I would like to have a reference for a proof of the rationality of the zeta function for a curve by Dwork's method. I would like to know if Dwork's proof simplifies for a curve.
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2$\begingroup$ Please use high-level tags like ag.algebraic-geometry and nt.number-theory. I added these tags now. Regarding high-level tags, see meta.mathoverflow.net/q/1075 $\endgroup$GH from MO– GH from MO2026-02-28 19:11:40 +00:00Commented 9 hours ago
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4$\begingroup$ The case of hypersurfaces, in particular plane curves, is done in Chapter V of Koblitz's "p-adic Numbers, p-adic Analysis, and Zeta-Functions", Springer, 1984. $\endgroup$F Zaldivar– F Zaldivar2026-02-28 20:37:17 +00:00Commented 8 hours ago
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For a detailed description, see the lecture notes M. Mustaţă, or by K.S.Kedlaya. For more expository discussions, see Tao's blog or S.W. Park's notes.
For the specialization to one-dimensional curves, chapter 2 of Topics in the Theory of Zeta Functions of Curves by S. Chan might be useful. As far as I can tell, the proof is not explicitly "for curves only", but it is set up so that the one-dimensional case is the intended primary example, worked out in the subsequent chapters.
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5$\begingroup$ OP has asked specifically about the case of curves, and whether Dwork's proof simplifies in this case. On a brief look, I don't think either of your references addresses that question. $\endgroup$Wojowu– Wojowu2026-02-28 20:04:58 +00:00Commented 8 hours ago
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$\begingroup$ Chan's thesis might go some way towards that. $\endgroup$Carlo Beenakker– Carlo Beenakker2026-02-28 20:41:33 +00:00Commented 8 hours ago
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2$\begingroup$ FWIW: My undergraduate thesis fits the category of expository discussions that also, unfortunately, do not meaningfully broach the case of curves. It is linked here. $\endgroup$Benjamin Dickman– Benjamin Dickman2026-02-28 21:40:53 +00:00Commented 7 hours ago