Selmer Groups And Ranks Of Hecke Rings

Abstract

In this work, we investigate congruences between modular cuspforms. Specifically, we start with a given cuspform and count the number of cuspforms congruent to it as we vary the weight or level. This counting problem is equivalent to understanding the ranks of certain completed Hecke rings. Using the deep modularity results of Wiles, et al., we investigate these Hecke rings by studying the deformation theory of the residual representation corresponding to our given cuspform. This leads us to consider certain Selmer groups attached to this residual representation. In this setting, we can apply standard theorems from local and global Galois cohomology to achieve our results.