Robert Harron : ${\mathcal L}$-invariants of symmetric powers of modular forms

Posted by Ramin Takloo-Bighash , part of the Number Theory Seminar.

At

Nov. 5, 2013, 1 p.m.

In

SEO 636

Abstract

A fruitful way to study the arithmetic significance of special values of L-functions is via their interpolation by p-adic L-functions. In this talk, I will discuss the phenomenon of L-invariants, which arise when the interpolation property provides no immediate information. Specifically, the value of the p-adic L-function may vanish even when the value of the original L-function does not. Beginning with the work of Mazur–Tate–Teitelbaum on a p-adic Birch–Swinnerton-Dyer conjecture, it has been conjectured that the value of the derivative of the p-adic L-function should relate to the original L-value, up to the introduction of a new factor: the${\mathcal L}$-invariant. I will give an overview of the subject and what is known before discussing joint work with Andrei Jorza where we obtain formulas for the ${\mathcal L}$-invariants of symmetric powers of modular forms.