The aim of the present paper is to prove some general results on abelian varieties with action by a finite group, and to deduce from those general facts a method to decompose the Jacobian of a curve in terms of a finite group action on that curve.
Let be a finite group acting on an abelian variety . Let denote the irreducible -representations of and , where for . Then the authors’ first main theorem states that there are abelian subvarieties of and an isogeny . This refined isotypical decomposition is then applied to Jacobians of smooth projective curves equipped with the action of a finite automorphism group. If is a finite group acting on such a curve of genus , then the quotient map induces, via the general main theorem, an isogeny decomposition
of into certain abelian varieties, the components of which can be linked to certain suitable Prym varieties arising from the -action. Finally, the authors work out several concrete examples of particular group actions on curves, including the symmetric group of degree 4, the alternating group of degree 5, the dihedral groups of order and and the quaternion group, which illustrate the general results by explicit computations.