Related papers: Computations of vector-valued Siegel modular forms

We survey the theory of vector-valued modular forms and their connections with modular differential equations and Fuchsian equations over the three-punctured sphere. We present a number of numerical examples showing how the theory in…

Vector-valued Siegel modular forms are the natural generalization of the classical elliptic modular forms as seen by studying the cohomology of the universal abelian variety. We show that for g>=4, a new class of vector-valued modular…

Hecke eigenvalues of classical modular forms often encode a wealth of arithmetic data. The Satake $p$-parameters of a Siegel modular form play a role analogous to the one played by Hecke eigenvalues in the characterization of classical…

This is a survey based on the construction of Siegel modular forms of degree 2 and 3 using invariant theory in joint work with Fabien Cl\'ery and Carel Faber.

We provide a power-saving bound for certain smoothed shifted convolution sums for Fourier coefficients of Siegel cusp forms. This result is the first nontrivial estimate for a shifted convolution sum with two cusp forms on a group of higher…

We prove that formal Fourier Jacobi expansions of degree 2 are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension 2, which were defined by Kudla. A second application is…

We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we…

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…

In this paper, we construct Hecke eigenforms for two families of quotient spaces of meromorphic cusp forms on $\mathrm{SL}_2(\mathbb{Z})$. We show that each quotient space in the first (resp. second family) is isomorphic as a Hecke module…

We formulate an explicit refinement of B\"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of L-functions. To achieve this, we compute the…

In this article, the authors give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. They also provide an explicit upper…

We establish an isomorphism between certain complex-valued and vector-valued modular form spaces of half-integral weight, generalizing the well-known isomorphism between modular forms for $\Gamma_0(4)$ with Kohnen's plus condition and…

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms $F,G$ for orthogonal groups of signature $(2,n+2)$. In the case when $F$ is a Hecke eigenform and $G$ is a Maass lift of a Poincar\'e series, we…

We study the L-functions associated to Siegel modular forms (equivalently, automorphic representations of ${\rm GSp}(4,\mathbb{A}_{\mathbb{Q}})$) both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform…

We say that a normalized modular form is of CM type modulo $\ell$ by an imaginary quadratic field $K$ if its Fourier coefficients $a_p$ are congruent to $0$ modulo a prime $\mathcal L\mid \ell$ for every prime $p$ that is inert in $K$. In…

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…

We consider some questions related to the signs of Hecke eigenvalues or Fourier coefficients of classical modular forms. One problem is to determine to what extent those signs, for suitable sets of primes, determine uniquely the modular…

H. Aoki showed that any symmetric formal Fourier-Jacobi series for the symplectic group Sp_2(Z) is the Fourier-Jacobi expansion of a holomorphic Siegel modular form. We prove an analogous result for vector valued symmetric formal…

We deduce a weighted equidistribution theorem of the Satake parameters of Siegel cusp forms on Sp_2({\mathbb Z})with growing even weights.

Let $\rho$ denote an irreducible two-dimensional representation of $\Gamma_{0}(2)$. The collection of vector-valued modular forms for $\rho$, which we denote by $M(\rho)$, form a graded and free module of rank two over the ring of modular…