### Mordell-Weil Groups and Selmer Groups of Two Types of Elliptic Curves, by DeRong QIU and XianKe ZHANG

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/Q), S^{(\varphi)}(E/Q), and
S^{(\widehat{\varphi})}(E/Q) are explicitly determined, e.g.,
S^{(2)}(E_{+1}/Q)= (Z/2Z)^2; (Z}}/2Z)^3; or
(Z/2Z)^4 when $p\equiv 5; 1 or 3; or 7 (mod 8) respectively.
(2) When p\equiv 5 (3 ,5 for \sigma =-1) (mod 8),
it is proved that the Mordell- Weil group E(Q) \cong
Z/2Z \oplus Z/2Z having rank 0, and
Shafarevich-Tate group \247\272(E/Q)[2]=0.
(3) In any case, the sum of
rankE(Q)and dimension of \247\272(E/Q)[2] is given,
e.g., 0; 1; 2 when p\equiv 5; 1 or 3; 7 (mod 8)
for \sigma =1.
(4) The Kodaira symbol, the torsion subgroup E(K)_{tors}
for any number field K , etc. are also obtained.

DeRong QIU and XianKe ZHANG <xianke@tsinghua.edu.cn , xzhang@math.tsinghua.edu.cn>