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Del Pezzo surfaces over finite fields and their Frobenius traces
Published online by Cambridge University Press: 10 April 2018
Abstract
Let S be a smooth cubic surface over a finite field $\mathbb{F}$q. It is known that #S($\mathbb{F}$q) = 1 + aq + q2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 35 - 60
- Copyright
- Copyright © Cambridge Philosophical Society 2018
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