On the μ-invariant of p-adic L-functions attached to elliptic curves with complex multiplication (original) (raw)
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A note on Iwasawa µ-invariants of elliptic curves
Bulletin of the Brazilian Mathematical Society, New Series, 2010
Suppose that E 1 and E 2 are elliptic curves defined over Q and p is an odd prime where E 1 and E 2 have good ordinary reduction. In this paper, we generalize a theorem of Greenberg and Vatsal [3] and prove that if E 1 [ p i ] and E 2 [ p i ] are isomorphic as Galois modules for i = μ(E 1), then μ(E 1) ≤ μ(E 2). If the isomorphism holds for i = μ(E 1) + 1, then both the curves have same μ-invariants. We also discuss one numerical example.
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.
Katz p-adic L-functions, congruence modules and deformation of Galois representations
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.
Iwasawa theory and p-adic L-functions over Zp^2-extensions
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.