This article is an introduction to the L~-cohomology of arithmetic symmetric spaces. We shall describe properties of automorphic representations which lead to a remarkable interplay among Hodge structures, Lefschetz structures, and Hecke operators. In order to make the discussion slightly more concrete, we shall focus on the special case of Siegel moduli space. Resume. Cet article est une intro...

We give an explicit conjectural formula for the motivic Euler characteristic of an arbitrary symplectic local system on the moduli space A3 of principally polarized abelian threefolds. The main term of the formula is a conjectural motive of Siegel modular forms of a certain type; the remaining terms admit a surprisingly simple description in terms of the motivic Euler characteristics for lower ...

We describe the moduli spaces of morphisms between polarized complex abelian varieties. The discrete invariants, derived from a Poincaré decomposition of morphisms, are the types of polarizations and of lattice homomorphisms occurring in the decomposition. For a given type of morphisms the moduli variety is irreducible, and is obtained from a product of Siegel spaces modulo the action of a disc...

We derive the explicit form of the (g − 2)(g − 3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of canonical curves of genus g ≥ 4. It turns out that Petri’s relations remarkably match in determinantal conditions. We explicitly express the volume form on the moduli space M̂g of canonical curves induced by the Siegel metri...

In his letter (Serre, 1996), J.-P. Serre proves that the systems of Hecke eigenvalues given by modular forms (mod p) are the same as the ones given by locally constant functions A×B/B × → Fp, where B is the endomorphism algebra of a supersingular elliptic curve. We generalize this result to Siegel modular forms, proving that the systems of Hecke eigenvalues given by Siegel modular forms (mod p)...

0 Introduction In this paper, we investigate the moduli space S g,L of supersingular abelian varieties X with endomorphism structure θ : O L → End(X) and O L-linear principal polarization, where O L is the ring of integers in an algebraic number field L with involution *. General existence of the coarse moduli scheme of polarized abelian varieties with endomorphism structure is shown by using t...

In this paper, under a weakened version of Hardy—Littlewood Conjecture on the number representations in Goldbach problem, we shall prove bounds for Siegel zeros real primitive Dirichlet characters composite moduli.

The result of Siegel that the Tamagawa number of SLr over a function field is 1 has an expression purely in terms of vector bundles on a curve, which is known as the Siegel formula. We prove an analogous formula for vector bundles with quasi-parabolic structures. This formula can be used to calculate the betti numbers of the moduli of parabolic vector bundles using the Weil conjucture.

In this paper we analyze the large genus asymptotics for intersection numbers between $$\psi $$ -classes, also called correlators, on moduli space of stable curves. Our proofs proceed through a combinatorial analysis recursive relations (Virasoro constraints) that uniquely determine these together with comparison coefficients in jump probabilities certain asymmetric simple random walk. As an ap...