Algebraic Statistics and the Study of Multistate Models: Theory and Applications, April 1-4, 2025, Aalto and U. Helsinki
Elliptic curves and modular forms bring Galois representations (don't worry, these terms will be introduced/recalled in the talk). One of the many versions of Taylor-Wiles modularity theorem rephrases as: "representations attached to rational elliptic curves come from modular forms". The power of this somewhat abstract statement is that it allows to prove Fermat's Last Theorem (with the aid of Ribet's level lowering theorem and Frey's curve, but that's other story). In 2007, Dieulefait gave the first proof of the Serre modularity conjecture, generalising Taylor-Wiles, namely: Any rational Galois representation which "looks like" modular, is indeed modular. This statement was generalised in full by Khare and Wintenberger in 2011 (and that's also another story...) Since then, a series of hits have established that any 2-dimensional "regular enough" Galois representation comes from modular-ish forms (namely, automorphic forms), so establishing a (conjectural) dictionary between arithmetic representations and automorphic ones. However, the following question remains openut (up to some minor results, includig the speaker's ones): to determine which group-theoretical representations preserve automorphicity. The present talk is an introductory account of these ideas, a discussion of what's open and, time permitting, a statement of two of the works by the speaker.
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