Intersection of a correspondence with a graph of Frobenius
Author:
Yakov Varshavsky
Journal:
J. Algebraic Geom. 27 (2018), 1-20
DOI:
https://doi.org/10.1090/jag/676
Published electronically:
February 17, 2017
MathSciNet review:
3722688
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Abstract |
References |
Additional Information
Abstract: The goal of this note is to give a short geometric proof of a theorem of Hrushovski asserting that an intersection of a correspondence with a graph of a sufficiently large power of Frobenius is non-empty.
References
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- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258
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- Najmuddin Fakhruddin, Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109–122. MR 1995861
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
- A. Grothendieck, rédigé par P. Deligne, La classe de cohomologie associées à un cycle, SGA $4\frac {1}{2}$, Lecture Notes in Math. 569, Springer-Verlag, New York, 1977, pp. 129–153.
- E. Hrushovski, The elementary theory of the Frobenius automorphism, preprint, available at http://www.ma.huji.ac.il/˜ehud/FROB.pdf.
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020
- Kazuya Kato and Takeshi Saito, Ramification theory for varieties over a perfect field, Ann. of Math. (2) 168 (2008), no. 1, 33–96. MR 2415398, DOI https://doi.org/10.4007/annals.2008.168.33
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- Gérard Laumon, Homologie étale, Séminaire de géométrie analytique (École Norm. Sup., Paris, 1974-75), Soc. Math. France, Paris, 1976, pp. 163–188. Astérisque, No. 36-37. MR 0444667
- Richard Pink, On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne, Ann. of Math. (2) 135 (1992), no. 3, 483–525. MR 1166642, DOI https://doi.org/10.2307/2946574
- Yakov Varshavsky, Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara, Geom. Funct. Anal. 17 (2007), no. 1, 271–319. MR 2306659, DOI https://doi.org/10.1007/s00039-007-0596-9
References
- Alexander Borisov and Mark Sapir, Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms, Invent. Math. 160 (2005), no. 2, 341–356. MR 2138070 (2006b:20033), DOI https://doi.org/10.1007/s00222-004-0411-2
- Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 0340258 (49 \#5013)
- Hélène Esnault and Vikram Mehta, Simply connected projective manifolds in characteristic $p>0$ have no nontrivial stratified bundles, Invent. Math. 181 (2010), no. 3, 449–465. MR 2660450 (2012h:14048), DOI https://doi.org/10.1007/s00222-010-0250-2
- Najmuddin Fakhruddin, Questions on self maps of algebraic varieties, J. Ramanujan Math. Soc. 18 (2003), no. 2, 109–122. MR 1995861 (2004f:14038)
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
- A. Grothendieck, rédigé par P. Deligne, La classe de cohomologie associées à un cycle, SGA $4\frac {1}{2}$, Lecture Notes in Math. 569, Springer-Verlag, New York, 1977, pp. 129–153.
- E. Hrushovski, The elementary theory of the Frobenius automorphism, preprint, available at http://www.ma.huji.ac.il/˜ehud/FROB.pdf.
- A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. 83 (1996), 51–93. MR 1423020 (98e:14011)
- Kazuya Kato and Takeshi Saito, Ramification theory for varieties over a perfect field, Ann. of Math. (2) 168 (2008), no. 1, 33–96. MR 2415398 (2009h:14033), DOI https://doi.org/10.4007/annals.2008.168.33
- Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241 (French, with English and French summaries). MR 1875184 (2002m:11039), DOI https://doi.org/10.1007/s002220100174
- Gérard Laumon, Homologie étale, Séminaire de géométrie analytique (École Norm. Sup., Paris, 1974-75), Soc. Math. France, Paris, 1976, pp. 163–188. Astérisque, No. 36-37. MR 0444667 (56 \#3017)
- Richard Pink, On the calculation of local terms in the Lefschetz-Verdier trace formula and its application to a conjecture of Deligne, Ann. of Math. (2) 135 (1992), no. 3, 483–525. MR 1166642 (93m:14017), DOI https://doi.org/10.2307/2946574
- Yakov Varshavsky, Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara, Geom. Funct. Anal. 17 (2007), no. 1, 271–319. MR 2306659 (2008d:14032), DOI https://doi.org/10.1007/s00039-007-0596-9
Additional Information
Yakov Varshavsky
Affiliation:
Institute of Mathematics, Hebrew University, Givat-Ram, Jerusalem, 91904, Israel
MR Author ID:
638793
Email:
vyakov@math.huji.ac.il
Received by editor(s):
October 29, 2014
Received by editor(s) in revised form:
November 28, 2014, and July 1, 2016
Published electronically:
February 17, 2017
Additional Notes:
This work was supported by The Israel Science Foundation (grant No. 1017/13)
Article copyright:
© Copyright 2017
Universal Press, Inc.
