Abstract
For the Artin–Schreier curve y q − y = f(x) defined over a finite field \({{\mathbb F}_q}\) of q elements, the celebrated Weil bound for the number of \({{\mathbb F}_{q^r}}\)-rational points can be sharp, especially in super-singular cases and when r is divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz’s work on ℓ-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra \({\sqrt{q}}\) factor in the error term.
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It is a pleasure to thank the referee for his careful reading of the first version and for his very helpful comments.
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The research of A. Rojas-León was partially supported by P08-FQM-03894 (Junta de Andalucía), MTM2007-66929 and FEDER. The research of D. Wan was partially supported by NSF.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Rojas-León, A., Wan, D. Improvements of the Weil bound for Artin–Schreier curves. Math. Ann. 351, 417–442 (2011). https://doi.org/10.1007/s00208-010-0606-3
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DOI: https://doi.org/10.1007/s00208-010-0606-3