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Mosc. Math. J., 2012, Volume 12, Number 3, Pages 497–514 (Mi mmj455)  

This article is cited in 13 scientific papers (total in 13 papers)

Finiteness of the extension of $\mathbb Q$ generated by Frobenius traces, in finite characteristic

Pierre Deligne

Institute for Advanced Study, School of Mathematics, 1 Einstein Drive, Princeton, NJ 08540 USA

Abstract: Let $\mathscr{F}_0$ be a $\bar{\mathbb{Q}}_l$-sheaf on a scheme $Z_0$ of finite type over $\mathbb{F}_q$. We show the existence of a finite type extension $E\subset\bar{\mathbb{Q}}_l$ of $\mathbb{Q}$ such that all local factors of the $L$-function of $\mathscr{F}_0$ have coefficients in $E$. When $Z_0$ is normal and connected, and $\mathscr{F}_0$ is an irreducible $l$-adic local system whose determinant is of finite order, $E$ can be taken to be a finite extension of $\mathbb{Q}$.

Key words and phrases: $l$-adic sheaves, Frobenius traces.

DOI: https://doi.org/10.17323/1609-4514-2012-12-3-497-514

Full text: http://www.ams.org/.../abst12-3-2012.html
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Bibliographic databases:

MSC: 14F20, 14G15
Received: June 21, 2011
Language:

Citation: Pierre Deligne, “Finiteness of the extension of $\mathbb Q$ generated by Frobenius traces, in finite characteristic”, Mosc. Math. J., 12:3 (2012), 497–514

Citation in format AMSBIB
\Bibitem{Del12}
\by Pierre~Deligne
\paper Finiteness of the extension of $\mathbb Q$ generated by Frobenius traces, in finite characteristic
\jour Mosc. Math.~J.
\yr 2012
\vol 12
\issue 3
\pages 497--514
\mathnet{http://mi.mathnet.ru/mmj455}
\crossref{https://doi.org/10.17323/1609-4514-2012-12-3-497-514}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3024820}
\zmath{https://zbmath.org/?q=an:1260.14022}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309366400003}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Vladimir Drinfeld, “On a conjecture of Deligne”, Mosc. Math. J., 12:3 (2012), 515–542  mathnet  mathscinet  zmath
    2. Fouvry E., Kowalski E., Michel Ph., “Algebraic Trace Functions Over the Primes”, Duke Math. J., 163:9 (2014), 1683–1736  crossref  mathscinet  zmath  isi  scopus
    3. M. Kerz, Sh. Saito, “Chow group of 0-cycles with modulus and higher-dimensional class field theory”, Duke Math. J., 165:15 (2016), 2811–2897  crossref  mathscinet  zmath  isi  scopus
    4. R. Weissauer, “Vanishing theorems for constructible sheaves on abelian varieties over finite fields”, Math. Ann., 365:1-2 (2016), 559–578  crossref  mathscinet  zmath  isi  scopus
    5. Sh. Sun, W. Zheng, “Parity and symmetry in intersection and ordinary cohomology”, Algebr. Number Theory, 10:2 (2016), 235–307  crossref  mathscinet  zmath  isi  scopus
    6. H. Esnault, “Survey on some aspects of Lefschetz theorems in algebraic geometry”, Rev. Mat. Complut., 30:2 (2017), 217–232  crossref  mathscinet  zmath  isi  scopus
    7. K. Shimizu, “Existence of compatible systems of lisse sheaves on arithmetic schemes”, Algebr. Number Theory, 11:1 (2017), 181–211  crossref  mathscinet  zmath  isi  scopus
    8. H. Esnault, “A remark on Deligne's finiteness theorem”, Int. Math. Res. Notices, 2017, no. 16, 4962–4970  crossref  mathscinet  isi
    9. T. Koshikawa, “Overconvergent unit-root $F$-isocrystals and isotriviality”, Math. Res. Lett., 24:6 (2017), 1707–1727  crossref  mathscinet  zmath  isi
    10. Sun Sh., “Independence of l For the Supports in the Decomposition Theorem”, Duke Math. J., 167:10 (2018), 1803–1823  crossref  mathscinet  zmath  isi  scopus
    11. Lafforgue V., “Chtoucas For Reductive Groups and Parameterization of Global Langlands”, J. Am. Math. Soc., 31:3 (2018), 719–891  crossref  mathscinet  zmath  isi  scopus
    12. Drinfeld V., “On the Pro-Semisimple Completion of the Fundamental Group of a Smooth Variety Over a Finite Field”, Adv. Math., 327 (2018), 708–788  crossref  mathscinet  zmath  isi  scopus
    13. Esnault H., “Some Fundamental Groups in Arithmetic Geometry”, Algebraic Geometry: Salt Lake City 2015, Pt 2, Proceedings of Symposia in Pure Mathematics, 97, no. 2, eds. DeFernex T., Hassett B., Mustata M., Olsson M., Popa M., Thomas R., Amer Mathematical Soc, 2018, 169–179  crossref  mathscinet  isi
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