Elliptic curve variants of the least quadratic nonresidue problem and Linnik’s theorem

2018 
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. Assuming the Generalized Riemann Hypothesis (GRH) for the convolved symmetric power L-functions L(s,SymiE 1 ⊗SymjE 2) 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)) ⋅log NE1NE2)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,SymkE 1) 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)/(2p) ∈ I and p < ((21 + o(1)) ⋅ μ−2log(N E1/μ))2.
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