Details

Autor(en) / Beteiligte

• Wang, ZhangJie

Titel

Congruent elliptic curves with non-trivial Shafarevich-Tate groups

Ist Teil von

• Science China. Mathematics, 2016-11, Vol.59 (11), p.2145-2166

Ort / Verlag

Beijing: Science China Press

Erscheinungsjahr

2016

Quelle

SpringerLink

Beschreibungen/Notizen

• We study 2-primary parts ⅢX(E（n）/Q)[2∞] of Shafarevich-Tate groups of congruent elliptic curves E（n）: y2= x3-n2x, n ∈Q×/Q×2. Previous results focused on finding sufficient conditions for ⅢX(E（n）/Q)[2∞]trivial or isomorphic to（Z/2Z）2. Our first result gives necessary and sufficient conditions such that the 2-primary part of the Shafarevich-Tate group of E（n）is isomorphic to（Z/2Z）～2 and the Mordell-Weil rank of E（n） is zero,provided that all prime divisors of n are congruent to 1 modulo 4. Our second result provides sufficient conditions for ⅢX(E（n）/Q)[2∞]■（Z/2Z）2k, where k≥2.