Iwasawa theory for tensor products of Hilbert modular forms

Abstract

The main conjecture of Iwasawa theory bridges two seemingly disjoint areas of mathematics: arithmetic and analysis. In particular, it provides a deep connection between the p-adic L-function which interpolates critical values of the complex L-series, and the Selmer group which is an important object used to control the growth of arithmetic data. In this thesis, we explore some questions that arise organically from the Iwasawa Main Conjecture, applied to Hilbert modular forms and their tensor products. Greenberg and Vatsal developed an approach to study the main conjecture for a large class of elliptic curves simultaneously. They showed that if a given pair of elliptic curves share the same residual Galois representation, then the main conjecture holds for one if and only if it does for the other. The first part of this thesis investigates whether the ideas of Greenberg and Vatsal work for elliptic curves twisted by a CM-Hecke character. The second part of this thesis then extends the method to treat non-ordinary classical modular forms (without any twist). The former permits one to study rational elliptic curves base-changed to an arbitrary number field, whilst the latter requires techniques crafted by Pollack and Kobayashi in the early 2000s. Finally, the third part of this thesis concerns Euler systems and their applications to the arithmetic of motives arising from modular forms—these objects are indispensable tools which can be used to prove half of the Iwasawa main conjecture. However they often give rise to additional “junk” error terms, as well as causing the p-adic L-function to be stripped away of certain bad Euler factors. For modular forms and their tensor products we devise a method to dispose of the error terms and to replenish the missing factors, allowing one to genuinely obtain half of the main conjecture free from any discrepancies.