• Corpus ID: 10563458

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

  title={Mordell-Weil groups and Selmer groups of two types of elliptic curves},
  author={Derong Qiu and Xianke Zhang},
  journal={arXiv: Number Theory},
  • D. Qiu, Xianke Zhang
  • Published 15 March 2001
  • Mathematics, Computer Science
  • arXiv: Number Theory
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… 

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