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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
