𝑝-class groups of certain extensions of degree 𝑝

Wittmann, Christian

doi:10.1090/S0025-5718-04-01725-9

$p$-class groups of certain extensions of degree $p$
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Math. Comp. 74 (2005), 937-947 Request permission

Abstract:

Let $p$ be an odd prime number. In this article we study the distribution of $p$-class groups of cyclic number fields of degree $p$, and of cyclic extensions of degree $p$ of an imaginary quadratic field whose class number is coprime to $p$. We formulate a heuristic principle predicting the distribution of the $p$-class groups as Galois modules, which is analogous to the Cohen-Lenstra heuristics concerning the prime-to-$p$-part of the class group, although in our case we have to fix the number of primes that ramify in the extensions considered. Using results of Gerth we are able to prove part of this conjecture. Furthermore, we present some numerical evidence for the conjecture.

References

Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
Henri Cohen, Advanced topics in computational number theory, Graduate Texts in Mathematics, vol. 193, Springer-Verlag, New York, 2000. MR 1728313, DOI 10.1007/978-1-4419-8489-0
H. Cohen and H. W. Lenstra Jr., Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Noordwijkerhout, 1983) Lecture Notes in Math., vol. 1068, Springer, Berlin, 1984, pp. 33–62. MR 756082, DOI 10.1007/BFb0099440
Henri Cohen and Jacques Martinet, Étude heuristique des groupes de classes des corps de nombres, J. Reine Angew. Math. 404 (1990), 39–76 (French). MR 1037430
S.D. Fisher and M.N. Alexander, Matrices over a finite field, Amer. Math. Monthly 73 (1966), 639–641.
Georges Gras, Sur les $l$-classes d’idéaux dans les extensions cycliques relatives de degré premier $l$. I, II, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 1–48; ibid. 23 (1973), no. 4, 1–44 (French, with English summary). MR 360519, DOI 10.5802/aif.471
Frank Gerth III, Counting certain number fields with prescribed $l$-class numbers, J. Reine Angew. Math. 337 (1982), 195–207. MR 676052, DOI 10.1515/crll.1982.337.195
Frank Gerth III, Densities for ranks of certain parts of $p$-class groups, Proc. Amer. Math. Soc. 99 (1987), no. 1, 1–8. MR 866419, DOI 10.1090/S0002-9939-1987-0866419-3
Frank Gerth III, On $p$-class groups of cyclic extensions of prime degree $p$ of quadratic fields, Mathematika 36 (1989), no. 1, 89–102. MR 1014203, DOI 10.1112/S0025579300013590
Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028, DOI 10.1007/978-1-4612-0987-4
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
L. Rédei, Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper, J. Reine Angew. Math. 171 (1935), 55–60.