Applications of the Square Sieve to a Conjecture of Lang and Trotter for a Pair of Elliptic Curves Over the Rationals

Abstract

Let E be an elliptic curve over ${\mathbb Q}$. Let p be a prime of good reduction for E. Then, for a prime $p \ne \ell$, the Frobenius automorphism associated with p (unique up to conjugation) acts on the $\ell$-adic Tate module of E. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of $\ell$. Its splitting field is called the Frobenius field of E at p. Let $E_1$ and $E_2$ be two elliptic curves defined over ${\mathbb Q}$ that are non-isogenous over ${\overline{{\mathbb Q}}}$ and also without complex multiplication over ${\overline{{\mathbb Q}}}$. In analogy with the well-known Lang-Trotter conjecture for a single elliptic curve, it is natural to consider the asymptotic behaviour of the function that counts the number of primes $p \le x$ such that the Frobenius fields of $E_1$ and $E_2$ at p coincide. In this short note, using Heath-Brown’s square sieve, we provide both conditional (upon the Generalized Riemann Hypothesis) and unconditional upper bounds.

To V. Kumar Murty: On the occasion of his sixtieth birthday