Related papers: Universal abelian variety and Siegel modular forms

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group $\mathrm{SU}_{2,2}(\mathcal{O}_K)$ where $K$ is the imaginary-quadratic number field…

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 two vanishing precisely along the diagonal of the Siegel upper half-plane. This is a…

We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular…

Let A be a supersingular abelian variety over a finite field k. We give an approximate description of the structure of the group A(k) of rational points of A over k in terms of the characteristic polynomial f of the Frobenius endomorphism…

We study a natural enlargement of the BGG Category O for a semisimple Lie algebra: the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We give a geometric…

We suggest to look at formal sentences describing complex algebraic varieties together with their universal covers as topological invariants. We prove that for abelian varieties and Shimura varieties this is indeed a complete invariant,…

We study the cone of moving divisors on the moduli space ${\mathcal A}_g$ of principally polarized abelian varieties. Partly motivated by the generalized Rankin-Cohen bracket, we construct a non-linear holomorphic differential operator that…

The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no…

We study Hodge bundles on Siegel varieties and their various extensions to smooth toroidal compactifications. Precisely, we construct a canonical Hodge bundle on an arbitrary Siegel variety so that the holomorphic tangent bundle can be…

Let $\mathcal{O}_K$ be a discrete valuation ring with fraction field $K$ of characteristic $0$ and algebraically closed residue field $k$ of characteristic $p > 0$. Let $A/K$ be an abelian variety of dimension $g$ with a $K$-rational point…

We survey old and new results about the cohomology of the moduli space $A_g$ of principally polarized abelian varieties of genus $g$ and its compactifications. The main emphasis lies on the computation of the cohomology for small genus and…

We consider the finite set of isogeny classes of $g$-dimensional abelian varieties defined over the finite field $\mathbb{F}_q$ with endomorphism algebra being a field. We prove that the class within this set whose varieties have maximal…

We compute Seidel's mirror map for abelian varieties by constructing the homogeneous coordinate rings from the Fukaya category of the symplectic mirrors. The computations are feasible as only linear holomorphic disks contribute to the…

This paper gives a canonical construction, in terms of additive cohomological functors, of the universal formal deformation of a compact complex manifold without vector fields (more generally of a faithful $g$-module, where $g$ is a sheaf…

We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type. More precisely, we first build on Kisin's work to construct integral canonical models over rings of integers of number…

We will give the graded ring of Siegel modular forms of degree two with respect to a non-split symplectic group explicitly.

We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level $\ell$ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way…

We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also…

Let A_2 be the moduli stack of principally polarized abelian surfaces and V a smooth l-adic sheaf on A_2 associated to an irreducible rational finite dimensional representation of Sp(4). We give an explicit expression for the cohomology of…

The purpose of this paper is to describe explicitly the modules of (Siegel-)Jacobi forms of degree two of index one of any scalar valued weight with respect to some congruence subgroups of small levels $N\leq 4$. Such a structure for the…