Siegel Modular Forms Of Genus Research Articles

CONGRUENCES OF SAITO–KUROKAWA LIFTS AND DENOMINATORS OF CENTRAL SPINORL-VALUES

AbstractFollowing Ryan and Tornaría, we prove that moduli of congruences of Hecke eigenvalues, between Saito–Kurokawa lifts and non-lifts (certain Siegel modular forms of genus 2), occur (squared) in denominators of central spinorL-values (divided by twists) for the non-lifts. This is conditional on Böcherer’s conjecture and its analogues and is viewed in the context of recent work of Furusawa, Morimoto and others. It requires a congruence of Fourier coefficients, which follows from a uniqueness assumption or can be proved in examples. We explain these factors in denominators via a close examination of the Bloch–Kato conjecture.

Minimal modularity lifting for nonregular symplectic representations

In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series at infinity.

Modular invariants for genus 3 hyperelliptic curves

In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary oc-tics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the value of these modular functions at CM points of the Siegel upper-half space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.

Modular forms for the $$A_{1}$$ A 1 -tower

In the 1960s 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 $${\text {Sp}}(2,\mathbb {Z})$$ . In 2010 Gritsenko found three towers of orthogonal type modular forms which are connected with certain series of root lattices. In this setting Siegel modular forms can be identified with the orthogonal group of signature (2, 3) for the lattice $$A_{1}$$ and Igusa’s form $$\chi _{5}$$ appears as the roof of this tower. We use this interpretation to construct a framework for this tower which uses three different types of constructions for modular forms. It turns out that our method produces simple coordinates.

Sturm’s operator for scalar weight in arbitrary genus

In contrast to the well-known cases of large weights, Sturm’s operator does not realize the holomorphic projection operator for lower weights. We prove its failure for arbitrary Siegel modular forms of genus [Formula: see text] and scalar weight [Formula: see text]. This generalizes a result for genus two in [K. Maurischat and R. Weissauer, Phantom holomorphic projections arising from Sturm’s formula, preprint (2016)].

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusa’s description of the relation between invariants of binary sextics and Siegel modular forms of degree 2 to a relation between covariants and vector-valued Siegel modular forms of degree 2. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree 2.

Byte weight enumerators of codes over 𝔽<SUB align="right">p and modular forms over a totally real field

In this paper, we obtain Jacobi forms over kp from byte weight enumerators of self-dual codes over Fp, where p is an odd prime, kp is the totally real field of the pth cyclotomic field and Fp is the finite field of order p. We also determine Siegel modular forms of genus g (g ≥ 1 is an integer) over kp by substituting certain theta series into byte weight enumerators in genus g of self-dual codes over Fp for all p ∈ P, where the set P consists of all those odd primes p for which the ring of algebraic integers of kp is a Euclidean domain. Further, we define some partial Epstein zeta functions and derive their functional equation using the Mellin transform of the theta series.

Level stripping for vector-valued Siegel modular forms of genus 2

In this paper, we present a method by which one can strip primes from the level of a vector-valued genus 2 Siegel modular form while preserving a congruence modulo this prime. An application of this result to four-dimensional Galois representations will also be presented.

K3 Surfaces, Modular Forms, and Non-Geometric Heterotic Compactifications

We construct non-geometric compactifications by using the F-theory dual of the heterotic string compactified on a two-torus, together with a close connection between Siegel modular forms of genus two and the equations of certain K3 surfaces. The modular group mixes together the K\"ahler, complex structure, and Wilson line moduli of the torus yielding weakly coupled heterotic string compactifications which have no large radius interpretation.

Siegel modular forms of genus 2 and level 2

We study vector-valued Siegel modular forms of genus 2 on the three level 2 groups Γ[2] ◁ Γ1[2] ◁ Γ0[2] ⊂ Sp(4, ℤ). We give generating functions for the dimension of spaces of vector-valued modular forms, construct various vector-valued modular forms by using theta functions and describe the structure of certain modules of vector-valued modular forms over rings of scalar-valued Siegel modular forms.

On the arithmetic of Shalika models and the critical values of L -functions for GL 2 n

Let $\Pi$ be a cohomological cuspidal automorphic representation of ${\rm GL}_{2n}(\Bbb{A})$ over a totally real number field $F$. Suppose that $\Pi$ has a Shalika model. We define a rational structure on the Shalika model of $\Pi_f$. Comparing it with a rational structure on a realization of $\Pi_f$ in cuspidal cohomology in top-degree, we define certain periods $\omega^{\epsilon}(\Pi_f)$. We describe the behavior of such top-degree periods upon twisting $\Pi$ by algebraic Hecke characters $\chi$ of $F$. Then we prove an algebraicity result for all the critical values of the standard $L$-functions $L(s,\Pi\otimes\chi)$; here we use the recent work of B. Sun on the non-vanishing of a certain quantity attached to $\Pi_\infty$. As applications, we obtain algebraicity results in the following cases: Firstly, for the symmetric cube $L$-functions attached to holomorphic Hilbert modular cusp forms; we also discuss the situation for higher symmetric powers. Secondly, for certain (self-dual of symplectic type) Rankin-Selberg $L$-functions for ${\rm GL}_3\times{\rm GL}_2$; assuming Langlands Functoriality, this generalizes to certain Rankin-Selberg $L$-functions of ${\rm GL}_n\times{\rm GL}_{n-1}$. Thirdly, for the degree four $L$-functions attached to Siegel modular forms of genus $2$ and full level. Moreover, we compare our top-degree periods with periods defined by other authors. We also show that our main theorem is compatible with conjectures of Deligne and Gross.

