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Mosc. Math. J., 2012, Volume 12, Number 3, Pages 515–542 (Mi mmj456)  

This article is cited in 17 scientific papers (total in 17 papers)

On a conjecture of Deligne

Vladimir Drinfeld

University of Chicago, Department of Mathematics, Chicago, IL 60637

Abstract: Let $X$ be a smooth variety over $\mathbb{F}_p$. Let $E$ be a number field. For each nonarchimedean place $\lambda$ of $E$ prime to $p$ consider the set of isomorphism classes of irreducible lisse $\overline{E}_{\lambda}$-sheaves on $X$ with determinant of finite order such that for every closed point $x\in X$ the characteristic polynomial of the Frobenius $F_x$ has coefficents in $E$. We prove that this set does not depend on $\lambda$.
The idea is to use a method developed by G. Wiesend to reduce the problem to the case where $X$ is a curve. This case was treated by L. Lafforgue.

Key words and phrases: $\ell$-adic representation, independence of $\ell$, local system, Langlands conjecture, arithmetic scheme, Hilbert irreducibility, weakly motivic.


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MSC: 14G15, 11G35

Citation: Vladimir Drinfeld, “On a conjecture of Deligne”, Mosc. Math. J., 12:3 (2012), 515–542

Citation in format AMSBIB
\by Vladimir~Drinfeld
\paper On a conjecture of Deligne
\jour Mosc. Math.~J.
\yr 2012
\vol 12
\issue 3
\pages 515--542

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    This publication is cited in the following articles:
    1. Pierre Deligne, “Finiteness of the extension of $\mathbb Q$ generated by Frobenius traces, in finite characteristic”, Mosc. Math. J., 12:3 (2012), 497–514  mathnet  mathscinet  zmath
    2. H. Esnault, L. Kindler, “Lefschetz theorems for tamely ramified coverings”, Proc. Amer. Math. Soc., 144:12 (2016), 5071–5080  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. Esnault H., Groechenig M., “Cohomologically Rigid Local Systems and Integrality”, Sel. Math.-New Ser., 24:5 (2018), 4279–4292  crossref  mathscinet  zmath  isi  scopus
    11. Abe T., “Langlands Correspondence For Isocrystals and the Existence of Crystalline Companions For Curves”, J. Am. Math. Soc., 31:4 (2018), 921–1057  crossref  mathscinet  zmath  isi  scopus
    12. 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
    13. Nagamachi I., “On a Good Reduction Criterion For Proper Polycurves With Sections”, Hiroshima Math. J., 48:2 (2018), 223–251  crossref  mathscinet  zmath  isi  scopus
    14. 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
    15. 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
    16. Boeckle G., Gajda W., Petersen S., “A Variational Open Image Theorem in Positive Characteristic”, J. Theor. Nr. Bordx., 30:3 (2018), 965–977  crossref  mathscinet  zmath  isi
    17. 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, ed. 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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