Related papers: Zeta elements for elliptic curves and applications

Let $E$ be an elliptic curve over $\mathbb Q$ and let $p\geq5$ be a prime of good supersingular reduction for $E$. Let $K$ be an imaginary quadratic field satisfying a modified "Heegner hypothesis" in which $p$ splits, write $K_\infty$ for…

If E is an elliptic curve over Q and K is an imaginary quadratic field, there is an Iwasawa main conjecture predicting the behavior of the Selmer group of E over the anticyclotomic Z_p-extension of K. The main conjecture takes different…

Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the…

Let $E/\mathbf{Q}$ be a semistable elliptic curve of analytic rank one, and let $p>3$ be a prime for which $E[p]$ is irreducible. In this note, following a slight modification of the methods of Jetchev-Skinner-Wan, we use Iwasawa theory to…

The formula of the title relates $p$-adic heights of Heegner points and derivatives of $p$-adic $L$-functions. It was originally proved by Perrin-Riou for $p$-ordinary elliptic curves over the rationals, under the assumption that $p$ splits…

We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the…

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these…

Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field in…

We characterize the possible groups $E(\mathbb{Z}/N\mathbb{Z})$ arising from elliptic curves over $\mathbb{Z}/N\mathbb{Z}$ in terms of the groups $E(\mathbb{F}_p)$, with $p$ varying among the prime divisors of $N$. This classification is…

Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law. We present two methods of…

In 1987, B. Perrin-Riou formulated a Heegner point main conjecture for elliptic curves at primes of ordinary reduction. In this paper, we formulate an analogue of Perrin-Riou's main conjecture for supersingular primes. We then prove this…

Let $E/\mathbb{Q}$ be an elliptic curve and let $p$ be an odd prime of good reduction for $E$. Let $K$ be an imaginary quadratic field satisfying the classical Heegner hypothesis and in which $p$ splits. The goal of this paper is two-fold:…

Let $E_1$ and $E_2$ be $\overline{\mathbb{Q}}$-nonisogenous, semistable elliptic curves over $\mathbb{Q}$, having respective conductors $N_{E_1}$ and $N_{E_2}$ and both without complex multiplication. For each prime $p$, denote by…

Let $E/\mathbb{Q}$ be an elliptic curve of level $N$ and rank equal to $1$. Let $p$ be a prime of ordinary reduction. We experimentally study conjecture $4$ of B. Mazur and J. Tate in his article "Refined Conjectures of the Birch and…

Let $C$ be a smooth projective curve over $\mathbb{F}_q$ with function field $K$, $E/K$ a nonconstant elliptic curve and $\phi:\mathcal{E}\to C$ its minimal regular model. For each $P\in C$ such that $E$ has good reduction at $P$, i.e., the…

Let $p$ be a prime and $\mathcal{K}$ be an imaginary quadratic field. In this paper we generalize a recent construction of a new type of $p$-adic $L$-function and $p$-adic Waldspurger formula by Andreatta-Iovita for $p$ non-split in…

We prove an analogue of Kida's formula for the Iwasawa invariants of the Mazur-Tate elements attached to elliptic curves over $\mathbb{Q}$. Let $p$ be an odd prime and let $L/K$ be a Galois extension of abelian number fields with $p$-power…

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 an elliptic curve over the rational number field and a prime number $p$, we study the structure of the classical Selmer group of $p$-power torsion points. In our previous paper \cite{Ku6}, assuming the main conjecture and the…

Let $\mathbb{F}_q$ be a finite field of odd characteristic $p$. We exhibit elliptic curves over the rational function field $K = \mathbb{F}_q(t)$ whose Tate-Shafarevich groups are large. More precisely, we consider certain infinite…