Title:
Efficient irreducibility of residual Galois representations of
absolutely simple abelian surfaces with RM over the rationals
Abstract:
Let A be an absolutely simple abelian surface with RM by O over the
rationals. The irreducibility of the residual Galois representation
\rho_p: Gal_\Q -> GL_2(O \otimes \Z_p) of the p-torsion module
A[p](\overline{\Q}) has implications on the p-part of the strong BSD
conjecture for A/\Q. It is known (even effectively by a recent result of
Davide Lombardo) that \rho_p is irreducible for all but finitely many p,
but so far there is no efficient algorithm to determine these p.
We show how to modify an algorithm of Dieulefait to determine these p
exactly for squarefree conductor of \rho and in the general case a good
cofinite subset, which we can compute exactly for the example of the
Jacobian of the Atkin-Lehner quotient X_0(125)/w_5.