Abstract
Let k be an algebraically closed field of characteristic p > 0. We consider the problem of lifting p-cyclic covers of \({\mathbb{P}^{1}_k}\) as p-cyclic covers C of the projective line over some discrete valuation field K under the condition that the wild monodromy is maximal. We answer positively the problem for covers birationally given by w p −w = t R(t) for any additive polynomial R(t). One gives further informations about the ramification filtration of the monodromy extension and in the case when p = 2, one computes the conductor exponent f (Jac(C)/K) and the Swan conductor sw(Jac(C)/K).
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References
Brumer A., Kramer K.: The conductor of an abelian variety. Compos. Math. 2(92), 227–248 (2005)
Chrétien P., Matignon M.: Maximal wild monodromy in unequal characteristic. J. Number Theory 133(4), 1389–1408 (2013)
Guardia J., Montes J., Nart E.: Higher Newton polygons in the computation of discriminants and prime ideal decomposition in number fields. J. de théorie des nombres de Bordeaux 2, 667–696 (2011)
Guralnick, R.: Monodromy groups of coverings of curves. In: Galois Groups and Fundamental Groups, Math. Sci. Res. Inst. Publ., vol.41, pp.1–46. Cambridge University Press, Cambridge (2003)
Huppert, B.: Endliche gruppen i. Grundlehren der Math. Wiss. 134, xii+793 (1967)
Kraus A.: Sur le défaut de semi-stabilité des courbes elliptiques à réduction additive. Manuscr. Math. 69, 353–385 (1990)
Lehr C., Matignon M.: Automorphism groups for p-cyclic covers of the affine line. Compos. Math. 5(141), 1213–1237 (2005)
Liu Q.: Algebraic Geometry and Arithmetic Curves. Oxford University Press, Oxford (2002)
Lockhart P., Rosen M.I., Silverman J.: An upper bound for the conductor of an abelian variety. J. Algebr. Geom. 2, 569–601 (1993)
Ogg A.P.: Elliptic curves and wild ramification. Am. J. Math. 89(1), 1–21 (1967)
Raynaud M.: p-groupes et réduction semi-stable des courbes. In: The Grothendieck Festschrift, vol. III. pp. 179–197 Birkhäuser, Basel (1990)
Serre J.-P.: Représentations linéaires des groupes finis. Hermann, Paris (1967)
Serre, J.-P.: Local fields. In: Graduate Text in Mathematics, vol. 67. Springer-Verlag, Newyork (1979)
Serre J.-P., Tate J.: Good reduction of abelian varieties. Ann. Math. 88, 492–517 (1968)
Stichtenoth H.: Über die Automorphismengruppe eines algebraischen Funktionenkörpers von Primzahlcharakteristik. I, II. Eine Abschätzung der Ordnung der Automorphismengruppe.. Arch. Math. 24, 527–544 (1973)
Suzuki, M.: Group theory II. In: Grundlehren der Mathematischen Wissenschaften, vol. 248 Springer-Verlag, New York (1986)
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Chrétien, P. Lifting Artin–Schreier covers with maximal wild monodromy. manuscripta math. 143, 253–271 (2014). https://doi.org/10.1007/s00229-013-0636-8
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DOI: https://doi.org/10.1007/s00229-013-0636-8