相关论文: Polarized superspecial simple abelian surfaces wit…

For any number $m \equiv 0,1 \, (4)$ we correct the generating function of Hurwitz class number sums $\sum_r H(4n - mr^2)$ to a modular form (or quasimodular form if $m$ is a square) of weight two for the Weil representation attached to a…

Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p.$ Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$ dimensional abelian varieties…

We introduce and study a new way to catagorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed, or fully minimal. The type of $A$ depends on the normalized Weil numbers of $A$ and its…

We compute divisors class groups of singular surfaces. Most notably we produce an exact sequence that relates the Cartier divisors and almost Cartier divisors of a surface to the those of its normalization. This generalizes Hartshorne's…

We consider the Siegel modular variety of genus 2 and a p-integral model of it for a good prime p>2, which parametrizes principally polarized abelian varieties of dimension two with a level structure. We consider cycles on this model which…

We present several new heuristic algorithms to compute class polynomials and modular polynomials modulo a prime $p$ by revisiting the idea of working with supersingular elliptic curves. The best known algorithms to this date are based on…

Fix primes $p$ and $\ell$ with $\ell\neq p$. If $(A,\lambda)$ is a $g$-dimensional principally polarized abelian variety, an $(\ell)^g$-isogeny of $(A,\lambda)$ has kernel a maximal isotropic subgroup of the $\ell$-torsion of $A$; the image…

We determine which codimension two Hodge classes on $J\times J$, where $J$ is a general abelian surface, deform to Hodge classes on a family of abelian fourfolds of Weil type. If a Hodge class deforms, there is in general a unique such…

This paper gives a conjectural characterization of those elliptic curves over the field of complex numbers which "should" be covered by standard modular curves. The elliptic curves in question all have algebraic j-invariant, so they can be…

We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral…

We study the rationality properties of the moduli space $\mathcal{A}_g$ of principally polarised abelian $g$-folds over $\mathbb{Q}$ and apply the results to arithmetic questions. In particular we show that any principally polarised abelian…

Compactifications of moduli spaces of (1,p)-polarized abelian surfaces with level structures of canonical type have been described in great detail by Hulek, Kahn and Weintraub. The aim of this paper is to determine some invariants of smooth…

We show that a character sum attached to a family of 3-isogenies defined on the fibers of a certain elliptic surface over $\mathbb{F}_p$ relates to the class number of the quadratic imaginary number field $\Q(\sqrt{-p})$. In this sense,…

We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a 9-dimensional vector space over an algebraically closed field of characteristic different from 2 and 3. To a general such…

In this short note we prove two theorems, the first one is a sharpening of a result of Lange and Sernesi: the discriminant curve W of a general Abelian surface $A$ endowed with an irreducible polarization $D$ of type $(1,3)$ is an…

Suppose $C$ is an isogeny class of abelian varieties over a finite field $k$. In this paper we give a partial answer to the question of which finite group schemes over $k$ occur as kernels of polarizations of varieties in $C$. We show that…

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their…

Let $P$ be a finite $p$-group and $p$ be an odd prime. Let $\mathcal{A}_p(P)_{\geq2}$ be a poset consisting of elementary abelian subgroups of rank at least 2. If the derived subgroup $P'\cong C_p\times C_p$, then the spheres occurring in…

Consider all moduli points corresponding with polarized abelian varieties in characteristic p such that the associated quasi-polarized p-divisible group is geometrically isomorphic with a given one. This defines a subset C of the moduli…

The moduli space of principally polarized abelian varieties with real structure and with level $N=4m$ structure (with $m \ge 1$) is shown to coincide with the set of real points of a quasi-projective algebraic variety defined over $\mathbb…