For zoom details contact Pieter Moree (moree@mpim-bonn.mpg.de).

In this talk I will explain new research on L-invariants of modular forms, including ongoing joint work with Robert Pollack. L-invariants, which are p-adic invariants of modular forms, were discovered in the 1980's, by Mazur, Tate, and Teitelbaum, who were formulating a p-adic analogue of Birch and Swinnerton-Dyer's conjecture on elliptic curves. In the decades since, L-invariants have shown up in a ton of places: p-adic L-series for higher weight modular forms or higher rank automorphic forms, the Banach space representation theory of GL(2,Qp), p-adic families of modular forms, Coleman integration on the p-adic upper half-plane, and Fontaine's p-adic Hodge theory for Galois representations. In this talk I will focus on recent numerical and statistical investigations of these L-invariants, which touch on at least four of the theories just mentioned. I will try to put everything into the overall context of practical questions in the theory of automorphic forms and Galois representations, keeping everything as concrete as possible, and explain what the future holds.

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