 Paulhus (x)
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Title

Completely Decomposable Jacobian Varieties in New Genera

Description

We present a new technique to study Jacobian variety decompositions using subgroups of the automorphism group of the curve and the corresponding intermediate covers. In particular, this new method allows us to produce many new examples of genera for which there is a curve with completely decomposable Jacobian. These examples greatly extend the list given by Ekedahl and Serre of genera containing such curves, and provide more evidence for a positive answer to two questions they asked. Additionally, we produce new examples of families of curves, all of which have completely decomposable Jacobian varieties. These families relate to questions about special subvarieties in the moduli space of principally polarized abelian varieties.

PID

grinnell:13253


Title

Elliptic Factors in Jacobians of Hyperelliptic Curves with Certain Automorphism Groups

Description

We decompose the Jacobian variety of hyperelliptic curves up to genus 10, defined over an algebraically closed field of characteristic zero, with reduced automorphism group A 4 , S 4 , or A 5 . Among those curves is a genus 4 curve with Jacobian variety isogenous to E 2 1 × E 2 2 and a genus 5 curve with Jacobian variety isogenous to E 5 for E and E i elliptic curves. These types of results have some interesting consequences to questions of ranks of elliptic curves and ranks of their twists.

PID

grinnell:3326


Title

Jacobian Varieties of Hurwitz Curves with Automorphism Group PSL(2,q)

Description

The size of the automorphism group of a compact Riemann surface of genus g > 1 is bounded by 84(g1). Curves with automorphism group of size this bound are called Hurwitz curves. In many cases the automorphism group of these curves is the projective special linear group PSL(2,q). We present a decomposition of the Jacobian varieties for all curves of this type and prove that no such Jacobian variety is simple.

PID

grinnell:13254