Moduli Spaces Of Abelian Varieties Research Articles

P-Adic Geometry

We discuss recent developments in p-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for “compact p-adic manifolds” over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the construction of “universal” p-adic cohomology theories. We finish with some speculations on how a theory that combines all primes p, including the archimedean prime, might look like.

Moduli of abelian surfaces, symmetric theta structures and theta characteristics

We study the birational geometry of some moduli spaces of abelian varieties with extra structure: in particular, with a symmetric theta structure and an odd theta characteristic. For a ($d\_1, d\_2$)-polarized abelian surface, we show how the parities of the $d\_i$ influence the relation between canonical level structures and symmetric theta structures. For certain values of $d\_1$ and $d\_2$, a theta characteristic is needed in order to define Theta-null maps. We use these Theta-null maps and preceding work of other authors on the representations of the Heisenberg group to study the birational geometry and the Kodaira dimension of these moduli spaces.

Topological methods in moduli theory

One of the main themes of this long article is the study of projective varieties which are K(H,1)’s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera–de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)’s, through Teichmuller theory. The main trhust of the paper is to show how in the case of K(H,1)’s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lonne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an algebraic surface S such that the Galois conjugate surface of S has fundamental group not isomorphic to the one of S.

Semi-Abelian Schemes and Heights of Cycles in Moduli Spaces of Abelian Varieties

Semi-Abelian Schemes and Heights of Cycles in Moduli Spaces of Abelian Varieties

The supersingular locus in Siegel modular varieties with Iwahori level structure

We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl–Oort stratification on the former, the Kottwitz–Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case g is even.

Geometry of the Siegel modular threefold with paramodular level structure

In this paper we extend some results of Norman and Oort and of de Jong, and give an explicit description of the geometry of the Siegel modular threefold with paramodular level structure. We also discuss advantages and restrictions of three standard methods for studying moduli spaces of abelian varieties.

On Weddle surfaces and their moduli

Abstract The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2 curve. First we construct a moduli space A 2(3)− parametrizing abelian surfaces with a symmetric theta structure and an odd theta characteristic. Such objects can in fact be seen as Weddle surfaces. We prove that A 2(3)− is rational. Then, given a genus 2 curve C, we give an interpretation of the Weddle surface as a moduli space of extensions classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf ω of C with ω −1. This in turn allows to see the Weddle surface as a hyperplane section of the secant variety Sec(C) of the curve C tricanonically embedded in ℙ4.

Foliations in moduli spaces of abelian varieties

We study moduli spaces of polarized abelian varieties in positive characteristic. Our final goal will be to understand Hecke orbits in such spaces. This paper provides one of the tools. For a given $p$-divisible group, all abelian varieties which give rise to this group have moduli points in a locally closed subset of the moduli space; we call an irreducible component of this subset a central leaf. Newton polygon strata are foliated by such leaves. Moreover, iterated $\alpha _p$-isogenies give a second leaf structure, which was already known under the name of Rapoport-Zink spaces. Any Newton polygon stratum is, up to a finite morphism, isomorphic to a product of an isogeny leaf and a finite cover of a central leaf. We conjecture that any Hecke-$\ell$-orbit is dense in the corresponding central leaf.

An isomorphism between moduli spaces of abelian varieties

AbstractIn 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 consequence we obtain an involution on the set of moduli spaces of polarized abelian varieties of dimension g. In particular, the moduli spaces $ {\cal A}_{(d_{1}, \ldots, d_{g})} $ and $ {\cal A}_{\left( d_1, {{d_{1}d_{g} \over {d_{g-1}}}, \ldots, {d_{1}d_{g} \over {d_{2}}}}, d_{g}\right)} $ are canonically isomorphic.

Kummer Varieties and the Moduli Spaces of Abelian Varieties

Kummer Varieties and the Moduli Spaces of Abelian Varieties

Subvarieties of moduli spaces

In this paper we try to decide along algebraic lines whether moduli spaces of abelian varieties or of algebraic curves contain complete subvarieties. In Theorem (1.1) we consider abelian varieties and curves of genus three in characteristic p :~ 0. In Section 2 we use monodromy in order to show certain families of abelian varieties up to a purely inseparable isogeny are isotrivial; probably the results (2.1) and (2.2) are special cases of more general facts about / -adic monodromy. In Section 4 we answer a question raised by Martin concerning a possible generalization of the fact that two supersingular elliptic curves over an algebraically closed field (of characteristic p) are isogenous. We abbreviate abelian variety(ies) by AV; we use X t for the dual, and 2 for the formal group of an AV X.

Explicit formulas for infinitely many Shimura curves in genus $4$

In this paper we construct infinitely many Shimura curves contained in the locus of Jacobians of genus four curves. All Jacobians in these families are ${\mathbb Z}/3$ covers of varying elliptic curves that appear in a geometric construction of Pirola, and include an example of a Shimura-Teichmuller curve that parameterizes Jacobians that are suitable ${\mathbb Z}/6$ covers of ${\mathbb P}^1$. We compute explicitly the period matrices of the Shimura curves we construct using the original construction of Shimura for moduli spaces of abelian varieties with automorphisms.