Seminar der Forschergruppe 'Symmetrie, Geometrie und Arithmetik'

Let K be a number field and ρ a Galois representation of K on a two-dimensional vector space V over an ell-adic field L. If ρ is residually irreducible, then there is a unique Galois invariant lattice λ in V, up to homothety. In this talk we consider the general problem of describing the structure of λ locally at finite primes p of K which are unramified in ρ. This amounts to finding the integral matrix describing the action of Frob_p on λ, for which the characteristic polynomial of ρ (Frob_p) will not suffice, in general. We focus on the ell-adic Tate module of an elliptic curve E over K, see how the problem is related to the study of prime splitting in the torsion fields K(E[ell ⁿ])/K, and give a solution of the problem for this class of representations. The main theorem, which really is a statement about elliptic curves over finite fields, emphasizes a role that Hilbert Class Polynomials and the j-invariant of E play in this context, and follows from classical results of Deuring.

Freitag, den 18. Oktober 2013 um 13:30 Uhr, in INF 288, HS2 Freitag, den 18. Oktober 2013 at 13:30, in INF 288, HS2