Cari tutti,
faccio pubblicità ad un seminario "interno" di un dottorando del nostro dipartimento, Lorenzo Furio, che si svolgerà martedì 27 giugno alle ore 11 in aula seminari. Trovate qui sotto il titolo e l'abstract.
(Nonostante il titolo in inglese, se non ci saranno richieste in senso contrario il seminario si svolgerà in italiano.)
Saluti a tutti, Davide
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Galois representations attached to elliptic curves and Serre's uniformity question
The study of Galois representations attached to elliptic curves is a very fruitful branch of number theory, leading to the solution of very difficult problems, such as Fermat's Last Theorem. Given a rational elliptic curve E, the representation \rho_{E,p} is described by the action of the absolute Galois group of \mathbb{Q} on the p-torsion points of E. In 1972 Serre proved that for every rational elliptic curve E without complex multiplication there exists an integer N_E such that, for every prime p>N_E, the Galois representation \rho_{E,p} is surjective onto GL_2(\mathbb{F}_p). In the same article, he asked whether the constant N_E can be taken to be independent of the curve, and this became known as Serre's Uniformity Question. In this talk, I will discuss the current progress towards an answer to this question, in particular the Runge method for modular curves developed by Bilu and Parent and the recent improvements obtained via this method by Le Fourn and Lemos, as well as explain how to solve some of the questions left open by the latter result. This is joint work with Davide Lombardo.
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