We will have three guests in March and April 2025. The seminar will take place in Aula INdAM at 12, with the exception of Perucca's talk on April 16.
2025.3.28. Lucile Devin (Université du Littoral Côte d'Opale)
Vertical distribution of zeros of L-functions, extended support
I will discuss the distribution of low-lying zeros of L-functions in families of degree two, for which, thanks to good trace formulas, we are able to extend the unconditional support in the Katz-Sarnak prediction. Joint with Martin Čech, Daniel Fiorilli, Kaisa Matomäki and Anders Södergren.
2025.4.11. Davide Lombardo (Università di Pisa)
A simple dichotomy in Serre's uniformity question
Let E be an elliptic curve defined over the field of rational numbers and suppose that E does not have (potential) complex multiplication. For every prime p, the action of the absolute Galois group of Q on the p-torsion points of E gives rise to a representation ρ_{E, p}: Gal(\overline{Q}/Q) --> Aut(E[p]) \cong GL_2(F_p). Serre's celebrated open image theorem shows that this representation is surjective for all p greater than some bound p_0=p_0(E), depending in principle on E. Serre has asked whether there is a universal p_0 that works for all (non-CM) elliptic curves over Q. Building on results by many authors, including most recently Le Fourn and Lemos, we prove that for p>37 there are at most two possibilities (up to conjugacy) for the image of ρ_{E, p}: the whole group GL_2(F_p) and the normaliser of a so-called non-split Cartan subgroup.
This is joint work with Lorenzo Furio.
2025.4.16. Antonella Perucca (University of Luxembourg)
[This talk will be part of the Seminar of Algebra and Geometry and will take place in Sala di Consiglio at 14:00]
Almost 100 years of Artin's Conjecture
In 1927, Artin formulated his famous conjecture on primitive roots. The most basic question, which is still open, is as follows. For an odd prime number p, we say that 2 is a primitive root modulo p if every integer that is not a multiple of p is congruent to a power of 2 modulo p. Are there infinitely many primes p such that 2 is a primitive root modulo p?
Artin's conjecture (that has been proven by Hooley in 1967 under GRH) says in particular that there is a positive density of primes p such that 2 is a primitive root modulo p. This density is conjectured to be Artin's constant, which is roughly 37%. Beyond the classical results, I will present very general results on the index map (joint work with Järviniemi and Sgobba), an unexpected lower bound on the density (joint work with Shparlinski) and a computational project (joint work with Tholl).