## EVENT DETAILS AND ABSTRACT

**Number Theory**
**Title:** The $p$-adic $L$-functions of critical Eisenstein series

**Speaker:** Samit Dasgupta

**Speaker Info:** UCSC

**Brief Description:**

**Special Note**: **There might be a preseminar talk in Lunt 102 at 11 am. Stay tuned!**

**Abstract:**

Let $f$ be a newform of weight $k+ 2$ on $\Gamma_1(N)$, and let $p
mid N$ be a prime.
For each root $\alpha$ of the Hecke polynomial of $f$ at $p$, there is a corresponding
$p$-stabilization $f_\alpha$ on $\Gamma_1(N) \cap \Gamma_0(p)$ with $U_p$-eigenvalue
equal to $\alpha$. The construction of $p$-adic $L$-functions associated to such forms $f_\alpha$
has been much studied. The non-critical case (when $\text{ord}_p(\alpha) < k+1$) was handled
in the 1970s via interpolation of the classical $L$-function in work of Mazur, Swinnerton-Dyer, Manin,
Visik, and Amice-V\'elu. Recently, certain critical cases were handled by Pollack and Stevens, and
the remaining cases were finished off by Bella\"iche. Many years prior to Bella\"iche's proof of their
existence, Stevens had conjectured a factorization formula for the $p$-adic $L$-functions of critical
Eisenstein series based on computational evidence. In this talk we describe a proof of Steven's factorization
formula. A key element of the proof is the theory of distribution-valued partial modular symbols.
This is joint work with Jo\"el Bella\"iche.

**Date:** Monday, April 04, 2011

**Time:** 3:00PM

**Where:** Lunt 107

**Contact Person:** Ellen Eischen

**Contact email:** eeischen@math

**Contact Phone:** 847-467-1891

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