Curves of genus 2 with group of automorphisms isomorphic to D₈ or D₁2

2002/03/17 by Gabriel Cardona, Cardona, Gabriel, Jordi Quer +1 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.math/0203165

26 pages

arxiv created 2002/03/17 · openalex publication_date 2002/03/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we classify curves of genus 2 with group of automorphisms isomorphic to D₈ or D₁2 over an arbitrary field k (of characteristic different from 2 in the D₈ case and from 2 and 3 in the D₁₂ case) up to k-isomorphism. As an application of the classification of curves of genus 2 obtained, we get precise arithmetic information on their elliptic quotients and on their jacobians. Over the field k=Q, we show that the elliptic quotients of the curves with automorphisms D₈ and D₁₂ are precisely the Q-curves of degrees 2 and 3, respectively, and we determine which curves have jacobians of GL₂-type.

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