Frobenius distributions in short intervals for CM elliptic curves
Let \(E/\mathbb{Q}\) be an elliptic curve, and for prime \(p\), let \(a_p\) be the trace of Frobenius, which is equal to \(\#E(\mathbb{F}_p)-p-1\), where \(\#E(\mathbb{F}_p)\) is the number of solutions to the curve over the finite field \(\mathbb{F}_p\). By Hasse's theorem, \(|a_p| < 2\sqrt p\) for all primes \(p\), and the distribution of \(a_p\) on the interval \([-2\sqrt p, 2\sqrt p]\) is an important subject in the theory of elliptic curves. Let \(A_p\) denote \(a_p/(2\sqrt p)\). Then, by Hasse's theorem, \(-1 < A_p < 1\). By the work of Hecke in 1920 and the work of Deuring in 1950's, the probability density of \(A_p\) for \(E\) with complex multiplication (CM) was fully understood. The density of \(A_p\) for \(E\) without CM, which is different from CM cases, was known as the Sato-Tate conjecture, and it was fully answered by the collaborative work of Clozel, Harris, Shepherd-Barron, and Taylor in 2006. The authors of the paper under review are interested in the distribution of \(A_p\) for subsets of ``smaller'' measure. A measurable function \(f : (0,\infty) \to (0,\infty)\) is called \textit{regularly varying} if \(f(\lambda x)/f(x) \to C(\lambda)\) as \(x\to\infty\) for all \(\lambda >0\). Polynomials and \(\log(x)\) are examples of regularly varying functions. The authors of the paper under review prove the following: Suppose that \(E\) has CM, and \(f\) is a convex, differentiable, and regularly varying function. If \(x^{-1/2}\ll f(x) = o(1)\), then \[ \#\{ p\leq x : A_p > 1 - f(p) \} \sim \frac{\sqrt 2}{\pi(2+\alpha)}\, \sqrt{f(x)}\, \frac{x}{\log(x)} \] where \(\alpha\) is determined by \(f\), and the same asymptotic holds for \(\#\{ p\leq x : A_p < -1 + f(p) \}\).
scientific article; zbMATH DE number 6855847