Rationality of twists of the Siegel modular variety of genus 2 and level 3

2020/09/01 by Calegari, Frank, Chidambaram, Shiva
#11F80 #11G18 (Primary) 14K10 #14E08 (Secondary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2009.00194

Abstract

Let ρ: G_Q → \rm GSp₄(F₃) be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the 3-torsion of a principally polarized abelian surface A/Q. We prove that the moduli space A₂(ρ) of principally polarized abelian surfaces B/Q admitting a symplectic isomorphism B[3] ≃ ρ of Galois representations is never rational over Q when ρ is surjective, even though it is both rational over C and unirational over Q via a map of degree 6.

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