Boundedness results for 2-adic Galois images associated to hyperelliptic\n Jacobians

2017/03/31 by Jeffrey Yelton, Yelton, Jeffrey · 1 citation
Computer Science · Mathematics · #14G20 #14H30 #14K15 #20G25 #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1703.10917

openalex publication_date 2017/03/31 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let K be a number field, and let C be a hyperelliptic curve over K with\nJacobian J. Suppose that C is defined by an equation of the form y² =\nf(x)(x - \λ) for some irreducible monic polynomial f \∈\n\O_K[x] of discriminant \Δ and some element \λ \∈\n\O_K. Our first main result says that if there is a prime\n mathfrakp of K dividing (f(\λ)) but not (2\Δ), then the\nimage of the natural 2-adic Galois representation is open in\n\GSp(T₂(J)) and contains a certain congruence subgroup of\n\Sp(T₂(J)) depending on the maximal power of mathfrakp\ndividing (f(\λ)). We also present and prove a variant of this result\nthat applies when C is defined by an equation of the form y² = f(x)(x -\n\λ)(x - \λ') for distinct elements \λ, \λ' \∈ K. We\nthen show that the hypothesis in the former statement holds for almost all\n\λ \∈ \O_K and prove a quantitative form of a uniform\nboundedness result of Cadoret and Tamagawa.\n

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