EVENT DETAILS AND ABSTRACT

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.