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

Let E1 and E2 be Q-nonisogenous, semistable elliptic curves over Q, having respective conductors NE1 and NE2 and both without complex multiplication. For each prime p, denote by aEi(p) := p+1−#Ei(Fp) the trace of Frobenius. Under the assumption of the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power L-functions L(s, SymE1 ⊗SymE2) where i, j ∈ {0, 1, 2}, we prove an explicit result that can be stated succinctly as follows: there exists a prime p ∤ NE1NE2 such that aE1(p)aE2(p) < 0 and p < ( (32 + o(1)) · logNE1NE2 )2 . This improves and makes explicit a result of Bucur and Kedlaya. Now, if I ⊂ [−1, 1] is a subinterval with Sato-Tate measure μ and if the symmetric power L-functions L(s, SymE1) are functorial and satisfy GRH for all k ≤ 8/μ, we employ similar techniques to prove an explicit result that can be stated succinctly as follows: there exists a prime p ∤ NE1 such that aE1(p)/(2 √ p) ∈ I and p < ( (21 + o(1)) · μ log(NE1/μ) )2 .