A Brauer-Siegel theorem for Fermat surfaces over finite fields
Abstract
We prove an analogue of the Brauer-Siegel theorem for Fermat surfaces over a finite field. Namely, letting Fd be the Fermat surface of degree d over Fq and pg(Fd) be its geometric genus, we consider the product of the order of the Brauer group Br(Fd) of Fd times the absolute value of a Gram determinant of the N\'eron-Severi group of Fd with respect to the intersection form (the regulator Reg(Fd) of Fd). We show that this product grows like qpg(Fd) when d tends to infinity: ( |Br(Fd)|· Reg(Fd)) qpg(Fd).
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