相关论文: Rationality and arithmetic of the moduli of abelia…

In a previous paper we showed that for every polarization on an abelian variety there is a dual polarization on the dual abelian variety. In this note we extend this notion of duality to families of polarized abelian varieties. As a main…

We prove that the moduli space A_{11}^{lev} of (1,11) polarized abelian surfaces with level structure of canonical type is birational to Klein's cubic hypersurface: a^2b+b^2c+c^2d+d^2e+e^2a=0 in P^4. Therefore, A_{11}^{lev} is unirational…

We consider the moduli space $A_{pol}(n)$ of (non-principally) polarised abelian varieties of genus $g\geq3$ with coprime polarisation and full level-$n$ structure. Based upon the analysis of the Tits building in math/0405321, we give an…

Let $\overline{\rho}: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the…

Let A be the moduli space of principally polarized abelian varieties of dimension 4 over an algebraically closed field k of characteristic different from 2,3. It is proved that the universal principally polarized abelian variety over A, as…

We introduce the concept of the modularity of an abelian variety defined over the rational number field extending the modularity of an elliptic curve. We discuss the modularity of an abelian variety over the rational number field. We…

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…

We describe birational models and decide the rationality/unirationality of moduli spaces AA_{d} (and AA^{lev}_{d}) of (1,d)-polarized abelian surfaces (with canonical level structure, respectively) for small values of d. The projective…

We determine the maximal dimension of compact subvarieties of $\mathcal{A}_g$, the moduli space of complex principally polarized abelian varieties of dimension $g$, and the maximal dimension of a compact subvariety through a very general…

The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows…

Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g \ge 3.$ In this paper we prove the connectedness of the singular sublocus of $\mathcal{A}_g$ consisting of those abelian varieties which…

We prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least 4 is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic…

For fixed $k<g$ and a family of polarized abelian varieties of dimension $g$ over $\mathbb{R}$, we give a criterion for the density in the parameter space of those abelian varieties over $\mathbb{R}$ containing a $k$-dimensional abelian…

We prove that the moduli space X(1,7) of (1,7)-polarized abelian surfaces with canonical level structure is birational to the Fano 3-fold V22 of polar hexagons of the Klein quartic X(7). In particular X(1,7) is rational and the birational…

We construct explicit dominant, rational morphisms from projective bundles over rational varieties to relevant moduli spaces, showing their unirationality. These constructions work for $U_{r,d,g}$; for all ranks, degrees and genus $2\leq g…

Refereed version to appear in Michigan Mathematical Journal. A mistake in the last section of the previous version has been corrected. The new title exactly describes the main result obtained. Building on the geometry of cubic surfaces and…

We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties.

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…

In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our…

We prove the Ax-Lindemann theorem for the coarse moduli space $\mathcal{A}_{g}$ of principally polarized abelian varieties of dimension $g\ge 1$, and affirm the Andr\'e-Oort conjecture unconditionally for $\mathcal{A}_{g}$ for $g\le 6$.