The asymptotic density of supersingular Fermat varieties
The arithmetic function \(\delta\) (m) gives the portion of primes modulo which the curve \(X^ m+Y^ m=1\) is supersingular; explicitly, \[ \delta(m)=2^{-d(m)}\quad if\quad 4| m,\quad \delta(m)=2^{- d(m)}\cdot \frac{2^{cs}-1}{2^ s-1}\quad otherwise, \] where \(2^{d(m)} \| \phi(m)\) and \(2^ c\) is the highest power of 2 dividing all the \(p_ i-1\), where \(p_ i (i=1,...,s)\) are the odd distinct primes dividing m. The second author (see the preceding review) proved that \((1/x)\sum_{m\leq x}\delta(m)\to 0,\) as \(x\to \infty\). Approximating \(\delta\) (m) by the multiplicative function \(g(m)=2^{- d(m)}\) in the mean, and applying a theorem of \textit{E. Wirsing} [Math. Ann. 143, 75-102 (1961; Zbl 0104.042)], the authors prove the asymptotic formula \[ \sum_{m\leq x}\delta(m)\quad \sim \quad D x (\log x)^{- 2/3}. \]
scientific article; zbMATH DE number 3867446