Related papers: Note on mod p property of Hermitian modular forms

In \cite{Shimura}, Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm's bounds give…

We calculate the action of some Hecke operators on spaces of modular forms spanned by the Siegel theta-series of certain genera of strongly modular lattices closely related to the Leech lattice. Their eigenforms provide explicit examples of…

In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\frak{p}$ in any ring of algebraic integers, mod $\frak{p}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application…

In the 1960's Igusa determined the graded ring of Siegel modular forms of genus two. He used theta series to construct $\chi_{5}$, the cusp form of lowest weight for the group $\operatorname{Sp}(2;Z)$. In 2010 Gritsenko found three towers…

Let $p$ be a prime number and $N$ an integer prime to $p$. We show that the operator $U_p$ on the space of cuspidal modular forms of level $pN$ and weight two is semi-simple. It follows from this that the Hecke algebra acting on the space…

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…

For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the ${\rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${\rm Leech}$. The harmonic Siegel theta series built from…

Let $\Delta= \sum_{m=0}^\infty q^{(2m+1)^2} \in \mathbb{F}_2[[q]]$ be the reduction mod 2 of the $\Delta$ series. A modular form $f$ modulo $2$ of level 1 is a polynomial in $\Delta$. If $p$ is an odd prime, then the Hecke operator $T_p$…

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…

We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p< 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over the rationals of…

Let p be a prime number. The Hasse invariant is a modular form modulo p that is often used to produce congruences between modular forms of different weights. We show how to produce such congruences between forms of weights 2 and p+1, in…

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's…

The theta cycle of a modular form modulo a prime $p\geq 5$ is well understood. By contrast, the theta cycle modulo a power of $p$ is still mysterious and experimentally erratic. Here we completely determine the theta cycle of a weight $k <…

Given a characteristic, we define a character of the Siegel modular group of level 2, the computations of their values are also obtained. By using our theorems, some key theorems of Igusa [1] can be recovered.

We discuss an arithmetic approach to some congruence properties of Siegel theta series of even positive definite unimodular quadratic forms.

The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this…

In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are…

We introduce an alternate set of generators for the Hecka algebra, and give an explicit formula for the action of these operators on Fourier coefficients. With this, we compute the eigenvalues of Hecke operators acting on average Siegel…

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp_4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit…

We propose to associate to a modular form (an infinite number of) complex valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We elaborate on the correspondence and study its consequence in terms of the Mellin…