Explicit computation of Galois representations occurring in families of curves

TL;DR

This paper develops a method to compute Galois representations associated with families of algebraic curves over Q, providing explicit examples and analyzing their behavior at bad reduction points.

Contribution

It extends division polynomial computations from individual Jacobians over Q to families over Q(t), enabling explicit construction of Galois representations in a family context.

Findings

Explicit families of Galois representations over P^1_Q are obtained.

The degeneration of these representations at bad reduction places is studied.

Division polynomials become too complex for some applications.

Abstract

We extend our method to compute division polynomials of Jacobians of curves over Q to curves over Q(t), in view of computing mod ell Galois representations occurring in the \'etale cohomology of surfaces over Q. Although the division polynomials which we obtain are unfortunately too complicated to achieve this last goal, we still obtain explicit families of Galois representations over P^1_Q, and we study their degeneration at places of bad reduction of the corresponding curve.