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

TL;DR

This paper establishes explicit bounds on the smallest prime where elliptic curve Frobenius traces exhibit certain behaviors, under GRH assumptions, improving previous results and providing concrete estimates.

Contribution

It derives explicit bounds for primes related to elliptic curve Frobenius traces and extends results to Sato-Tate distributions under GRH, improving prior work.

Findings

Existence of a prime p with specific Frobenius trace properties below explicit bounds.

Improved explicit bounds compared to previous results of Bucur and Kedlaya.

Extension of bounds to Sato-Tate measure intervals under functoriality and GRH.

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 \[ p < \left((21+o(1)) \cdot \mu^{-2}\log (N_{E_1}/\mu)\right)^2. \]