On the geometry of the action of finite linked groups: Isogenous Jacobian varieties via intermediate covering.

Abstract

Let G be a nite group acting on a compact Riemann surface X. This action induces the so called group algebra decomposition of the corresponding Jacobian variety JX. Moreover, consider a subgroup H G of G and the intermediate quotient X=H arising from this action restricted to H. The group algebra decomposition of JX determines a decomposition of the Jacobian variety J(X=H) of X=H. In this work, we prove a condition under which two intermediate quotients, X=H and X=K for H;K G, correspond to isogenous Jacobian varieties. The condition is that they induce the same permutation character, a concept that has been widely studied in the context of Representation Theory, where it is said that H and K are linked subgroups in G. For every (odd) prime p 3, we study a family of groups Gp = (Z=p2Z Z=pZ) o (Z=pZ Z=pZ) having two linked subgroups which are not conjugate. We describe their elements, irreducible complex (and rational) representations, di erent signatures for their actions on Riemann surfaces, and the corresponding impact on the group algebra decomposition of the associated Jacobian varieties.

Sea G un grupo nito actuando en una super cie de Riemann compacta X. Esta acci on induce la llamada descomposici on seg un el algebra de grupo de la variedad Jacobiana JX correspondiente a X. M as a un, considere H G subgrupo de G y la super cie cuociente (intermedia) X=H determinada por la acci on restringida a H. La descomposici on de JX determina una descomposici on de la Jacobiana de X=H, J(X=H). En este trabajo demostramos una condici on bajo la cual las variedades Jacobianas de dos cubrientes intermedios, X=H y X=K para H;K G, son is ogenas. Esta condici on es que H y K inducen la misma representaci on permutacional. Ello ha sido ampliamente estudiado en el contexto de Teor a de Representaciones, donde se dice que H y K son subgroups ligados en G. Para todo primo (impar) p 3, estudiamos una familia de grupos Gp = (Z=p2Z Z=pZ)o (Z=pZ Z=pZ) que tienen dos subgrupos ligados no conjugados. Describimos sus elementos, caracteres irreducibles complejos (y racionales), diferentes rmas y acciones, y las consecuencias en la descomposici on de las variedades Jacobianas asociadas.