Comparing residually reducible semisimple Galois representationsMPIM

by Ignasi Sánchez Rodríguez

Thursday Aug 14, 2025, 5:00 PM → 5:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Number theory lunch seminar

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 of Frobenius at those primes is enough to
ensure that the semisimplification of both \rho_1 and \rho_2 are isomorphic
? Is the list T finite? Can it be computed? 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. 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}.