In order to apply the ideas of Iwasawa theory to the symmetric square of a newform, we need to be able to define non‐archimedean analogues of its complex L‐series. The interpolated p‐adic L‐function is closely connected via a “Main Conjecture” with certain Selmer groups over the cyclotomic Zp‐extension of Q. In the p‐ordinary case these functions are well understood. In this article we extend the interpolation to an arbitrary set S of good primes (not necessarily satisfying ordinarity conditions). The corresponding S‐adic functions can be characterised in terms of certain admissibility criteria. We also allow interpolation at particular primes dividing the level of the newform. One interesting application is to the symmetric square of a modular elliptic curve E defined over Q. Our constructions yield p‐adic L‐functions at all primes of stable or semi‐stable reduction. If p is ordinary or multiplicative the corresponding analytic function is bounded; if p is supersingular our function behaves like log2(1 + T). 1991 Mathematics Subject Classification: 11F67, 11F66, 11F33, 11F30

We prove that the p-adic zeta-function constructed by Kubota and Leopoldt has the Dirichlet series expansion

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We define a topological space over the p-adic numbers, in which Euler products and Dirichlet series converge. We then show how the classical Riemann zeta function has a (p-adic) Euler product structure at the negative integers. Finally, as a corollary of these results, we derive a new formula for the non-Archimedean Euler-Mascheroni constant.

The arithmetic properties of modular forms and elliptic curves lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula. Three main steps are outlined: the first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. Finiteness results for big Selmer groups are then established. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture. As the first book on the subject, the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases.

We prove weak forms of Kato's K1-congruences for elliptic curves with complex multiplication, subject to two technical hypotheses. We next use Magma to calculate the µ-invariant measuring the discrepancy between the "motivic" and "automorphic" p-adic L-functions. Via the two-variable main conjecture, one can then estimate growth in this µ-invariant using arithmetic of the Z 2 p -extension.

Let E /Q be an elliptic curve, p > 3 a good ordinary prime for E, and K ∞ a p-adic Lie extension of a number field k. Under some standard hypotheses, we study the asymptotic growth in both the Mordell-Weil rank and Shafarevich-Tate group for E over a tower of extensions K n /k inside K ∞ ; we obtain lower bounds on the former, and upper bounds on the latter's size.

Let$E_{/{\mathbb{Q}}}$be a semistable elliptic curve, andp≠ 2 a prime of bad multiplicative reduction. For each Lie extension$\mathbb{Q}$FT/$\mathbb{Q}$with Galois groupG∞≅$\mathbb{Z}$p⋊$\mathbb{Z}$p×, we constructp-adicL-functions interpolating Artin twists of the Hasse–WeilL-series of the curveE. Through the use of congruences, we next prove a formula for the analytic λ-invariant over the false Tate tower, analogous to Chern–Yang Lee's results on its algebraic counterpart. If one assumes the Pontryagin dual of the Selmer group belongs to the$\mathfrak{M}_{\mathcal{H}}$(G∞)-category, the leading terms of its associated Akashi series can then be computed, allowing us to formulate a non-commutative Iwasawa Main Conjecture in the multiplicative setting.