A Brauer-Siegel theorem for Fermat surfaces over finite fields

摘要

We prove an analogue of the Brauer-Siegel theorem for Fermat surfaces over a finite field Fq. Namely, letting Fd be the Fermat surface of degree d over Fq and pg(Fd) be its geometric genus, we show that, for d ranging over the set of integers coprime with q, one has |Br(Fd)|boldReg/bold(Fd)approximate to logqpg(Fd)approximate to Here, Br(Fd) denotes the Brauer group of Fd and Reg(Fd) the absolute value of a Gram determinant of the Neron-Severi group NS(Fd) with respect to the intersection form.