Seminario di Teoria dei Numeri

Venerdì 19 luglio 2024 alle ore 14:30 Steven J. Miller (Williams College) terrà un seminario di Teoria dei Numeri dal titolo "Machine Learning in Elliptic Curves and Beyond: From Conjectures to Theorems to Conjectures".

Abstract: Many delicate problems in number theory have only recently become amenable to numerical exploration due to the painfully slow convergence rate. Quantities associated to elliptic curves often converge at the scale of the logarithm of the conductor; thus while we may have millions of curves with conductors at most $10^20$, this translates to less than 50 (and one would never conjecture on properties of primes from integers up to 50!). Improvements in computing power have led to larger data sets, which in conjunction with machine learning (ML) techniques have found new behavior. Recently He, Le and Oliver used ML to distinguish numerous curves based on standard invariants, and discovered oscillatory behavior in the coefficients of the associated $L$-functions, which agrees with recently developed theoretical models. We report on work of the author and his colleagues on lower order terms in coefficients in families, describing an open conjecture where the "nice" term is hard to extract due to large, fluctuating terms, in the hopes of forming collaborations with audience members. These lower order terms have implications for the distribution of zeros of elliptic curve $L$-functions at and near the central point.

Il seminario avrà luogo in presenza presso il Dipartimento di Matematica e Fisica
Via della Vasca Navale 84 - Aula A

Per informazioni, rivolgersi a: fabrizio.barroero@uniroma3.it