Related papers: Paramodular Cusp Forms

We use Borcherds products to give a new construction of the weight $3$ paramodular nonlift eigenform $f_N$ for levels $N = 61, 73, 79$. We classify the congruences of $f_N$ to Gritsenko lifts. We provide techniques that compute eigenvalues…

We prove a formula of Petersson's type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual Fourier coefficient. The method in this…

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…

A precise and testable modularity conjecture for rational abelian surfaces A with trivial endomorphisms, End_Q A = Z, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on…

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…

This is a report on recent work, with Wen-Ching Winnie Li and Ling Long. In that work explicit formulas are given, involving hypergeometric character sums, for the traces of Hecke operators $T_p$ acting spaces of cusp forms $S_k(\Gamma)$ of…

Given two Siegel eigenforms of different weights, we determine explicit sets of Hecke eigenvalues for the two forms that must be distinct. In degree two, and under some additional conditions, we determine explicit sets of Fourier…

Given a prime $p$ and cusp forms $f_1$ and $f_2$ on some $\Gamma_1(N)$ that are eigenforms outside $Np$ and have coefficients in the ring of integers of some number field $K$, we consider the problem of deciding whether $f_1$ and $f_2$ have…

We construct a maximal discrete extension of the paramodular group with a full level-2 structure. The corresponding Siegel variety parametrizes (birationally) the space of Kummer surfaces associated to (1,p)-polarized abelian surfaces with…

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of…

We define an algebraic set in $23$~dimensional projective space whose $\mathbb Q$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on…

For a point $x_0$ in a Shimura variety attached to a Shimura datum of Hodge type $(G,X)$, we have an associated abelian scheme $A_0$. Fixing a non-empty finite set $\mathcal{S}$ of primes, we consider the simultaneous supersingular…

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…

Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f),…

In this paper we construct an infinite family of paramodular forms of weight $2$ which are simultaneously Borcherds products and additive Jacobi lifts. This proves an important part of the theta-block conjecture of Gritsenko--Poor--Yuen…

There is a lifting from a non-CM elliptic curve $E/\mathbb{Q}$ to a paramodular form $f$ of degree $2$ and weight $3$ given by the symmetric cube map. We find the level of $f$ in an explicit way in terms of the coefficients of the…

We generalize some of the results of Andreatta, Iovita, and Pilloni and the author to Hodge type Shimura varieties having non-empty ordinary locus. For any $p$-adic weight $\kappa$, we give a geometric definition of the space of…

By using new techniques with the degenerate Whittaker functions found by Ikeda-Yamana, we construct higher level cusp form on $E_{7,3}$, called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no…

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 investigate the (classical) weights of mod $p$ Siegel modular forms of degree 2 toward studying Serre's conjecture for $GSp_4$. We first construct various theta operators on the space of such forms a la Katz and define the…