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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.

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