Conférence "Zeta Functions"

Brauer-Siegel theorems for families of abelian varieties

Marc Hindry (Paris)

Lundi 21 juin, 10:00 - 11:00

Résumé

The analog of the classical Brauer-Siegel theorem we want to discuss is largely conjectural; it states that, for a family of abelian varieties of fixed dimension over a global field, the product of the order of the Tate-Shafarevic group by the Neron-Tate regulator should behave asymptotically like the (exponential) height of the abelian variety, as the height or conductor goes to infinity. The situation over number fields is almost entirely conjectural, the situation over function fields of finite characteristic is much better. We will explain a proof of the conjecture for examples like the elliptic curves y^2+xy=x^3-t^d when d goes to infinity and what is missing in general. Essentially two missing ingredients: 1) a proof of the finiteness of the Tate-Shafarevic group [a well- known hard problem] 2) a better lower bound for the value of the associated L-function at the center of the critical strip [ there is a lower bound log L^*(1)> -ch and we want log L^*(1)> -epsilon h]. The work over function fields is joint with Amilcar Pacheco (Rio, Brazil)