Eigenvarieties and p-adic L-functions

The project explores a deep and old question in algebra and number theory: how hidden structures inside numbers are connected to the behavior of special mathematical objects in complex analysis called L-functions and having a geometrical nature. These L-functions play a central role in algebra and geometry because they encode important arithmetic information and invariants. Understanding their values can reveal fundamental properties of (algebraic) diophantine polynomial equations, like elliptic curves. A major development in the last decades has been the Langlands program, an ambitious network of difficult conjectures suggesting that several areas of mathematics (complex analysis, algebra and geometry) are connected in a profound and natural way. One part of this program explains how certain functions in complex analysis verifying some symmetries (called automorphic forms) are directly connected to objects in algebra and number theory Galois representations, and also to multivariable algebraic polynomial equations. Famous breakthroughs, such as the proof of Fermats Last Theorem by Andrew Wiles, rely on these ideas. The project focuses on a modern tool in number theory and arithmetic geometry called eigenvarieties. These are geometric objects (spaces) that organize large families of automorphic forms in a continuous way, but with respect to the p-adic distance rather than the usual Archimedean one. By studying the shape of these spaces, one can detect some patterns relating the geometry to the values of p-adic L-functions, a p-adic analogue of classical L-functions that carries deep arithmetic information. In this project we aim to construct new types of eigenvarieties, and investigate their local geometry at certain crossing points related to zeros of p-adic L- functions. Understanding this phenomenon is crucial and our goal is to compute the derivative, and relate it to arithmetic invariants occurring in number theory. By combining methods from geometry, p-adic analysis, and Galois theory, this research aims to advance our understanding of how fundamental arithmetic invariants are connected to the values of p-adic L-functions. These insights contribute to central open problems in modern number theory.