On Jacobians with group action and coverings

Abstract

Let $S$ be a compact Riemann surface and let $H$ be a finite group. It is known that if $H$ acts on $S$ then there is a $H$-equivariant isogeny decomposition of the Jacobian variety $JS$ of $S,$ called the group algebra decomposition of $JS$ with respect to $H.$ If $S_1\to S_2$ is a regular covering map, then it is also known that the group algebra decomposition of $J{S}_1$ induces an isogeny decomposition of $J{S}_2.$ In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions.

18 pages

References in corpus (5)

• On the one-dimensional family of Riemann surfaces of genus $q$ with $4q$ automorphisms
• Quasiplatonic curves with symmetry group ${\mathbb{Z}}_2^2⋊{\mathbb{Z}}_m$ are definable over $\mathbb{Q}$
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Cited by in corpus (8)

• On the one-dimensional family of Riemann surfaces of genus $q$ with $4q$ automorphisms
• A note on large automorphism groups of compact Riemann surfaces
• On families of Riemann surfaces with automorphisms
• On Riemann surfaces of genus $g$ with $4g-4$ automorphisms
• On $pq$-fold regular covers of the projective line
• On large prime actions on Riemann surfaces
• Nilpotent groups of automorphisms of families of Riemann surfaces
• Abelian varieties and Riemann surfaces with generalized quaternion group action