COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES
Glasgow mathematical journal, Tome 59 (2017) no. 1, pp. 77-108

Voir la notice de l'article provenant de la source Cambridge University Press

We give an explicit description of the stable reduction of superelliptic curves of the form y n =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.
BOUW, IRENE I.; WEWERS, STEFAN. COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES. Glasgow mathematical journal, Tome 59 (2017) no. 1, pp. 77-108. doi: 10.1017/S0017089516000057
@article{10_1017_S0017089516000057,
     author = {BOUW, IRENE I. and WEWERS, STEFAN},
     title = {COMPUTING {L-FUNCTIONS} {AND} {SEMISTABLE} {REDUCTION} {OF} {SUPERELLIPTIC} {CURVES}},
     journal = {Glasgow mathematical journal},
     pages = {77--108},
     year = {2017},
     volume = {59},
     number = {1},
     doi = {10.1017/S0017089516000057},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000057/}
}
TY  - JOUR
AU  - BOUW, IRENE I.
AU  - WEWERS, STEFAN
TI  - COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES
JO  - Glasgow mathematical journal
PY  - 2017
SP  - 77
EP  - 108
VL  - 59
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000057/
DO  - 10.1017/S0017089516000057
ID  - 10_1017_S0017089516000057
ER  - 
%0 Journal Article
%A BOUW, IRENE I.
%A WEWERS, STEFAN
%T COMPUTING L-FUNCTIONS AND SEMISTABLE REDUCTION OF SUPERELLIPTIC CURVES
%J Glasgow mathematical journal
%D 2017
%P 77-108
%V 59
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000057/
%R 10.1017/S0017089516000057
%F 10_1017_S0017089516000057

[1] 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] 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] 3. Arzdorf, K. and Wewers, S., Another proof of the semistable reduction theorem. Preprint, arXiv:1211.4624 (2012). Google Scholar

[4] 4. Börner, M., Bouw, I. I. and Wewers, S., The functional equation for L-functions of hyperelliptic curves, arXiv:1504.00508 (2015). Google Scholar | DOI

[5] 5. Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Number 21 in Ergebnisse der Mathematik und ihrer Grenzgebiete (Springer-Verlag, Berlin, 1990). Google Scholar | DOI

[6] 6. Chênevert, G., Some remarks on Frobenius and Lefschetz in étale cohomology, Unpublished seminar notes (2004). Google Scholar

[7] 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. Google Scholar | DOI

[8] 8. Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus, Publ. Math. IHES 36 (1969), 75–109. Google Scholar | DOI

[9] 9. Dokchitser, T., Computing special values of motivic L-functions, Experiment. Math., 13 (2) (2004), 137–149. Google Scholar | DOI

[10] 10. Dokchitser, T., De Jeu, R. and Zagier, D., Numerical verification of Beilinson's conjecture for K of hyperelliptic curves, Compositio Math. 142 (2) (2006) 339–373. Google Scholar | DOI

[11] 11. Gerritzen, L., Herrlich, F. and Van Der Put, M., Stable n-pointed trees of projective lines, Indag. Math. 91 (2) (1988), 131–163. Google Scholar | DOI

[12] 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] 13. Grothendieck, A. and Raynaud, M., Revêtements étales et groupe fondamental (SGA 1), Number 224 in LNM (Springer-Verlag, Berlin, 1971). Google Scholar | DOI

[14] 14. Harris, J. and Mumford, D., On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23–86. Google Scholar | DOI

[15] 15. Knudsen, F. F., The projectivity of the moduli space of stable curves, ii, Math. Scand. 52 (2) (1983), 161–199. Google Scholar | DOI

[16] 16. Liu, Q., Conducteur et discriminant minimal de courbes de genre 2, Compositio Math. 94 (1) (1994), 51–79. Google Scholar

[17] 17. Liu, Q., Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète, Trans. Amer. Math. Soc. 348 (1996), 4577–4610. Google Scholar | DOI

[18] 18. Liu, Q. and Lorenzini, D., Models of curves and finite covers, Compositio Math. 118 (1999), 61–102. Google Scholar | DOI

[19] 19. Milne, J. S., Étale cohomology (Princeton University Press, Princeton, NJ, 1980). Google Scholar

[20] 20. Ogg, A. P., Elliptic curves and wild ramification, Amer. J. Math. 89 (1) (1967), 1–21. Google Scholar | DOI

[21] 21. Saito, T., Conductor, discriminant, and the Noether formula of arithmetic surfaces, Duke Math. J. 57 (1) (1988), 151–173. Google Scholar | DOI

[22] 22. Serre, J-P., Cohomologie galoisienne, Number 5 in LNM (Springer, Berlin, 1964). Google Scholar

[23] 23. Serre, J-P., Corps locaux, Troisième édition, Publications de l'Université de Nancago, No. VIII (Hermann, Paris, 1968). Google Scholar

[24] 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] 25. Serre, J-P. and Tate, J., Good reduction of abelian varieties, Annals of Math. 88 (3) (1968), 492–517. Google Scholar | DOI

[26] 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] 27. Wiese, G., Galois representations, Lecture notes, (2008), available at math.uni.lu/~wiese. Google Scholar

Cité par Sources :