P-Adic L-functions and Zeta Element

Abstract

One of the most fruitful developments in the last couple of decades, has been the use of K -theory in the study of special values of L -functions of modular forms. This link is manifested in several ways. The first is through the work of Beilinson, who constructed zeta-elements living in K 2 of a modular curve. He managed to prove that their image under the regulator map yielded the residue of the L -series at s = 0. In an orthogonal direction, Coates and Wiles found a non-archimedean approach to relating the L -function of a CM elliptic curve at s = 1, with a certain Euler system of elliptic units. They were then able to prove the first concrete theorems in the direction of the Birch and Swinnerton-Dyer conjecture. It was Kato who realised the approach of Beilinson and that of Coates-Wiles could be combined, with spectacular results. He noticed that the zeta-elements considered by Beilinson satisfied norm-compatibility relations, reminiscent of those satisfied by elliptic units. Moreover, he devised a purely p -adic method whereby these elements could be used to study L -values. Underlying his discoveries was the ground-breaking work of Perrin-Riou in the early nineties, which extended the local Iwasawa theory used by Coates and Wiles to a very general framework (required for modular forms and non-CM elliptic curves). We shall review some of the highlights of their work, and in Chapters III, V and VI we will generalise it to modular symbols and Λ-adic families of modular forms.