Computation of the 𝑝-part of the ideal class group of certain real abelian fields

Sumida-Takahashi, Hiroki

doi:10.1090/S0025-5718-07-01926-6

Computation of the $p$-part of the ideal class group of certain real abelian fields
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Math. Comp. 76 (2007), 1059-1071 Request permission

Abstract:

Under Greenberg’s conjecture, we give an efficient method to compute the $p$-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the $p$-part of the ideal class group of the maximal real subfield of $\mathbf {Q}(\sqrt {-f},\zeta _{p^{n+1}})$ in the range $1 <f<200$ and $5 \le p <100000$. In order to explain our method, we show an example whose ideal class group is not cyclic.

References

M. Aoki and T. Fukuda, An algorithm for computing $p$-class groups of abelian number fields, Lecture Notes in Comput. Sci. 4076, Springer, Berlin, 2006, pp. 56–71.
Joe Buhler, Richard Crandall, Reijo Ernvall, Tauno Metsänkylä, and M. Amin Shokrollahi, Irregular primes and cyclotomic invariants to 12 million, J. Symbolic Comput. 31 (2001), no. 1-2, 89–96. Computational algebra and number theory (Milwaukee, WI, 1996). MR 1806208, DOI 10.1006/jsco.1999.1011
Bruce Ferrero and Ralph Greenberg, On the behavior of $p$-adic $L$-functions at $s=0$, Invent. Math. 50 (1978/79), no. 1, 91–102. MR 516606, DOI 10.1007/BF01406470
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
Roland Gillard, Remarques sur les unités cyclotomiques et les unités elliptiques, J. Number Theory 11 (1979), no. 1, 21–48 (French). MR 527759, DOI 10.1016/0022-314X(79)90018-0
Ralph Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), no. 1, 263–284. MR 401702, DOI 10.2307/2373625
Humio Ichimura, Local units modulo Gauss sums, J. Number Theory 68 (1998), no. 1, 36–56. MR 1492887, DOI 10.1006/jnth.1997.2206
Humio Ichimura and Hiroki Sumida, On the Iwasawa invariants of certain real abelian fields, Tohoku Math. J. (2) 49 (1997), no. 2, 203–215. MR 1447182, DOI 10.2748/tmj/1178225147
Kenkichi Iwasawa, Lectures on $p$-adic $L$-functions, Annals of Mathematics Studies, No. 74, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0360526
Kenkichi Iwasawa, On $\textbf {Z}_{l}$-extensions of algebraic number fields, Ann. of Math. (2) 98 (1973), 246–326. MR 349627, DOI 10.2307/1970784
James S. Kraft and René Schoof, Computing Iwasawa modules of real quadratic number fields, Compositio Math. 97 (1995), no. 1-2, 135–155. Special issue in honour of Frans Oort. MR 1355121
Tomio Kubota and Heinrich-Wolfgang Leopoldt, Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen, J. Reine Angew. Math. 214(215) (1964), 328–339 (German). MR 163900
B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
Manabu Ozaki, On the cyclotomic unit group and the ideal class group of a real abelian number field. I, II, J. Number Theory 64 (1997), no. 2, 211–222, 223–232. MR 1453211, DOI 10.1006/jnth.1997.2086
Hiroki Sumida-Takahashi, Computation of the Iwasawa invariants of certain real abelian fields, J. Number Theory 105 (2004), no. 2, 235–250. MR 2040156, DOI 10.1016/j.jnt.2003.11.002
Hiroki Sumida-Takahashi, The Iwasawa invariants and the higher $K$-groups associated to real quadratic fields, Experiment. Math. 14 (2005), no. 3, 307–316. MR 2172709
Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7