A remark on the Tate Conjecture
Author:
Ben Moonen
Journal:
J. Algebraic Geom. 28 (2019), 599-603
DOI:
https://doi.org/10.1090/jag/720
Published electronically:
April 24, 2019
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Abstract |
References |
Additional Information
Abstract: The strong version of the Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic $0$, (T) implies (S). In characteristic $p$ an analogous result is true under stronger assumptions.
References
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References
- Yves André, Pour une théorie inconditionnelle des motifs, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 5–49 (French). MR 1423019
- Gerd Faltings, $p$-adic Hodge theory, J. Amer. Math. Soc. 1 (1988), no. 1, 255–299. MR 924705, DOI https://doi.org/10.2307/1990970
- Jean-Marc Fontaine, Représentations $p$-adiques semi-stables, with with an appendix by Pierre Colmez, Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 113–184 (French). MR 1293972
- Lei Fu, On the semisimplicity of pure sheaves, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2529–2533. MR 1676348, DOI https://doi.org/10.1090/S0002-9939-99-05414-3
- J. S. Milne, Values of zeta functions of varieties over finite fields, Amer. J. Math. 108 (1986), no. 2, 297–360. MR 833360, DOI https://doi.org/10.2307/2374676
- Jean-Pierre Serre, Œuvres. Collected papers. IV, Springer-Verlag, Berlin, 2000 (French). 1985–1998. MR 1730973
- Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, 3rd ed., with translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, with a foreword by Brown and Serre, Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1997. MR 1757192
Additional Information
Ben Moonen
Affiliation:
IMAPP, Radboud University, P.O. Box 9010, 6500GL Nijmegen, The Netherlands
MR Author ID:
254842
Email:
b.moonen@science.ru.nl
Received by editor(s):
September 29, 2017
Received by editor(s) in revised form:
January 25, 2018
Published electronically:
April 24, 2019
Article copyright:
© Copyright 2019
University Press, Inc.
