Abstract: Galois representations have become one of the most powerful tools in number theory. They give new proofs of classical results, such as the fact that Ramanujan’s tau function is divisible by 23 for exactly half of the primes, and they were key to proving deep theorems like Fermat’s Last Theorem and the Sato-Tate conjecture.
`Local' versions of these objects, which look at one prime at a time, are easier to study, yet basic questions remain open even in the case of dimension 2. One of them: if we reduce such an object modulo the same prime, does it stay intact (irreducible) or does it split into simpler pieces (reducible)? The answer seems to depend on two numbers, called the slope and the weight.
I will describe recent joint work with Sudipta Majumder (arXiv:2610.19293) that answers this question for slope 2, for all weights outside one congruence class. The talk will be a gentle introduction to this result.