On the Frobenius endomorphisms of Fermat and Artin-Schreier curves. (English) Zbl 0666.14014
The author shows that in the ring of correspondences of a Fermat (resp. Artin-Schreier) curve, Frobenius can be identified, in a suitable sense, with a Jacobi (resp. a Gauss) sum. This gives a strikingly simple interpretation of Weil’s results on the eigenvalues of Frobenius on the \(\ell\)-adic cohomology of those curves.
The author then deduces from the same identification a very short proof, in the case of Fermat curves of the Brumer-Stark conjecture for function fields, proven in general by P. Deligne [Cf. J. Tate, “Les conjectures de Stark sur les fonctions L d’Artin en \(s=0\)”, Prog. Math. 47 (1984; Zbl 0545.12009)].
The case of Artin-Schreier curves can be treated similarly. The proof is analogous to Stickelberger’s proof of Stickelberger’s theorem.
This paper is a jewel of clarity.
We should point out a few misprints on page 465 of the text:
line 19, \(``(u,v)\to u^ m\)” instead of “(u,v)\(\to u\)”; line 19 and 24, \(``G_ m\)” instead of “G”; line 20, \(``G_ m\)” instead of \(``F_ m\)”; line 25, “image” instead of “kernel”.
The author then deduces from the same identification a very short proof, in the case of Fermat curves of the Brumer-Stark conjecture for function fields, proven in general by P. Deligne [Cf. J. Tate, “Les conjectures de Stark sur les fonctions L d’Artin en \(s=0\)”, Prog. Math. 47 (1984; Zbl 0545.12009)].
The case of Artin-Schreier curves can be treated similarly. The proof is analogous to Stickelberger’s proof of Stickelberger’s theorem.
This paper is a jewel of clarity.
We should point out a few misprints on page 465 of the text:
line 19, \(``(u,v)\to u^ m\)” instead of “(u,v)\(\to u\)”; line 19 and 24, \(``G_ m\)” instead of “G”; line 20, \(``G_ m\)” instead of \(``F_ m\)”; line 25, “image” instead of “kernel”.
Reviewer: F.Baldassarri
MSC:
14H25 | Arithmetic ground fields for curves |
14G15 | Finite ground fields in algebraic geometry |
11R42 | Zeta functions and \(L\)-functions of number fields |
14G20 | Local ground fields in algebraic geometry |
14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
Citations:
Zbl 0545.12009References:
[1] | K. F. Gauss, Disquisitiones arithmeticae, Werke, Vol. I, 1901. |
[2] | H. Davenport and H. Hasse, Die Nullstellen der Kongruenzzetafunktionen in gewessin zyklischen Fällen, Crelles J. 172 (1935), 151. · JFM 60.0913.01 |
[3] | David R. Hayes, Stickelberger elements in function fields, Compositio Math. 55 (1985), no. 2, 209 – 239. · Zbl 0569.12008 |
[4] | John Tate, Les conjectures de Stark sur les fonctions \? d’Artin en \?=0, Progress in Mathematics, vol. 47, Birkhäuser Boston, Inc., Boston, MA, 1984 (French). Lecture notes edited by Dominique Bernardi and Norbert Schappacher. · Zbl 0545.12009 |
[5] | A. Weil, Courbes algébriques et variétés abéliennes, Hermann, Paris, 1971. · Zbl 0208.49202 |
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