The density of supersingular Fermat varieties
Let \(\delta\) (m) be the number of congruence classes mod m that have some power congruent to -1. This number measures the portion of primes modulo which the Jacobian of \(X^ m+Y^ m=1\) is supersingular [see e.g. \textit{N. Yui}, J. Algebra 65, 1-35 (1980; Zbl 0437.14015)]. The author first derives Yui's elementary formula for \(\delta\) (m) as a special case of a theorem on finite Abelian groups. The main theorem then is that, though \(\delta\) (m) takes on values arbitrarily close to 1, the average value goes to 0. This is shown by observing that \(\delta(m)\leq 4/2^{\omega(m)}\), where \(\omega\) (m) is the number of distinct primes dividing m. The size of \((1/N)\sum_{m\leq N}2^{-\omega(m)}\) is then estimated by standard techniques. In a subsequent paper (see the following review), \textit{W. Schwarz} and the author have determined the asymptotic behavior of \(\sum_{m\leq N}\delta(m)\).
scientific article; zbMATH DE number 3867445