##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Romyar Sharifi

#### McMaster University

## Cup products and $p$-adic $L$-functions of cusp forms

##### Abstract:

In Iwasawa theory, one attempts to relate arithmetic objects arising from Galois cohomology groups to analytic objects, specifically $p$-adic $L$-functions, usually at the level of a cyclotomic $Z_p$-extension of a number field. For some time, I have been interested in operations in the Galois cohomology of number fields with restricted ramification, particularly cup products. I will explain various applications of these operations, for instance to Iwasawa theory over certain nonabelian $p$-adic Lie extensions of number fields. I will then discuss the broad outline of a conjecture relating an inverse limit of such cup products up the cyclotomic tower to the two-variable $p$-adic $L$-function for Hida families of cusp forms congruent to Eisenstein series modulo $p$.

Host: W. Stein

### November 3, 2005

### 1:00 PM

### AP&M 7321

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