Abstract
We prove an analogue of the Brauer–Siegel theorem for Fermat surfaces over a finite field F q . Namely, letting F d be the Fermat surface of degree d over F q and p g ( F d ) be its geometric genus, we show that, for d → ∞ ranging over the set of integers coprime with q, one has log | Br ( F d ) | · Reg ( F d ) ∼ log q p g ( F d ) ∼ log q 6 · d 3 . Here, Br ( F d ) denotes the Brauer group of F d and Reg ( F d ) the absolute value of a Gram determinant of the Néron–Severi group NS ( F d ) with respect to the intersection form.
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