Abel–Prym Maps for Isotypical Components of Jacobians

Abstract

Let C be a smooth non-rational projective curve over the complex field $$\mathbb {C}$$ . If A is an abelian subvariety of the Jacobian J(C), we consider the Abel-Prym map $$\varphi _A : C \rightarrow A$$ defined as the composition of the Abel map of C with the norm map of A. The goal of this work is to investigate the degree of the map $$\varphi _A$$ in the case where A is one of the components of an isotypical decomposition of J(C). In this case we obtain a lower bound for $$\deg (\varphi _A)$$ and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of Abel-Prym maps in a few examples. In particular, these examples show that both bounds are sharp.