Congruent Elliptic Curves with Non-trivial Shafarevich-Tate Groups: Distribution Part

2015/11/12 by Wang, Zhangjie
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1511.03813

Abstract

We study the distribution of a subclass congruent elliptic curve E⁽ⁿ⁾: y²=x³-n²x, 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 (\mathbb Z /2\mathbb Z)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 (\mathbb Z /2\mathbb Z)⁴.

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