The Iwasawaμ-invariant ofp-adic HeckeL-functions

Compositio Mathematica, 2011

We prove vanishing of theμ-invariant of thep-adic KatzL-function in N. M. Katz [p-adic L-functions for CM fields, Invent. Math.49(1978), 199–297].

Nagoya Mathematical Journal, 1977

Letpbe a prime. If one adjoins toQallpn-th roots of unity forn= 1,2,3, …, then the resulting field will contain a unique subfieldQ∞such thatQ∞is a Galois extension ofQwith Gal (Q∞/Q)Zp, the additive group ofp-adic integers. We will denote Gal (Q∞/Q) byΓ. In a previous paper [6], we discussed a conjecture relatingp-adicL-functions to certain arithmetically defined representation spaces forΓ. Now by using some results of Iwasawa, one can reformulate that conjecture in terms of certain other representation spaces forΓ. This new conjecture, which we believe may be more susceptible to generalization, will be stated below.

L-Functions and Galois Representations

Archiv der Mathematik, 2004

We prove that the submodule in K-theory which gives the exact value (up to Z * (p) ) of the L-function by the Beilinson regulator map at noncritical values for Hecke characters of imaginary quadratic fields K with cl(K) = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soulé regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsen's conjecture, an upper bound for #H 2 et (OK [1/S], Vp(m)) in terms of the valuation of these p-adic L-functions, where V p denotes the p-adic realization of a Hecke motive. *

Journal of Number Theory, 1976

An elementary proof is given for the existence of the Kubota-Leopoldt padic L-functions. Also, an explicit formula is obtained for these functions, and a relationship between the values of the padic and classical L-functions at positive integers is discussed.

2011

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of Qp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2.

Nagoya Mathematical Journal, 1975

Let p be a prime. If one adjoins to Q all pn -th roots of unity for n = 1, 2, 3, …, then the resulting field will contain a unique subfield Q ∞ such that Q ∞ is a Galois extension of Q with Gal the additive group of p-adic integers. We will denote Gal(Q ∞/Q ) by Γ.

2006

L-Functions and Arithmetic, 1991

Although the two-variable main conjecture for imaginary quadratic fields has been successfully proven by Rubin [R] using brilliant ideas found by Thaine and Kolyvagin, we still have some interest in studying the new proof of a special case of the conjecture, i.e., the anticyclotomic case given by Mazur and the second named author of the present article ([M-T], [Tl]). Its interest lies firstly in surprizing amenability of the method to the case of CM fields in place of imaginary quadratic fields and secondly in its possible relevance for non-abelian cases. In this short note, we begin with a short summary of the result in [M-T] and [Tl] concerning the Iwasawa theory for imaginary quadratic fields, and after that, we shall give a very brief sketch of how one can generalize every step of the proof to the general CM-case. At the end, coming back to the original imaginary quadratic case, we remove some restriction of one of the main result in [M-T]. The idea for this slight amelioration to [M-T] is to consider deformations of Galois representations not only over finite fields but over any finite extension of Q p. Throughout the paper, we assume that p > 2.

Pacific Journal of Mathematics, 2012