##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Joseph Ferrara

#### U.C. Santa Cruz

## A $p$-adic Stark conjecture for Hecke characters of quadratic fields

##### Abstract:

In the 1970's Stark made precise conjectures about the leading term of the Taylor series expansion at $s=0$ of Artin $L$-functions, refining Dirichlet's class number formula. Around the same time Barsky, Cassou-Nogu\`{e}s, and Deligne and Ribet for totally real fields, along with Katz for CM fields defined $p$-adic $L$-functions of ray class characters. Since then Stark-type conjectures for these $p$-adic $L$-functions have been formulated, and progress has been made in some cases. The goal of this talk is to discuss a new definition of a $p$-adic $L$-function and Stark conjecture for a mixed signature character of a real quadratic field. After stating the definition and conjecture, theoretical and numerical evidence will be discussed.

Host: Cristian Popescu

### January 12, 2018

### 1:00 PM

### AP&M 6402

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