Abstract
For a wildly ramified p-extension E/F of algebraic function fields of one variable in an algebraically closed field k of characteristic p with Galois Group G, Nakajima obtained two exact sequences which determined implicitly the structure of the holomorphic semisimple differentials as k[G]-module. In this paper, in many cases, e.g., if there is a fully ramified prime, the structure is determined explicitly. Analogous results are obtained for p-extensions of ℤp-fields of CM-type. In the latter situation, if E/F is unramified, the structure of the minus part of the p-class group of E is determined as ℤp[G]-module.
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References
C. Chevalley and A. Weil, Über das Verhalten der Integrale 1. Gattung bei Automorphismen des Funktionenkörpers; Abh. Math. Sem. Univ. Hamburg 10, 358–361, (1934)
J. D'Mello and M. Madan, Class Group Rank Relations in ℤℓ-extensions, Manuscripta Math. 41, 75–107, (1983)
G. Ellingsrud and K. Lønsted, An Equivariant Lefschetz Formula for Finite Reductive Groups, Math. Ann. 251, 253–261, (1980)
R. Gold and M. Madan, An application of a Theorem of Deuring and Šafarevič, Math. Z. 191, 247–251, (1986)
R. Gold and M. Madan, Galois Representation of Iwasawa Modules, Preprint (1984)
A. Heller, Indecomposable Representations and the Loop-Space Operation, Amer. Math. Soc. Proceeding 12, 640–643, (1961)
A. Heller, The Loop-Space Functor in Homological Algebra, Amer. Math. Soc. Transactions 96, 382–394, (1960)
K. Iwasawa, Riemann-Hurwitz Formula and p-Adic Galois Representation for Number Fields, Tôhoku Math. Journ. 33, 263–288, (1981)
K. Iwasawa, On ℤℓ-extensions of Algebraic Number Fields, Ann. J. Math. 98, 246–326, (1973)
G. Janusz, Algebraic Number Fields, Academic Press 1977
E. Kani, The Galois-module structure of the space of holomorphic differentials of a curve, J. Reine Angew. Math. 367, 187–206, (1986)
Y. Kida, I-Extensions of CM-Fields and Cyclotomic Invariants, Journal of Number Theory 12, 519–528, (1980)
M. Madan, On a Theorem of M. Deuring and I. R. Šafarevič, Manuscripta Math. 23, 91–102, (1977)
S. Nakajima, On Galois module structure of the cohomology groups of an algebraic variety, Invent. Math. 75, 1–8, (1984)
S. Nakajima, Equivariant Form of the Deuring-Šafarevič Formula for Hasse-Witt Invariants, Math. Z. 190, 559–566, (1985)
S. Nakajima, Galois Module Structure of Cohomology Groups for Tamely Ramified Coverings of Algebraic Varieties, Journal of Number Theory 22, 115–123, (1986)
M. Rosen, Ambiguous Divisor Classes in Function Fields, Journal of Number Theory 9, 160–174, (1977)
J-P. Serre, Sur la Topologie des Variétés Algébriques en Caractéristique p, Symposium Internacional de Topología Algebraica, Universidad Nacional Autónoma de México y UNESCO 24–53, (1956)
J-P. Serre, Local Fields, Springer-Verlag, GTM 67, 1979.
D. Subrao, The p-Rank of Artin-Schreier Curves, Manuscripta Math. 16, 169–193, (1975)
T. Tamagawa, On Unramified Extensions of Algebraic Function Fields, Proc. Japan Acad. 27, 548–551, (1951)
R. Valentini, Some p-adic Galois Representations for Curves in Characteristic p, Math. Z. 192, 541–546, (1986)
R. Valentini and M. Madan, Automorphisms and Holomorphic Differentials in Characteristic p, Journal of Number Theory 13, 106–115, (1981)
R. Valentini, Representations of automorphisms on differentials of function fields of characteristic p, J. Reine Angew. Math. 335, 164–79, (1982)
K. Wingberg, Die Einseinheitengruppe von p-Erweiterungen regulärer ℘-adischer Zahlkörper als Galoismodul, J. Reine Angew. Math. 305, 206–214, (1979)
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Villa Salvador, G.D., Madan, M.L. Structure of semisimple differentials and p-class groups in ℤp-extensions. Manuscripta Math 57, 315–350 (1987). https://doi.org/10.1007/BF01437486
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DOI: https://doi.org/10.1007/BF01437486