Gassmann Equivalence and Decompositions of Jacobians

This thesis deals with the concept of Gassmann equivalence and its application in obtaining isogenous and isomorphic products of Jacobians of algebraic curves.

We study Gassmann equivalent G-sets with a particular emphasis on rationally, locally integral and integrally Gassmann equivalent G-sets. We develop MAGMA functions that verify the only known example of transitive integral Gassmann equivalent G-sets due to Leonard L. Scott and could potentially be used to obtain new intransitive examples.

Our main results generalize theorems of D. Prasad and C. S. Rajan, D. Prasad and D. Arapura et al. In particular, we show that if C is an algebraic curve, G <= Aut(C) a finite group and X,Y rationally Gassmann equivalent G-sets then the Jacobians J[(C x X)/G] and J[(C x Y)/G] are isogenous. Moreover, if instead the G-sets X,Y are integrally Gassmann equivalent the above isogeny becomes an isomorphism.