Seminario Teoría de Números
Comparing residually reducible semisimple Galois representations
Ponente: Ignasi Sánchez (Universidad de Barcelona)Fecha: jueves 30 de octubre de 2025 - 14:30Lugar: 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 $\{2,7,19,73\}$
