Article

# Abelian varieties with l-adic Galois representation of Mumford's type

ABSTRACT This paper is devoted to the study of 4-dimensional abelian varieties over number fields with the property that the Lie algebra of the image of some associated -adic Galois representation is Q -isomorphic to c ⊕ (sl 2) 3 . Such varieties will be referred to as 'abelian varieties with -adic Galois representation of Mumford's type'. We show that such abelian varieties have potentially good reduction at all prime ideals and we determine the possible Newton polygons and the possible isogeny types of these reductions.

0 0
·
0 Bookmarks
·
20 Views
• Source
##### Article:The p-rank strata of the moduli space of hyperelliptic curves
[show abstract] [hide abstract]
ABSTRACT: We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p > 2. Using this, we prove that the Z/\ell-monodromy of every irreducible component of the stratum H_g^f of hyperelliptic curves of genus g and p-rank f is the symplectic group Sp_{2g}(Z/\ell) if g > 2, f > 0 and \ell is an odd prime distinct from p. These results yield applications about the generic behavior of hyperelliptic curves of given genus and p-rank. The first application is that a generic hyperelliptic curve of genus g > 2 and p-rank 0 is not supersingular. Other applications are about absolutely simple Jacobians and the generic behavior of class groups and zeta functions of hyperelliptic curves of given genus and $p$-rank over finite fields. Comment: Slight reorganization, improved results for p-rank zero locus
02/2009;
• Source
##### Article:The Torelli locus and special subvarieties
[show abstract] [hide abstract]
ABSTRACT: We study the Torelli locus T_g in the moduli space A_g of abelian varieties. We consider special subvarieties (Shimura subvarieties) contained in the Torelli locus. We review the construction of some non-trivial examples, and we discuss some conjectures, techniques and recent progress.
12/2011;

Available from

### Keywords

'abelian varieties

4-dimensional abelian varieties

abelian varieties

number fields

possible isogeny types

possible Newton polygons

prime ideals

varieties