相关论文: Arithm\'etique des courbes elliptiques a r\'educti…

Let $p\ge 5$ be a prime number and $E/\mathbf{Q}$ an elliptic curve with good supersingular reduction at $p$. Under the generalized Heegner hypothesis, we investigate the $p$-primary subgroups of the Tate--Shafarevich groups of $E$ over…

This is a translation of a research announcement by Anas G. Nasybullin from 1976, in which he states formulas for the p-primary part of the Tate-Shafarevich group of an elliptic curve in cyclotomic $\Z_p$-extensions of number fields.

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 let $p$ be a prime of good supersingular reduction. Attached to $E$ are pairs of Iwasawa invariants $\mu_p^\pm$ and $\lambda_p^\pm$ which encode arithmetic properties of $E$ along the cyclotomic…

We study this subject by first proving that the p-primary subgroup of the classical Selmer group for an elliptic curve with good, ordinary reduction at a prime p has a very simple and elegant description which involves just the Galois…

A genus one curve of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We describe a general method for investigating the invariant theory of such models. We use it to explain how we found our…

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb…

Let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_2$-extension over $\mathbb{Q}$. For each integer $n\geq1$, let $\mathbb{Q}_n$ denote the unique subfield in $\mathbb{Q}_\infty$ such that $[\mathbb{Q}_\infty:\mathbb{Q}]=2^n$. Denote by…

Fix an elliptic curve $E$ over $\mathbb Q$ of rank $1$. In this paper, we develop an explicit numerical criterion, comparable to Gold's criterion, that determines whether the Iwasawa invariants of the elliptic curve at a good (ordinary or…

In this paper, for every prime $p$ and every $0\le n\le \infty$, we classify the structure of the torsion subgroup of the group of $\mathbb{Q}_p(\mu_{p^n})$-rational points of elliptic curves over $\mathbb{Q}_p$ with good reduction, where…

Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes $p\le x$ such…

In this paper, we study the Fitting ideals of Selmer groups over finite subextensions in the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of an elliptic curve over $\mathbb{Q}$. Especially, we present a proof of the "weak main…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be an imaginary quadratic field. Consider an odd prime $p$ at which $E$ has good supersingular reduction with $a_p(E)=0$ and which is inert in $K$. Under the assumption…

For CM elliptic curve over rational field with analytic rank one, for any potential good ordinary prime p, not dividing the number of roots of unity in the complex multiplication field, we show the p-part of its Shafarevich-Tate group has…

We study the asymptotic growth of the p-primary component of the Shafarevich-Tate group in the cyclotomic direction at any odd prime of good supersingular reduction, generalizing work of Kobayashi. This explains formulas obtained by…

Let $p$ be an odd prime number, $E$ an elliptic curve defined over a number field. Suppose that $E$ has good reduction at any prime lying above $p$, and has supersingular reduction at some prime lying above $p$. In this paper, we construct…

We show that the number of copies of ${\Bbb Q}_p/{\Bbb Z}_p$ in the Tate-Shafarevich group of an elliptic curve $E$ over ${\Bbb Q}$ with complex multipication, is at most $2p - g$, where $g$ is the rank of $E({\Bbb Q})$, and for all…

Let $p$ be an odd prime and let $E$ be an elliptic curve defined over a number field $F$ with good reduction at primes above $p$. In this survey article, we give an overview of some of the important results proven for the fine Selmer group…

Let p be an odd prime. Suppose that E is a modular elliptic curve/Q with good ordinary reduction at p. Let Q_{oo} denote the cyclotomic Z_p-extension of Q. It is conjectured that Sel_E(Q_{oo}) is a cotorsion Lambda-module and that its…

We investigate the $\lambda$-invariants of Mazur--Tate elements of elliptic curves defined over the field of rational numbers at primes of additive reduction. We explain their growth and how these invariants relate to other better…