# Mordell-Weil groups and Selmer groups of two types of elliptic curves

@article{Qiu2001MordellWeilGA, title={Mordell-Weil groups and Selmer groups of two types of elliptic curves}, author={Derong Qiu and Xianke Zhang}, journal={arXiv: Number Theory}, year={2001} }

Consider elliptic curves $ E=E_\sigma: y^2 = x (x+\sigma p) (x+\sigma q), $ where$ \sigma =\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(\phi)}(E/{\mathbf{Q})}$, and $\ S^{(\hat{\phi})}(E/{\mathbf{Q})} $ are explicitly determined, e.g., $\ S^{(2)}(E_{+1}/{\mathbf{Q}})= $ $({\mathbf{Z}}/2{\mathbf{Z}})^2; $ $ ({\mathbf{Z}}/2{\mathbf{Z}})^3; $ or $ ({\mathbf{Z}}/2{\mathbf{Z}})^4 $ when $p\equiv 5; 1 $ or $3; $ or $ 7 ({\mathrm{mod}} 8…

## 4 Citations

### On the Selmer groups and Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one

- Mathematics
- 2012

Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $\varphi -$Selmer groups, Shafarevich-Tate groups ($ \varphi - $ and $ 2-$part) and their dual ones as well the Mordell-Weil groups of…

### On several families of elliptic curves with arbitrary large Selmer groups

- Mathematics
- 2010

AbstractIn this paper, we calculate the $$
\phi (\hat \phi )
$$-Selmer groups S(φ)(E/ℚ) and $$
S^{(\hat \phi )} (E'/\mathbb{Q})
$$ of elliptic curves y2=x(x+εpD)(x+εqD) via the descent method. In…

### Elliptic curves of twin-primes over Gauss field and Diophantine Equations

- Mathematics
- 2000

Let $p, q$ be twin prime numbers with $q-p=2$ . Consider the elliptic curves E=E_\sigma: y^2 = x (x+\sigma p)(x+\sigma q) . (\sigma =\pm 1). E=E_\sigma is also denoted as E_+ or E_- when \sigma =…

### A graphical method to calculate Selmer groups of several families of non-CM elliptic curves

- Mathematics
- 2009

In this paper, we extend the ideas of Feng [F1], Feng-Xiong [FX] and Faulkner-James [FJ] to calculate the Selmer groups of elliptic curves $ y^{2} = x (x + \varepsilon p D) (x + \epsilon q D). $

## References

SHOWING 1-10 OF 12 REFERENCES

### On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3

- Mathematics
- 1985

The elliptic curve y2 = 4x3 28x + 25 has rank 3 over Q. Assuming the WeilTaniyama conjecture for this curve, we show that its L-series L(s) has a triple zero at s = 1 and compute lim, _I L(s)/(s 1)3…

### On the Equation Y 2 = X(X 2 + p)

- Mathematics
- 1984

Generators are found for the group of rational points on the title curve for all primes p = 5 (mod 8) less than 1,000. The rank is always 1 in accordance with conjectures of Seltner and Mordell. Some…

### Explicit 4-descents on an elliptic curve

- Mathematics
- 1996

where f(X,Z) is a binary quartic form (or quartic for short) with integer coefficients. One wishes to know whether equation (1) has a Q-rational point and if so to exhibit one. One can often show…

### ON THE EQUATION Y(2)=(X+P)(X(2)+P(2))

- Mathematics
- 1994

textabstractIn this paper the family of elliptic curves over Q given by the equation y2 = (x + p)(x2 + p2) is studied. It is shown that for p a prime number = ±3 mod 8, the only rational solution to…

### Advanced Topics in the Arithmetic of Elliptic Curves

- Mathematics
- 1994

In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational…

### Lectures on elliptic curves

- Mathematics
- 1991

Introduction 1. Curves of genus: introduction 2. p-adic numbers 3. The local-global principle for conics 4. Geometry of numbers 5. Local-global principle: conclusion of proof 6. Cubic curves 7.…

### Algorithms for Modular Elliptic Curves

- Computer Science, Mathematics
- 1992

This book presents a thorough treatment of many algorithms concerning the arithmetic of elliptic curves with remarks on computer implementation and an extensive set of tables giving the results of the author's implementations of the algorithms.