Integral Representation of p -Class Groups In Zp -Extensions and the Jacobian Variety

López-Bautista, Pedro Ricardo ; Villa-Salvador, Gabriel Daniel

Canadian journal of mathematics, Tome 50 (1998) no. 6, pp. 1253-1272

Résumé

For an arbitrary finite Galois $p$ -extension $L/K$ of ${{\mathbb{Z}}_{p}}$ -cyclotomic number fields of $\text{CM}$ -type with Galois group $G=\text{Gal}(L/K)$ such that the Iwasawa invariants $\mu _{K}^{-},\,\mu _{L}^{-}$ are zero, we obtain unconditionally and explicitly the Galois module structure of $C_{L}^{-}\,(p)$ , the minus part of the $p$ -subgroup of the class group of $L$ . For an arbitrary finite Galois $p$ -extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G=\text{Gal}(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$ -torsion part of the Jacobian variety ${{J}_{L}}(p)$ associated to $L/k$ .

Détail

DOI : 10.4153/CJM-1998-061-8

Mots-clés : 11R33, 11R23, 11R58, 14H40, Zp-extensions, Iwasawa’s theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure

Bibliographie

[1] 1. Anderson, F.W. and Fuller, K.R., Rings and Categories of Modules. Graduate Texts in Math. , Springer-Verlag, Berlin-Heidelberg-New York, 1974. Google Scholar

[2] 2. Chevalley, C., Introduction to the Theory of Algebraic Functions of One Variable. Mathematical Surveys , Amer. Math. Soc., 1951. Google Scholar

[3] 3. Curtis, C.W. and Reiner, I., Methods of Representation Theory with Applications to Finite Groups and Orders, Vol. I. Pure and Appl. Math., Wiley-Interscience, New York, 1981. Google Scholar

[4] 4. Gold, R. and Madan, M. Galois representation of Iwasawa modules. Acta Arith. 46(1986), 243–255. Google Scholar

[5] 5. Iwasawa, K., On Γ-extensions of Algebraic Number Fields. Bull. Amer. Math. Soc. 65(1959), 183–226. Google Scholar

[6] 6. Iwasawa, K., On ℤ ℓ-extensions of Algebraic Number Fields. Ann. of Math. 98(1973), 246–326. Google Scholar

[7] 7. Iwasawa, K., Riemann-Hurwitz Formula and p-adic Galois Representation for Number Fields. Tôhoku Math. J. 33(1981), 263–288. Google Scholar

[8] 8. Karpilovsky, G., Group Representations, Vol. I. North-Holland Mathematics Studies 175, North-Holland, Amsterdam-London-New York-Tokyo, 1992. Google Scholar

[9] 9. López, R., Ph.D. Dissertation. Centro de Investigación y de Estudios Avanzados del I.P.N., February, 1998. Google Scholar

[10] 10. Nakajima, S., Equivariant Form of the Deuring- Šafarevič Formula for Hasse-Witt Invariants. Math. Z. 190(1985), 559–566. Google Scholar

[11] 11. Rzedowski, M., Villa, G. and Madan, M., Galois module structure of Tate modules. Math. Z. 224(1997), 77–101. Google Scholar

[12] 12. Sehgal, Sudarshan K., Topics in Group Rings. Pure and Appl. Math. 50, Marcel Dekker, Inc., New York, 1978. Google Scholar

[13] 13. Serre, J.P., Local Fields, Graduate Texts in Math. 67, Springer-Verlag, New York, 1979. Google Scholar

[14] 14. Serre, J.P., Algebraic Groups and Class Fields. Graduate Texts in Math. 117, Springer-Verlag, Berlin- Heidelberg-New York, 1988. Google Scholar

[15] 15. Suzuki, M., Group Theory I. Grundlehren Math. Wiss. 247, Springer-Verlag, Berlin-Heidelberg-New York, 1980. Google Scholar

[16] 16. Valentini, R., Some p-adic Galois Representations for Curves in Characteristic p. Math. Z. 192(1986), 541–545. Google Scholar

[17] 17. Villa, G. and Madan, M., Structure of Semisimple Differentials and p-Class Groups in ℤ p-extensions. Manuscripta Math. 57(1987), 315–350. Google Scholar

[18] 18. Villa, G., Integral representations of p-class groups in ℤ p-extensions, semisimple differentials and Jacobians. Arch.Math. 56(1991), 254–269. Google Scholar