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. They 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 `$\mathrm{GL}(2,\mathbb{Q}_p)$`

, `$p$`

-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 many of the theories just mentioned. I will try to put everything into the context of practical questions in the theory of automorphic forms and Galois representations and explain what the future holds.

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- Seminars for 2022

Last updated: 28 Nov 2023