular…

For positive integers $g$ and $N$, let $\mathcal{F}_N$ be the field of meromorphic Siegel modular functions of genus $g$ and level $N$ whose Fourier coefficients belong to the $N$th cyclotomic field. We present explicit generators of…

It is proved that the theta series of an even lattice whose level is a power of a prime $\ell$ is congruent modulo $\ell$ to an elliptic modular form of level~1. The proof uses arithmetic and algebraic properties of lattices rather than…

We continue our study of Yoshida's lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms…

We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar…

We study a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor…

We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose "companions" in the lower half-plane can be also realized both as double Eichler integrals…

Assuming the Riemann hypothesis for $L$-functions attached to primitive Dirichlet characters, modular cusp forms, and their tensor products and symmetric squares, we write down explicit finite sets of Hecke operators that span the Hecke…

We introduce a method for producing vector-valued automorphic forms on unitary groups from scalar-valued ones. As an application, we construct an explicit example. Our strategy employs certain differential operators. It is inspired by work…

In this work we construct an eigencurve for p-adic modular forms attached to an indefinite quaternion algebra over Q. Our theory includes the definition, both as rules on test objects and sections of line bundle, of p-adic modular forms,…

We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a…

We study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms…

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the…

In the spirit of the geometric approach to two-dimensional conformal field theory, we explicitly associate to every holomorphic vertex operator algebra a section of a power of Hodge line bundle on the moduli space of curves of arbitrary…

We present a new proof of the transformation law of $\vartheta_1$ under the action of the generator of the full modular group $\Gamma$ using Siegel's method.

In this paper we will describe all vector-valued Siegel modular forms of degree 2 and weight ${\rm Sym}^6({\rm St}) \otimes \det^{k}({\rm St})$ with $k$ odd. These vector-valued forms constitute a module over the ring of classical Siegel…

We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$-theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we…