Genus 3 curves and explicit realisations of symplectic groups as Galois groups over Q

Abstract: Let n be a natural number and l a prime number. Given a genus n curve C defined over Q, the group of l-torsion points defined over an algebraic closure of Q of its Jacobian variety J_C is endowed with an action of the absolute Galois group G_Q , giving rise to a Galois representation ρ: G_Q → GSp(2n, l). When ρ is surjective, it provides us with a realisation of GSp(2n, l) as a Galois group over Q. To study Galois realisations (over Q) with particular ramification properties at l, it is of great interest to have conditions at auxiliary primes different from l that ensure surjectivity, while allowing great flexibility in the behaviour at the prime l. In this talk we focus on the case n = 3, and provide an explicit construction of curves C defined over Q such that ρ is surjective for a prefixed prime l. This is joint work with Cécile Armana, Valentijn Karemaker, Marusia Rebolledo, Lara Thomas and Núria Vila, and was initiated as a working group in the Conference Women in Numbers Europe (CIRM, 2013)