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On the weak Leopoldt conjecture and Iwasawa \(\mu \)-invariants. (English) Zbl 07118310
Summary: Let \(k_\infty/k\) be a \(\mathbb Z_p\)-extension of a number field \(k\). We show that \(\mu\)-invariants of Iwasawa modules naturally attached to \(k_\infty/k\) are closely related to the weak Leopoldt conjecture and the primes of \(k\) which split completely in \(k_\infty\).
MSC:
11R23 Iwasawa theory
11R34 Galois cohomology
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[1] A. Brumer, On the units of algebraic number fields, Mathematika 14 (1967), 121-124. · Zbl 0171.01105
[2] B. Ferrero and L. Washington, The Iwasawa invariant µpvanishes for abelian number fields, Ann. of Math. 109 (1979), 377-395. · Zbl 0443.12001
[3] R. Gillard, Fonctions L p-adiques des corps quadratiques imaginaires et de leurs extensions abéliennes, J. Reine Angew. Math. 358 (1985), 76-91. · Zbl 0551.12011
[4] K. Iwasawa, On Zl-extensions of algebraic number fields, Ann. of Math. 98 (1973), 246-326. · Zbl 0285.12008
[5] U. Jannsen, Iwasawa modules up to isomorphism, in: Algebraic Number Theory—in Honor of K. Iwasawa, Adv. Stud. Pure Math. 17, Academic Press, 1989, 171-207.
[6] J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., Grundlehren Math. Wiss. 323, Springer, 2008. · Zbl 1136.11001
[7] O. Neumann, On p-closed algebraic number fields with restricted ramification, Math. USSR-Izv. 9 (1975), 243-254. · Zbl 0352.12011
[8] K. Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, in: Arithmetic Theory of Elliptic Curves (Cetraro, 1997), Lecture Notes in Math. 1716, Springer, 1999, 167-234. · Zbl 0991.11028
[9] L. Schneps, On the µ-invariant of p-adic L-functions attached to elliptic curves with complex multiplication, J. Number Theory 25 (1987), 20-33. · Zbl 0615.12018
[10] L. Washington, Introduction to Cyclotomic Fields, 2nd ed., Grad. Texts in Math. 83, Springer, 1997. · Zbl 0966.11047
[11] K. Wingberg, Galois groups of number fields generated by torsion points of elliptic curves, Nagoya Math. J. 104 (1986), 43-53. · Zbl 0621.12011
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