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On rational points of varieties over local fields having a model with tame quotient singularities


Author: Annabelle Hartmann
Journal: Trans. Amer. Math. Soc. 367 (2015), 8199-8227
MSC (2010): Primary 14D10; Secondary 14G05, 14B05, 14B10
DOI: https://doi.org/10.1090/S0002-9947-2015-06291-6
Published electronically: February 13, 2015
MathSciNet review: 3391914
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Abstract: We study rational points on a smooth variety $ X$ over a complete local field $ K$ with algebraically closed residue field, and models $ \mathcal {X}$ of $ X$ with tame quotient singularities. If $ \mathcal {X}$ is the quotient of a Galois action on a weak Néron model of the base change of $ X$ to a tame Galois extension of $ K$, then we construct a canonical weak Néron model of $ X$ with a map to $ \mathcal {X}$, and examine its special fiber. As an application we get examples of singular models $ \mathcal {X}$ such that there are $ K$-rational points of $ X$ specializing to a singular point of $ \mathcal {X}$. Moreover we obtain formulas for the motivic Serre invariant and the rational volume, and the existence of $ K$-rational points on certain $ K$-varieties with potential good reduction.


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  • [1] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
  • [2] P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin-New York, 1977. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41\over2; Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier. MR 0463174
  • [3] Bas Edixhoven, Néron models and tame ramification, Compositio Math. 81 (1992), no. 3, 291–306. MR 1149171
  • [4] Hélène Esnault and Johannes Nicaise, Finite group actions, rational fixed points and weak Néron models, Pure Appl. Math. Q. 7 (2011), no. 4, Special Issue: In memory of Eckart Viehweg, 1209–1240. MR 2918159, https://doi.org/10.4310/PAMQ.2011.v7.n4.a7
  • [5] 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
  • [6] Ulrich Görtz and Torsten Wedhorn, Algebraic geometry I, Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden, 2010. Schemes with examples and exercises. MR 2675155
  • [7] A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228. MR 0217083
    A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222. MR 0217084
    A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 0217085
  • [8] Alexander Grothendieck, Revêtements étales et groupe fondamental. Fasc. I: Exposés 1 à 5, Séminaire de Géométrie Algébrique, vol. 1960/61, Institut des Hautes Études Scientifiques, Paris, 1963. MR 0217087
    Alexander Grothendieck, Revêtements étales et groupe fondamental. Fasc. II: Exposés 6, 8 à 11, Séminaire de Géométrie Algébrique, vol. 1960/61, Institut des Hautes Études Scientifiques, Paris, 1963. MR 0217088
  • [9] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. 32 (1967), 361 (French). MR 0238860
  • [10] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
  • [11] Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
  • [12] Johannes Nicaise, Private communication, 2012.
  • [13] Johannes Nicaise and Julien Sebag, Motivic invariants of rigid varieties, and applications to complex singularities, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume I, London Math. Soc. Lecture Note Ser., vol. 383, Cambridge Univ. Press, Cambridge, 2011, pp. 244–304. MR 2885338
  • [14] Johannes Nicaise and Julien Sebag, The Grothendieck ring of varieties, Motivic integration and its interactions with model theory and non-Archimedean geometry. Volume I, London Math. Soc. Lecture Note Ser., vol. 383, Cambridge Univ. Press, Cambridge, 2011, pp. 145–188. MR 2885336
  • [15] Jean-Pierre Serre, Groupes finis d’automorphismes d’anneaux locaux réguliers, Colloque d’Algèbre (Paris, 1967) Secrétariat mathématique, Paris, 1968, pp. 11 (French). MR 0234953
  • [16] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237
  • [17] Jean-Pierre Serre, How to use finite fields for problems concerning infinite fields, Arithmetic, geometry, cryptography and coding theory, Contemp. Math., vol. 487, Amer. Math. Soc., Providence, RI, 2009, pp. 183–193. MR 2555994, https://doi.org/10.1090/conm/487/09532

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Additional Information

Annabelle Hartmann
Affiliation: Department of Mathematics, KU Leuven, 3001 Leuven, Belgium
Email: annabelle.hartmann@wis.kuleuven.be

DOI: https://doi.org/10.1090/S0002-9947-2015-06291-6
Received by editor(s): March 28, 2013
Received by editor(s) in revised form: September 11, 2013
Published electronically: February 13, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.