Crafting Euler Systems: Beyond the Motivic Mold

Abstract

This dissertation studies Euler Systems and their arithmetic applications. Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of $L$-functions. However, despite their importance in foundational conjectures in number theory like the Bloch--Kato conjecture, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle in constructing Euler Systems lies in producing candidate Galois cohomology classes. This thesis presents a method to overcome this obstacle without relying on rare motivic classes. In joint work with C. Skinner, I use Eisenstein classes on Siegel threefolds to construct a cyclotomic Euler System for the adjoint of an elliptic modular form. I also construct integral absolute \'etale classes on modular curves associated to theta series at inert primes. These theta classes should give new insight into the Iwasawa main conjecture for modular forms over quadratic imaginary fields at inert primes.