On the μ-invariant of p-adic L-functions attached to elliptic curves with complex multiplication (original) (raw)
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.
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.
L-Invariant of the Symmetric Powers of Tate Curves
Publications of the Research Institute for Mathematical Sciences, 2009
Contents §1. Symmetric Tensor L-Invariant §1.1. Selmer groups §1.2. Greenberg's L-invariant §1.3. Factorization of L-invariants §2. Proof of Conjecture 0.1 under Potential Modularity When n=1 References In my earlier paper [H07] and in my talk at the workshop on "Arithmetic Algebraic Geometry" at RIMS in September 2006, we made explicit a conjectural formula of the L-invariant of symmetric powers of a Tate curve over a totally real field (generalizing the conjecture of Mazur-Tate-Teitelbaum, which is now a theorem of Greenberg-Stevens). In this paper, we prove the formula for Greenberg's L-invariant when the symmetric power is of adjoint type, assuming a standard conjecture (see Conjecture 0.1) on the ring structure of a Galois deformation ring of the symmetric powers. Let p be an odd prime and F be a totally real field of degree d < ∞ with integer ring O. Order all the prime factors of p in O as p 1 ,. .. , p e. Throughout this paper, we study an elliptic curve E /F over O with split multiplicative reduction at p j |p for j = 1, 2,. .. , b and ordinary good reduction at p j |p for j > b. Write F j = F p j for the p j-adic completion of F and q j ∈ F × j with j ≤ b for the Tate period of E /F j. Put Q j = N F p j /Q p (q j). When b = 0, as a convention, we assume that E /F has good ordinary reduction at every p-adic place of F. We assume throughout the paper that E does not have complex Communicated by A. Tamagawa.
p-Adic Aspects of Modular Forms, 2016
These are the expanded notes of a mini-course of four lectures by the same title given in the workshop "p-adic aspects of modular forms" held at IISER Pune, in June, 2014. We give a brief introduction of p-adic L-functions attached to certain types of automorphic forms on GLn with the specific aim to understand the p-adic symmetric cube L-function attached to cusp forms on GL 2 over rational numbers. Contents 1. What is a p-adic L-function? 2 2. The symmetric power L-functions 11 3. p-adic L-functions for GL 4 16 4. p-adic L-functions for GL 3 × GL 2 22 References 27 The aim of this survey article is to bring together some known constructions of the p-adic L-functions associated to cohomological, cuspidal automorphic representations on GL n /Q. In particular, we wish to briefly recall the various approaches to construct p-adic L-functions with a focus on the construction of the p-adic L-functions for the Sym 3 transfer of a cuspidal automorphic representation π of GL 2 /Q. We note that p-adic L-functions for modular forms or automorphic representations are defined using p-adic measures. In almost all cases, these p-adic measures are constructed using the fact that the L-functions have integral representations, for example as suitable Mellin transforms. Candidates for distributions corresponding to automorphic forms can be written down using such integral representations of the L-functions at the critical points. The well-known Prop. 2 is often used to prove that they are indeed distributions, which is usually a consequence of the defining relations of the Hecke operators. Boundedness of these distributions are shown by proving certain finiteness or integrality properties, giving the sought after p-adic measures. In Sect. 1, we discuss general notions concerning p-adic L-functions, including our working definition of what we mean by a p-adic L-function. As a concrete example, we discuss the construction of the p-adic L-functions that interpolate critical values of L-functions attached to modular forms. Manin [47]