zbMATH — the first resource for mathematics

Trivial Selmer groups and even partitions of a graph. (English) Zbl 1114.11051
The elliptic curve corresponding to the congruent number problem is \(E_n:y^2= x^3-n^2x\). If one uses descent via 2-isogeny one produces Selmer groups \(S_n, S_n'\). When \(n\equiv\pm 3\pmod 8\), a theorem of K. Feng and M. Xiong [J. Number Theory, 109, No. 1, 1–26 (2004; Zbl 1076.11036)] gives a criterion for these to contain only elements arising from the 2-torsion of \(E_n\). This criterion requires that a certain directed graph \(G(n)\) should be “odd”. The paper investigates the probability that a random graph on \(k\) vertices is “odd”, giving an exact formula for this probability. This is then related to the probability that descent via 2-isogeny will show that \(E_n\) has rank zero. The paper refers to, but mis-quotes, the reviewer’s paper [Invent. Math. 111, No. 1, 171–195 (1993; Zbl 0808.11041)]. This latter work concerns the “full” 2-dismit producing a different, Selmer group, which is generally smaller and therefore more likely to prove that the curve has rank zero.

11G05 Elliptic curves over global fields
05C30 Enumeration in graph theory
11N45 Asymptotic results on counting functions for algebraic and topological structures
Full Text: EuDML Link