Fischbacher-Weitz, Helena; Köck, Bernhard; Marmora, Adriano

Galois-Module Theory for Wildly Ramified Covers of Curves over Finite Fields (with an Appendix by Bernhard Köck and Adriano Marmora)

Doc. Math. 24, 175-208 (2019)
DOI: 10.25537/dm.2019v24.175-208
Communicated by Takeshi Saito

Summary

Given a Galois cover of curves over \(\mathbb{F}_p\), we relate the \(p\)-adic valuation of epsilon constants appearing in functional equations of Artin L-functions to an equivariant Euler characteristic. Our main theorem generalises a result of Chinburg from the tamely to the weakly ramified case. We furthermore apply Chinburg's result to obtain a `weak' relation in the general case. In the Appendix, we study, in this arbitrarily wildly ramified case, the integrality of \(p\)-adic valuations of epsilon constants.

Mathematics Subject Classification

11R58, 14G10, 14G15, 11R33, 14H30

Keywords/Phrases

Galois cover of curves, weakly ramified, epsilon constant, equivariant Euler characteristic

References

  • 1. A. Abbes and T. Saito, Local Fourier transform and epsilon factors, Compos. Math. 146, (2010), no. 6, 1507-1551. DOI 10.1112/S0010437X09004631; zbl 1206.14034; MR2735372; arxiv 0809.0180.
  • 2. T. Abe and A. Marmora, Product formula for $p$-adic epsilon factors. J. Inst. Math. Jussieu 14 (2015), no. 2, 275-377. DOI 10.1017/S1474748014000024; zbl 1319.14025; MR3315058; arxiv 1104.1563.
  • 3. W. Bley and A. Cobbe, Equivariant epsilon constant conjectures for weakly ramified extensions, Math. Z. 283 (2016), no. 3-4, 1217-1244. DOI 10.1007/s00209-016-1640-y; zbl 1378.11097; MR3520002; arxiv 1406.3168.
  • 4. N. Borne, Une formule de Riemann-Roch équivariante pour les courbes. Canad. J. Math. 55 (2003), no. 4, 693-710. DOI 10.4153/CJM-2003-029-2; zbl 1066.14052; MR1994069.
  • 5. C. J. Bushnell and G. Henniart, The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2006. zbl 1100.11041; MR2234120.
  • 6. L. Caputo and S. Vinatier, Gauss sums, Jacobi sums and cyclotomic units related to torsion Galois modules. arxiv 1402.3795.
  • 7. T. Chinburg, Galois structure of de Rham cohomology of tame covers of schemes. Ann. of Math. (2) 139 (1994), no. 2, 443-490. DOI 10.2307/2946586; zbl 0828.14007; MR1274097.
  • 8. C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics XI, Interscience Publishers, a division of John Wiley \& Sons, New York (1962). zbl 0131.25601; MR0144979.
  • 9. P. Deligne, Les constantes des équations fonctionnelles des fonctions L, in: Modular functions of one variable II (Proc. Internat. Summer School, Univ. Antwerp, 1972), Lecture Notes in Math. 349, Springer, Berlin, (1973), 501-597. zbl 0271.14011; MR0349635.
  • 10. P. Deligne, Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie SGA $4 \frac{1}{2}$. Avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier, Lecture Notes in Math., Vol. 569. Springer-Verlag, Berlin, 1977. DOI 10.1007/BFb0091516; zbl 0345.00010; MR0463174.
  • 11. B. Erez, Geometric trends in Galois module theory, in: A. J. Scholl (ed.) et al., Galois representations in arithmetic algebraic geometry (Proceedings of the symposium, Durham, UK, July 9-18, 1996), Lond. Math. Soc. Lect. Note Ser. 254, Cambridge University Press, Cambridge, (1998), 115-145. zbl 0931.11047; MR1696473.
  • 12. A. Fröhlich, Galois module structure of algebraic integers. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Band 1. Springer-Verlag, Berlin, 1983. zbl 0501.12012; MR0717033.
  • 13. C. Greither, A note on normal bases for the square root of the codifferent in local fields, preprint (2015).
  • 14. R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. zbl 0367.14001; MR0463157.
  • 15. H. Johnston, Explicit integral Galois module structure of weakly ramified extensions of local fields, Proc. Am. Math. Soc. 143 (2015), 5059-5071. DOI 10.1090/proc/12634; zbl 1331.11104; MR3411126; arxiv 1204.2133.
  • 16. N. M. Katz, Local-to-global extensions of representations of fundamental groups, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 69-106. DOI 10.5802/aif.1069; zbl 0564.14013; MR0867916.
  • 17. B. Köck, Galois structure of Zariski cohomology for weakly ramified covers of curves. Amer. J. Math. 126 (2004), no. 5, 1085-1107. DOI 10.1353/ajm.2004.0037; zbl 1095.14027; MR2089083; arxiv math/0207124.
  • 18. B. Köck and J. Tait, Faithfulness of actions on Riemann-Roch spaces, Can. J. Math. 67 (2015), no. 4, 848-869. DOI 10.4153/CJM-2014-015-2; zbl 1320.14046; MR3361016; arxiv 1404.3135.
  • 19. S. Lang, Algebraic number theory. Springer, New York, 1994. zbl 0811.11001; MR1282723.
  • 20. G. Laumon, Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131-210. DOI 10.1007/BF02698937; zbl 0641.14009; MR0908218.
  • 21. A. Marmora, Facteurs epsilon $p$-adiques. Compos. Math. 144 (2008), no. 2, 439-483. DOI 10.1112/S0010437X07002990; zbl 1162.12003; MR2406118.
  • 22. J. S. Milne, Étale cohomology. Princeton Mathematical Series, 33. Princeton University Press, Princeton, N.J., 1980. zbl 0433.14012; MR0559531.
  • 23. J. Neukirch, Algebraische Zahlentheorie. Springer-Verlag, Berlin, 1992. zbl 0747.11001; MR3444843.
  • 24. R. Pink, Euler-Poincaré formula in equal characteristic under ordinariness assumptions, Manuscripta Math. 102 (2000), 1-24. DOI 10.1007/PL00005849; zbl 1010.11034; MR1771225.
  • 25. D. Ramakrishnan and R. J. Valenza, Fourier analysis on number fields. Graduate Texts in Mathematics, 186. Springer-Verlag, New York, 1999. zbl 0916.11058; MR1680912.
  • 26. J.-P. Serre, Courps locaux. Publications de l'Institut de Mathématique de l'Université de Nancago VIII, Hermann, 1962. MR0150130.
  • 27. J.-P. Serre, Représentations linéaires des groupes finis. Deuxième édition, Hermann, Paris, 1971. zbl 0223.20003; MR0352231.
  • 28. J. Tate, Residues of differentials on curves. Ann. Sci. École Norm. Sup. 1 (1968), 149-159. DOI 10.24033/asens.1162; zbl 0159.22702; MR0227171.
  • 29. A. Weil, Basic number theory. Third edition. Die Grundlehren der Mathematischen Wissenschaften, 144. Springer-Verlag, New York, 1974. zbl 0326.12001; MR0427267.

Affiliation

Fischbacher-Weitz, Helena
Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
Köck, Bernhard
Mathematical Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
Marmora, Adriano
Institut de Recherche Mathématique Avancée, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg, France

Downloads