On the algebra of Siegel modular forms of genus $2$

On the algebra of Siegel modular forms of genus $2$

AUTOMORPHIC BLACK HOLE ENTROPY

Over the past few years the understanding of the microscopic theory of black hole entropy has made important conceptual progress by recognizing that the degeneracies are encoded in partition functions which are determined by higher rank automorphic representations, in particular in the context of Siegel modular forms of genus two. In this review, some of the elements of this framework are highlighted. One of the surprising aspects is that the Siegel forms that have appeared in the entropic context are geometric in origin, arising from weight two cusp forms, hence from elliptic curves.

Root multiplicities of hyperbolic Kac–Moody algebras and Fourier coefficients of modular forms

In this paper we consider the hyperbolic Kac–Moody algebra \(\mathcal {F}\) associated with the generalized Cartan matrix Open image in new window. Its connection to Siegel modular forms of genus 2 was first studied by A. Feingold and I. Frenkel. The denominator function of \(\mathcal{F}\) is not an automorphic form. However, Gritsenko and Nikulin extended \(\mathcal{F}\) to a generalized Kac–Moody algebra whose denominator function is a Siegel modular form. Using the Borcherds denominator identity, the denominator function can be written as an infinite product. The exponents that appear in the product are given by Fourier coefficients of a weak Jacobi form. P. Niemann also constructed a generalized Kac–Moody algebra which contains \(\mathcal {F}\) and whose denominator function is related to a product of Dedekind η-functions. In particular, root multiplicities of the generalized Kac–Moody algebra are determined by Fourier coefficients of a modular form. As the main results of this paper, we compute asymptotic formulas for these Fourier coefficients using the method of Hardy–Ramanujan–Rademacher, and obtain an asymptotic bound for root multiplicities of the algebra \(\mathcal{F}\). Our method can be applied to other hyperbolic Kac–Moody algebras and to other modular forms as demonstrated in the later part of the paper.

Symplectic Kloosterman sums

We give optimal bounds for Kloosterman sums that arise in the estimation of Fourier coefficients of Siegel modular forms of genus 2.

Sturm type theorem for Siegel modular forms of genus 2 modulo p

Suppose that $f$ is an elliptic modular form with integral coefficients. Sturm obtained bounds for a nonnegative integer $n$ such that every Fourier coefficient of $f$ vanishes modulo a prime $p$ if the first $n$ Fourier coefficients of $f$ are zero modul

Families of Siegel modular forms, L-functions and modularity lifting conjectures

For a prime p and a positive integer g, by making use of certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus g and associated p-adic L-functions. Describing L-functions attached to Siegel modular forms and their analytic properties from the point of view of motivic L-functions studied by Deligne and Yoshida, we discuss critical values of the L-functions and p-adic interpolation problems. In particular, we formulate a general conjecture on the existence of the modularity lifting from GSpr × GSp2m to GSpr+2m for some positive integers r and m.

Siegel modular forms of genus 2 with the simplest divisor

We prove that there exist exactly eight Siegel modular forms with respect to the congruence subgroups of Hecke type of the paramodular groups of genus 2 vanishing precisely along the diagonal of the Siegel upper half-plane. This is a solution of a question formulated during the conference ‘Black holes, Black Rings and Modular Forms’ (ENS, Paris, August 2007). These modular forms generalize the classical Igusa form and the forms constructed by Gritsenko and Nikulin in 1998.

Two modularity lifting conjectures for families of Siegel modular forms

For a prime p and a positive integer n, using certain lifting procedures, we study some constructions of p-adic families of Siegel modular forms of genus n. Describing L-functions attached to Siegel modular forms and their analytic properties, we formulate two conjectures on the existence of the modularity liftings from GSpr × GSp2m to GSpr+2m for some positive integers r and m.

Relèvement de formes modulaires de Siegel

(Lifting Siegel modular forms). In this paper, we give explicit conditions under which cuspidal Siegel modular forms of genus 2 or 3 with coefficients in a finite field lift to cuspidal modular forms with coefficients in a ring of characteristic 0. This result extends a classical theorem proved by Katz for genus 1 modular forms. We use ampleness results due to Shepherd-Barron, Hulek and Sankaran, and vanishing theorems due to Deligne, Illusie, Raynaud, Esnault and Viehweg.