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COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES

Part of: Arithmetic algebraic geometry Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  10 June 2016

IRENE I. BOUW  and
STEFAN WEWERS
IRENE I. BOUW
Affiliation:
Institut für Reine Mathematik, Universität Ulm, Helmholtzstr. 18, 89081 Ulm e-mails: irene.bouw@uni-ulm.de, stefan.wewers@uni-ulm.de
STEFAN WEWERS
Affiliation:
Institut für Reine Mathematik, Universität Ulm, Helmholtzstr. 18, 89081 Ulm e-mails: irene.bouw@uni-ulm.de, stefan.wewers@uni-ulm.de
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Abstract

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We give an explicit description of the stable reduction of superelliptic curves of the form yn=f(x) at primes $\mathfrak{p}$ whose residue characteristic is prime to the exponent n. We then use this description to compute the local L-factor and the exponent of conductor at $\mathfrak{p}$ of the curve.

MSC classification

Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Secondary: 14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture) 11G20: Curves over finite and local fields
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 1 , January 2017 , pp. 77 - 108
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Abbes, A., Réduction semi-stable des courbes, in Courbes semi-stables et groupe fondamental en géométrie algébrique (Loeser, F., Bost, J-B. and Raynaud, M., Editors) number 187 in Progress in Math. (Birkhäuser, Basel, 2000), 59–110.Google Scholar
2. Arzdorf, K., Semistable reduction of cyclic covers of prime power degree, PhD Thesis (Leibniz Universität Hannover, 2012). http://edok01.tib.uni-hannover.de/edoks/e01dh12/716096048.pdf.Google Scholar
3. Arzdorf, K. and Wewers, S., Another proof of the semistable reduction theorem. Preprint, arXiv:1211.4624 (2012).Google Scholar
4. Börner, M., Bouw, I. I. and Wewers, S., The functional equation for L-functions of hyperelliptic curves, arXiv:1504.00508 (2015).CrossRefGoogle Scholar
5. Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Number 21 in Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer-Verlag, Berlin, 1990).CrossRefGoogle Scholar
6. Chênevert, G., Some remarks on Frobenius and Lefschetz in étale cohomology, Unpublished seminar notes (2004).Google Scholar
7. Deligne, P., Formes modulaires et représentation ℓ-adiques, in Séminaire Bourbaki, vol. 1968/69, number 179 in Lecture Notes in Math. (Springer Verlag, Berlin, 1971), 139–172.CrossRefGoogle Scholar
8. Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. IHES 36 (1969), 75–109.CrossRefGoogle Scholar
9. Dokchitser, T., Computing special values of motivic L-functions, Experiment. Math., 13 (2) (2004), 137–149.CrossRefGoogle Scholar
10. Dokchitser, T., de Jeu, R. and Zagier, D., Numerical verification of Beilinson's conjecture for K 2 of hyperelliptic curves, Compositio Math. 142 (2) (2006) 339–373.CrossRefGoogle Scholar
11. Gerritzen, L., Herrlich, F. and van der Put, M., Stable n-pointed trees of projective lines, Indag. Math. 91 (2) (1988), 131–163.CrossRefGoogle Scholar
12. Grothendieck, A., Groupes de Monodromie en Géometrie Algébrique (SGA7 I), Number 288 in Lecture Notes in Math. (Springer-Verlag, Berlin, 1972).Google Scholar
13. Grothendieck, A. and Raynaud, M., Revêtements étales et groupe fondamental (SGA 1), Number 224 in LNM (Springer-Verlag, Berlin, 1971).CrossRefGoogle Scholar
14. Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–86.CrossRefGoogle Scholar
15. Knudsen, F. F., The projectivity of the moduli space of stable curves, ii, Math. Scand. 52 (2) (1983), 161–199.CrossRefGoogle Scholar
16. Liu, Q., Conducteur et discriminant minimal de courbes de genre 2, Compositio Math. 94 (1) (1994), 51–79.Google Scholar
17. Liu, Q., Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète, Trans. Amer. Math. Soc. 348 (1996), 4577–4610.CrossRefGoogle Scholar
18. Liu, Q. and Lorenzini, D., Models of curves and finite covers, Compositio Math. 118 (1999), 61–102.CrossRefGoogle Scholar
19. Milne, J. S., Étale cohomology (Princeton University Press, Princeton, NJ, 1980).Google Scholar
20. Ogg, A. P., Elliptic curves and wild ramification, Amer. J. Math. 89 (1) (1967), 1–21.CrossRefGoogle Scholar
21. Saito, T., Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. J. 57 (1) (1988), 151–173.CrossRefGoogle Scholar
22. Serre, J-P., Cohomologie galoisienne, Number 5 in LNM (Springer, Berlin, 1964).Google Scholar
23. Serre, J-P., Corps locaux, Troisième édition, Publications de l'Université de Nancago, No. VIII (Hermann, Paris, 1968).Google Scholar
24. Serre, J-P., Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), Number 19 in Séminaire Delange-Pisot-Poitou (Théorie des Nombres), (1970), 1–15.Google Scholar
25. Serre, J-P. and Tate, J., Good reduction of abelian varieties, Annals of Math. 88 (3) (1968), 492–517.CrossRefGoogle Scholar
26. Wewers, S., Deformation of tame admissible covers of curves, in Aspects of Galois theory (Völklein, H., Editor) number 256 in LMS Lecture Note Series (Cambridge University Press, Cambridge, 1999), 239–282.Google Scholar
27. Wiese, G., Galois representations, Lecture notes, (2008), available at math.uni.lu/~wiese.Google Scholar