Related papers: Siegel theta series for indefinite quadratic forms

A subvariety of a quasi-projective complex variety $X$ is called ``universally irreducible'' if its preimage inside the universal cover of $X$ is irreducible. In this paper we investigate sufficient conditions for universal irreducibility.…

We examine an elliptic curve constructed in an earlier paper from a certain representation of $\SL(2,\Z)$ on the space of convergent Dirichlet series. The curve is observed to be a modular curve for $\Gamma^1(15)$ and a certain orbit of…

The Hodge bundle $\omega$ over a modular curve is a square-root of the canonical bundle twisted by the cuspidal divisor, or a theta characteristic, due to the Kodaira--Spencer isomorphism. We prove that, in most cases, a section of a theta…

We give a new method for constructing vector-valued modular forms from singular scalar-valued ones. As an application we prove the identity between two remarkable spaces of vector-valued modular forms which seem to be unrelated at a first…

We prove that the representations numbers of a ternary definite integral quadratic form defined over F_q[t], where F_q is a finite field of odd characteristic, determine its integral equivalence class when q is large enough with respect to…

This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties;…

Let $k \geq 2$ and $N$ be positive integers and let $\chi$ be a Dirichlet character modulo $N$. Let $f(z)$ be a modular form in $M_k(\Gamma_0(N),\chi)$. Then we have a unique decomposition $f(z)=E_f(z)+S_f(z)$, where $E_f(z) \in…

Suppose that $\ell \geq 5$ is prime. For a positive integer $N$ with $4 \mid N$, previous works studied properties of half-integral weight modular forms on $\Gamma_0(N)$ which are supported on finitely many square classes modulo $\ell$, in…

We show how to use divisors on the projectivized Hodge bundle to construct special vector-valued modular forms and then apply invariant theory to construct all vector-valued Siegel modular forms of level two and degree two. Thus we…

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. Also, the tensor product of an indecomposable bilinear module $(U, \gamma)$ with…

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…

We study expansions of Drinfeld modular forms of rank \(r \geq 2\) along the boundary of moduli varieties. Product formulas for the discriminant forms \(\Delta_{\mathfrak{n}}\) are developed, which are analogous with Jacobi's formula for…

Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel…

In this paper, we obtain an explicit formula for the theta correspondence of unipotent principal-series representations between an even orthogonal and a symplectic group or between general linear groups over a finite field. The formula is…

We use the gradients of theta functions at odd two-torsion points --- thought of as vector-valued modular forms --- to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to…

By some SL(2, Z) modular forms introduced in [4] and [10], we construct some modular forms over SL2(Z) and some modular forms over {\Gamma}^0(2) and {\Gamma}_0(2) in odd dimensions. In parallel, we obtain some new cancellation formulas for…

The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several…

We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense…

We establish a Siegel-Weil formula for classical groups over a function field with odd characteristic, which asserts in many cases that the Siegel Eisenstein series is equal to an integral of a theta function. This is a function-field…

In this series of papers, we introduce higher level versions of the theta group $\Gamma_{\theta}.$ In this paper, we treat the theta group of level $5$, $\Gamma_{\theta,5},$ and construct modular forms on $\Gamma_{\theta,5}$. Moreover we…