The $p$-adic properties of L-functions

Nike Vatsal

(U. of Toronto).

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\centerline{\large The $p$-adic properties of L-functions}

\centerline{V. Vatsal, University of Toronto}


The origins of Iwasawa theory can be traced back to the famous class number formula of Dirichlet, which states that the number of inequivalent binary quadratic forms of a given fundamental discriminant $D$ is determined by the value $L(1,\chi_D)$ of the associated quadratic L-function. In modern terms, the class number formula is a very special case of a {\em Main Conjecture in Iwasawa theory}, which relates analytic information (as given by L-values) to algebraic information given by cohomology (the so-called Selmer group, which reduces in the classical case to quadratic forms and genus theory). The goal of this talk is to give an introduction to the theory of $p$-adic L-functions, and to describe some recent results in the subject.

A more detailed description is as follows. We will introduce the classical theory of $p$-adic L-functions attached to cyclotomic fields. We will define the L-functions and Selmer groups, and show how classical questions on the zeroes and poles of complex L-functions have precise $p$-adic analogues. We will state the main conjecture, which provides a precise link between the algebra and the analysis. We illustrate the theory by describing the proof of the $p$-adic Artin conjecture. In conclusion, we introduce some new results in the Iwasawa theory of modular forms and elliptic curves. \end{document}