Generalized Fermat equations, Darmon's program and Hypergeometric motives

Objective

This project addresses the Generalized Fermat Conjecture, one of the most challenging open problems in number theory. While the modular method has led to remarkable breakthroughs, including the proof of Fermat’s Last Theorem, its application to more general equations remains limited. Recently, a new framework was introduced by the supervisor through the use of hypergeometric motives, opening an unexplored direction for research. At the same time, the ER has contributed to Darmon’s program, a complementary approach aiming at the same goal. Building on these two advances, the project will pursue three main objectives: to develop a general strategy to apply the modular method to an arbitrary generalized Fermat equation, to establish modularity results for totally real hypergeometric motives, and to prove new cases of the Generalized Fermat Conjecture involving signatures with three distinct prime exponents.