Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem

Abstract

Let \(E_1\) and \(E_2\) be \(\overline{\mathbb{Q}}\)-nonisogenous, semistable elliptic curves over \(\mathbb{Q}\), having respective conductors \(N_{E_1}\) and \(N_{E_2}\) and both without complex multiplication. For each prime \(p\), denote by \(a_{E_i}(p) := p+1-\#E_i(\mathbb{F}_p)\) the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power \(L\)-functions \(L(s, \mathrm{Sym}^i E_1\otimes\mathrm{Sym}^j E_2)\) where \(i,j\in\{0,1,2\}\), we prove an explicit result that can be stated succinctly as follows: there exists a prime \(p\nmid N_{E_1}N_{E_2}\) such that \(a_{E_1}(p)a_{E_2}(p)<0\) and \[ p < \big( (32+o(1))\cdot \log N_{E_1} N_{E_2}\big)^2. \] This improves and makes explicit a result of Bucur and Kedlaya. Now, if \(I\subset[-1,1]\) is a subinterval with Sato-Tate measure \(\mu\) and if the symmetric power \(L\)-functions \(L(s, \mathrm{Sym}^k E_1)\) are functorial and satisfy GRH for all \(k \le 8/\mu\), we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime \(p\nmid N_{E_1}\) such that \(a_{E_1}(p)/(2\sqrt{p})\in I\) and \[ p < \left((21+o(1)) \cdot \mu^{-2}\log (N_{E_1}/\mu)\right)^2. \]