P-ADIC REGULATORS AND P-ADIC FAMILIES OF MODULAR FORMS

The theme of this Thesis is Iwasawa theory of Hida p-adic analytic families of modular forms. Our main goal is to describe special values of Hida\u2019s p-adic L-functions in the context of a p-adic Birch and Swinnerton-Dyer conjecture for the weight variable

Diagonal classes and the Bloch–Kato conjecture

The aim of this note is twofold. Firstly, we prove an explicit reciprocity law for certain diagonal classes in the \ub4etale cohomology of the triple product of a modular curve, stated in [8] and used there as a crucial ingredient in the proof of the main results. Secondly, we apply the aforementioned reciprocity law to address the rank-zero case of the equivariant Bloch\u2013Kato conjecture for the self-dual motive of an elliptic newform of weight k > 2. In the special case k = 2, our result gives a self-contained and simpler proof of the main result of [15]

Congruences between modular forms and the birch and swinnerton-dyer conjecture

We prove the p-part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank one for most ordinary primes