Between the l-adic and p-adic Langlands programmes

The Langlands programme, initiated in the 1970s, is a broad web of ideas that connect the worlds of analysis, geometry, algebra, and number theory. It predicts that sequences of numbers coming from counting the solutions to integer equations also arise in a completely different way, from analytic functions with a very large amount of symmetry. This is both surprising and powerful; the confirmation of a very special case of this prediction led to the proof in 1995 of Fermat's Last Theorem, a problem that had stood for over 300 years. Congruences play a key role in much of the recent progress in the Langlands programme and its applications. These arise when two sequences of integers leave the same remainders when they are divided by a prime number. The study of these congruences divides into two cases: l-adic congruences, where the prime l that we are dividing by is distinct from the prime that is being varied in the data producing the sequences, and p-adic congruences, where these primes are the same. The p-adic case in particular has received a large amount of attention in recent years. The aim of this project is to exhibit a great deal of unity between these two apparently quite different cases. We will show that ideas from one case can be used to produce new conjectures and new proofs in the other case. For example, the Breuil-Mézard conjecture originally arose on the p-adic side but was previously shown by the PI to have an l-adic analogue; we will generalise the scope of this l-adic analogue (to arbitrary "reductive groups") which will then provide a foundation for generalising the p-adic analogue. We will also show that p-adic results of Emerton-Gee-Savitt on the integral cohomology of certain highly symmetric varieties has an l-adic analogue; and, in the other direction, find a p-adic version of recent l-adic results showing that endomorphism rings of projective modules appear in deformation spaces of Galois representations.