On the 𝑙-adic Iwasawa 𝜆-invariant in a 𝑝-extension

Friedman, Eduardo; Sands, Jonathan W.

doi:10.1090/S0025-5718-1995-1308453-8

On the $l$-adic Iwasawa $\lambda$-invariant in a $p$-extension
HTML articles powered by AMS MathViewer

by Eduardo Friedman and Jonathan W. Sands PDF

Math. Comp. 64 (1995), 1659-1674 Request permission

Abstract:

For distinct primes l and p, the Iwasawa invariant $\lambda _l^ -$ stabilizes in the cyclotomic ${\mathbb {Z}_p}$-tower over a complex abelian base field. We calculate this stable invariant for $p = 3$ and various l and K. Our motivation was to search for a formula of Riemann-Hurwitz type for $\lambda _l^ -$ that might hold in a p-extension. From our numerical results, it is clear that no formula of such a simple kind can hold. In the course of our calculations, we develop a method of computing $\lambda _l^ -$ for an arbitrary complex abelian field and, for $p = 3$, we make effective Washington’s theorem on the stabilization of the l-part of the class group in the cyclotomic ${\mathbb {Z}_p}$-extension. A new proof of this theorem is given in the appendix.

References

J. Buhler, R. Crandall, R. Ernvall, and T. Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp. 61 (1993), no. 203, 151–153. MR 1197511, DOI 10.1090/S0025-5718-1993-1197511-5
Nancy Childress, Examples of $\lambda$-invariants, Manuscripta Math. 68 (1990), no. 4, 447–453. MR 1068267, DOI 10.1007/BF02568777
D. S. Dummit, D. Ford, H. Kisilevsky, and J. W. Sands, Computation of Iwasawa lambda invariants for imaginary quadratic fields, J. Number Theory 37 (1991), no. 1, 100–121. MR 1089792, DOI 10.1016/S0022-314X(05)80027-7
R. Ernvall and T. Metsänkylä, Cyclotomic invariants for primes between $125\,000$ and $150\,000$, Math. Comp. 56 (1991), no. 194, 851–858. MR 1068819, DOI 10.1090/S0025-5718-1991-1068819-7
R. Ernvall and T. Metsänkylä, Cyclotomic invariants for primes to one million, Math. Comp. 59 (1992), no. 199, 249–250. MR 1134727, DOI 10.1090/S0025-5718-1992-1134727-7
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
Eduardo C. Friedman, Ideal class groups in basic $\textbf {Z}_{p_{1}}\times \cdots \times \textbf {Z}_{p_{s}}$-extensions of abelian number fields, Invent. Math. 65 (1981/82), no. 3, 425–440. MR 643561, DOI 10.1007/BF01396627
Robert Gold, Examples of Iwasawa invariants. II, Acta Arith. 26 (1974/75), no. 3, 233–240. MR 374085, DOI 10.4064/aa-26-3-233-240
Robert Gold and Manohar Madan, Kida’s theorem for a class of nonnormal extensions, Proc. Amer. Math. Soc. 104 (1988), no. 1, 55–60. MR 958043, DOI 10.1090/S0002-9939-1988-0958043-X

Sur les invariants lambda d’Iwasawa des corps abéliennes

Kenkichi Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258. MR 83013, DOI 10.1007/BF03374563

On

extensions of algebraic number fields

65

Kenkichi Iwasawa, Riemann-Hurwitz formula and $p$-adic Galois representations for number fields, Tohoku Math. J. (2) 33 (1981), no. 2, 263–288. MR 624610, DOI 10.2748/tmj/1178229453
J.-F. Jaulent and A. Michel, Classes des corps surcirculaires et des corps de fonctions, Séminaire de Théorie des Nombres, Paris, 1989–90, Progr. Math., vol. 102, Birkhäuser Boston, Boston, MA, 1992, pp. 141–162 (French). MR 1476734, DOI 10.1007/978-1-4757-4269-5_{1}1
Yûji Kida, $l$-extensions of CM-fields and cyclotomic invariants, J. Number Theory 12 (1980), no. 4, 519–528. MR 599821, DOI 10.1016/0022-314X(80)90042-6
L. V. Kuz′min, Some duality theorems for cyclotomic $\Gamma$-extensions over algebraic number fields of CM-type, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 3, 483–546, 733–734 (Russian). MR 541659
Heinrich-Wolfgang Leopoldt, Eine Verallgemeinerung der Bernoullischen Zahlen, Abh. Math. Sem. Univ. Hamburg 22 (1958), 131–140 (German). MR 92812, DOI 10.1007/BF02941946
Hans-Georg Rück, Hasse-Witt-invariants and dihedral extensions, Math. Z. 191 (1986), no. 4, 513–517. MR 832808, DOI 10.1007/BF01162340
W. Sinnott, On the $\mu$-invariant of the $\Gamma$-transform of a rational function, Invent. Math. 75 (1984), no. 2, 273–282. MR 732547, DOI 10.1007/BF01388565
Warren M. Sinnott, On $p$-adic $L$-functions and the Riemann-Hurwitz genus formula, Compositio Math. 53 (1984), no. 1, 3–17. MR 762305
W. Sinnott, On a theorem of L. Washington, Astérisque 147-148 (1987), 209–224, 344. Journées arithmétiques de Besançon (Besançon, 1985). MR 891429
Samuel S. Wagstaff Jr., The irregular primes to $125000$, Math. Comp. 32 (1978), no. 142, 583–591. MR 491465, DOI 10.1090/S0025-5718-1978-0491465-4
Lawrence C. Washington, The class number of the field of $5^{n}$th roots of unity, Proc. Amer. Math. Soc. 61 (1976), no. 2, 205–208, (1977). MR 424752, DOI 10.1090/S0002-9939-1976-0424752-6
Lawrence C. Washington, Class numbers and $\textbf {Z}_{p}$-extensions, Math. Ann. 214 (1975), 177–193. MR 364182, DOI 10.1007/BF01352651
Lawrence C. Washington, The non-$p$-part of the class number in a cyclotomic $\textbf {Z}_{p}$-extension, Invent. Math. 49 (1978), no. 1, 87–97. MR 511097, DOI 10.1007/BF01399512
Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2

On Sinnott ’s proof on the vanishing of the Iwasawa invariant