Congruent Elliptic Curves Research Articles

On the signed Selmer groups of congruent elliptic curves with semistable reduction at all primes above $p$

Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $\lambda $-invariants of these signed Selmer groups for tw

Congruent number problem and Goldfeld conjecture

We introduce recent progress on the congruent number problem, in particular, on Goldfeld conjecture for congruent elliptic curves.

Congruent number problem and elliptic curves

Let $n$ be a positive integer congruent to 5 modulo 8. Assume that all prime factors of $n$ are congruent to 1 modulo 4 and that $\\BQ(\\sqrt{-n})$ has no ideal class of order 4. In this article, we prove that $n$ is a congruent number, i.e., it is the area of a right triangle with rational side-lengths (see Theorem thm6.1 ). Moreover, we also survey results on BSD conjecture and Goldfeld conjecture for congruent elliptic curves $ny^2=x^3-x$.

The non-congruent numbers via Monsky’s formula

This paper gives some new families of non-congruent numbers with arbitrarily many prime divisors. The main idea is based on Monsky’s formula for the 2-Selmer rank of congruent elliptic curves.

Congruent elliptic curves with non-trivial Shafarevich-Tate groups: Distribution part

We study the distribution of a subclass congruent elliptic curve $E^{(n)}: y^2=x^3-n^2x$, where $n$ is congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We prove an independence of residue symbol property. Consequently we get the distribution of rank zero such $E^{(n)}$ with $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z /2\mathbb Z\big)^2$. We also obtain a lower bound of the number of such $E^{(n)}$ with rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z /2\mathbb Z\big)^{4}$.

Congruent elliptic curves with non-trivial Shafarevich-Tate groups

We study a subclass of congruent elliptic curves $E^{(n)}: y^2=x^3-n^2x$, where $n$ is a positive integer congruent to $1\pmod 8$ with all prime factors congruent to $1\pmod 4$. We characterize such $E^{(n)}$ with Mordell-Weil rank zero and $2$-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^2$. We also discuss such $E^{(n)}$ with 2-primary part of Shafarevich-Tate group isomorphic to $\big(\mathbb Z/2\mathbb Z \big)^{2k}$ with $k\ge2$.

On families of 9-congruent elliptic curves

We compute equations for the families of elliptic curves $9$-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of $9$-congruent elliptic curves over ${\mathbb Q}$, i.e. pairs of non-isogenous elliptic curves over

On second 2-descent and non-congruent numbers

We use the so-called second $2$-descent method to find several series of non-congruent numbers. We consider three different $2$-isogenies of the congruent elliptic curves and their duals, and find a necessary condition to estimate the size of the images o

On non-congruent numbers with 1 modulo 4 prime factors

In this paper, we use the 2-descent method to find a series of odd non-congruent numbers ≡ 1 (mod 8) whose prime factors are ≡ 1 (mod 4) such that the congruent elliptic curves have second lowest Selmer groups, which include Li and Tian’s result as special cases.