Related papers: Counting polarizations on abelian varieties with g…

Given a polarized abelian scheme with action by a ring, and a projective finitely presented module over that ring, Serre's tensor construction produces a new abelian scheme. We show that to equip these abelian schemes with polarizations…

In this note it is shown that associating to a principally polarized abelian variety its de Rham cohomology defines a faithfully flat morphism of the moduli space of principally polarized abelian varieties in positive characteristics to the…

We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties.

Given a reductive algebraic group $G$ and a finite dimensional algebraic $G$-module $V$, we study how close is the algebra of $G$-invariant polynomials on $V^{\oplus n}$ to the subalgebra generated by polarizations of $G$-invariant…

The moduli space of abelian surfaces with polarisation of type (1,6) and a bilevel structure has positive Kodaira dimension. By contrast, Mukai has shown that the moduli space of bilevel-t abelian sufaces is rational for t=2,3,4,5.

For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety $A/\F_{p^n}$ of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of $A/\F_{p^n}$ of $l$-power degree is canonically a dense…

Let $\mathcal{A}_g$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. The arithmetic height, or arithmetic volume, of $\overline{\mathcal{A}}_g$, is defined to be the arithmetic degree of the metrized…

Let $A/K$ be an abelian variety with real multiplication defined over a $p$-adic field $K$ with $p>2$. We show that $A/K$ must have either potentially good or potentially totally toric reduction. In the former case we give formulas of the…

We consider a family of abelian surfaces over $\mathbb{Q}$ arising as Prym varieties of double covers of genus-$1$ curves by genus-$3$ curves. These abelian surfaces carry a polarization of type $(1,2)$ and we show that the average size of…

Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This…

We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties. We also give a generalization to higher genera of the famous formula $12…

Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two…

We consider a smooth Poisson affine variety with the trivial canonical bundle over complex numbers. For such a variety the deformation quantization algebra A_h enjoys the conditions of the Van den Bergh duality theorem and the corresponding…

We show that there are uncountably many algebraic extensions of $\mathbb{Q}$ containing at most finitely many moduli of CM simple principally polarized abelian varieties of any fixed dimension $g\geqslant1$, generalizing a result of…

Let $G$ be a connected semisimple algebraic group of adjoint type over the field $\mathbb{C}$ of complex numbers and $B$ be a Borel subgroup of $G.$ Let $F$ be an irreducible projective $B$-variety. Then consider the variety…

We give an effective procedure to explicitly find the decomposition of a polarized abelian variety into its simple factors if a period matrix is known. Since finding this datum is not easy, we also provide two methods to compute the period…

We give archimedean and non-archimedean constructions of Darmon points on modular abelian varieties attached to automorphic forms over arbitrary number fields and possibly non-trivial central character. An effort is made to present a…

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…

We show that certain abelian varieties over $\Q$ with bad reduction at one prime only are modular by using methods based on the tables of Odlyzko and class field theory.

Let $V$ be a subvariety of codimension $\leq g$ of the moduli space $\cA_g$ of principally polarized abelian varieties of dimension $g$ or of the moduli space $\tM_g$ of curves of compact type of genus $g$. We prove that the set $E_1(V)$ of…