An algebraic version of a theorem of Kurihara

Let E/Q be an elliptic curve and let p be an odd supersingular prime for E. In this article, we study the simplest case of Iwasawa theory for elliptic curves, namely when E(Q) is finite, III(E/Q) has no p-torsion and the Tamagawa factors for E are all prime to p. Under these hypotheses, we prove that E(Q_n) is finite and make precise statemens about the size and structure of the p-power part of III(E/Q_n). Here Q_n is the n-th step in the cyclotomic Z_p-extension of Q

Arithmetic properties of Fredholm series for -adic modular forms

We study the relationship between recent conjectures on slopes of overconvergent -adic modular forms "near the boundary" of -adic weight space. We also prove in tame level 1 that the coeffcients of the Fredholm series of the U operator never vanish modulo , a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo using a modular interpretation of the mod reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.Accepted manuscrip

Kida's formula and congruences

We prove a formula (analogous to that of Kida in classical Iwasawa theory and generalizing that of Hachimori-Matsuno for elliptic curves) giving the analytic and algebraic p-adic Iwasawa invariants of a modular eigenform over an abelian p-extension of Q to its p-adic Iwasawa invariants over Q. On the algebraic side our methods, which make use of congruences between modular forms, yield a Kida-type formula for a very general class of ordinary Galois representations. We are further able to deduce a Kida-type formula for elliptic curves at supersingular primes

Arithmetic properties of Fredholm series for p-adic modular forms

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.Comment: Final version. Numbering in main body different different from previous version. To appear in Proc. Lon. Math. Soc. 25 pages, 7 table

Mazur-Tate elements of non-ordinary modular forms

We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope . We give examples illustrating the strange behavior which can occur in the high weight, high slope case

Slopes of modular forms and the ghost conjecture

We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".Comment: 17 pages. 2 figures. Minor changes from v1. Final version. To appear in IMRN. arXiv admin note: text overlap with arXiv:1607.0465

Variation of Iwasawa invariants in Hida families

Let r : G_Q -> GL_2(Fpbar) be a p-ordinary and p-distinguished irreducible residual modular Galois representation. We show that the vanishing of the algebraic or analytic Iwasawa mu-invariant of a single modular form lifting r implies the vanishing of the corresponding mu-invariant for all such forms. Assuming that the mu-invariant vanishes, we also give explicit formulas for the difference in the algebraic or analytic lambda-invariants of modular forms lifting r. In particular, our formula shows that the lambda-invariant is constant on branches of the Hida family of r. We further show that our formulas are identical for the algebraic and analytic invariants, so that the truth of the main conjecture of Iwasawa theory for one form in the Hida family of r implies it for the entire Hida family

pp-adic Gross-Zagier formula at critical slope and a conjecture of Perrin-Riou

Let $p$ be an odd prime. Given an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-D_K})$ where $p$ splits with $D_K>3$, and a $p$-ordinary newform $f \in S_k(\Gamma_0(N))$ such that $N$ verifies the Heegner hypothesis relative to $K$, we prove a $p$-adic Gross-Zagier formula for the critical slope $p$-stabilization of $f$ (assuming that it is non-$\theta$-critical). In the particular case when $f=f_A$ is the newform of weight $2$ associated to an elliptic curve $A$ that has good ordinary reduction at $p$, this allows us to verify a conjecture of Perrin-Riou. The $p$-adic Gross-Zagier formula we prove has applications also towards the Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one.Comment: 35 pages, minor updates. Comments most welcome