Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p)=p+1-\#E({\mathbb{F}}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p)=2\sqrt{p}\cos {\theta }_p$ for a unique ${\theta }_p\in [0,\pi ]$. In this paper, we prove that the least prime $p$ such that ${\theta }_p\in [\alpha ,\beta ]\subset [0,\pi ]$ satisfies \[ p \ll \left(\frac{N_E}{β- α}\right)^A, \] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A>2$ are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's Theorem for arithmetic progressions, which states that the least prime $p\equiv a(\mathrm{mod}q)$ for $(a,q)=1$ satisfies $p\ll q^L$ for an absolute constant $L>0$.

arXiv comment11 pages; made minor modifications