On elliptic curves y=x−nx with rank zero |
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| Authors: | Keqin Feng Maosheng Xiong |
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| Institution: | a Department of Mathematical Sciences, Tsinghua University, Beijing 10084, China
b Department of Mathematics, University of Illinois at Urbana-Champaign, USA |
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| Abstract: | In this paper we determine all elliptic curves En:y2=x3−n2x with the smallest 2-Selmer groups Sn=Sel2(En(Q))={1} and Sn′=Sel2(En′(Q))={±1,±n}(En′:y2=x3+4n2x) based on the 2-descent method. The values of n for such curves En are described in terms of graph-theory language. It is well known that the rank of the group En(Q) for such curves En is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers. |
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| Keywords: | Elliptic curve Rank Selmer group 2-descent method Odd graph |
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