Seminario Teoría de Números

Comparing residually reducible semisimple Galois representations

Ponente:  Ignasi Sánchez (Universitat de Barcelona)
Fecha:  jueves 30 de octubre de 2025 - 14:30
Lugar:  Aula 420, Módulo 17, Departamento de Matemáticas, UAM

Resumen:

Let $n geq 2$ and $p$ be a prime. Let $K$ be a number field and consider two Galois representations $rho_1, rho_2 : Gal(Kbar / K) to GL_n(Q_p)$ having residual image a $p$-group. Is there a list $T$ of primes of $K$ such that comparing traces at those primes is enough to ensure that the semisimplification of both $rho_1$ and $rho_2$ are conjugate one from another? Is the list $T$ finite? How does one compute such $T$? How small (in norm) can the primes in $T$ be?

Loïc Grenié gave answers to these questions in his work in 2007.  In this talk we present a fully automatic implementation of Grenié's work that returns the minimal list~$T$. Moreover, we use the method to prove the following result: Let~$K=Q(sqrt{-3})$ and let $rho_1, rho_2 : G_K to GL_2(Z_3)$ be continuous representations unramified outside~$3$ having the same determinant trivial modulo~$3$. Then~$rho_1$ and~$rho_2$ have isomorphic semisimplifications if and only if~$rho_1(Frob_t)$ and~$rho_2(Frob_t) have the same trace for every~$t$ in~$K$ above the primes ${2,7,19,73}$