Modular forms and L-functions have played a central role in Number Theory in the last decades: they have been the source of inspiration for several central conjectures that have been the fuel of the research since they have been formulated up to now. Between them, the Birch and Swinnerton-Dyer conjecture and the Riemann Hypothesis, two of the six still open problems of the Millennium Prize Problems set by the Clay Mathematics Institute, Hilbert’s Twelfth and Nine Problem, the Stark’s conjectures and other important related conjectures such as the Main Conjectures of Iwasawa theory and Leopoldt’s conjecture. They also provide key insights towards deep elementary conjectures in Number Theory. An emblematic example is Fermat’s Last Theorem, which states that there are no positive integers a, b, and c satisfying the equation aⁿ+bⁿ=cⁿ for n≥3, proved by A. Wiles. It was the result of the efforts of several mathematicians which had the insight to apply the theory of elliptic curves, modular forms and Galois representations to this problem.

As explained, these conjectures are very deep and modular forms and L-functions appear sometime in an amazing way. Another unexpected feature of them is that they present p-adic analogues which can be approached and, sometime, even imply instances of the original conjectures. The aim of the conference is to join together mathematicians working on two fundamental arithmetic aspects of the theory, namely p-adic modular forms and p-adic L-functions, and report on the most recent developments and impact that these results have on the above mentioned conjectures. The conference will take place from May 20 to May 24, 2019.