A Brauer-Siegel theorem for Fermat surfaces over finite fields
The classical Brauer-Siegel theorem gives asymptotic upper and lower bounds on the product of the class-number times the regulator of units of a number field in terms of its discriminant as the latter grows. In this talk, I will describe an analogous result in a more geometric context. Namely, for a Fermat surface F over a given finite field, we consider the product of the order of its Brauer group (which is known to be finite) by the Gram determinant of a basis of its Néron-Severi group for the intersection form, and we give a very precise estimate of the growth of that product in terms of the geometric genus of F when the latter grows to infinity. As in the classical setting, the proof of the asymptotic estimate is rather analytic: it relies on obtaining asymptotic bounds on the size of the "residue" of the zeta function of F at its pole at s=1.