The distribution of invariants of modular forms has been studied in many contexts. The Sato-Tate conjecture makes a precise prediction on the distribution of normalized Hecke-eigenvalues for modular forms. Here one fixes a form and varies the eigenvalue. One could also fix the eigenvalue and vary the form and still this invariant has a beautifully predictable distribution.

In this talk, we will discuss p-adic variants of these questions and investigate the distribution of the p-adic size of Hecke-eigenvalues leading to Gouvea's conjecture. Further, we will study a more mysterious p-adic invariant of a modular form, namely the L-invariant. We will give an overview of this invariant and ultimately state a conjecture about its p-adic distribution. This work is joint with John Bergdall.

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