Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $\ell \not= p$, the Frobenius automorphism (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism is defined over $\mathbb{Z}$ 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 $\bar{Q}$ that are non-isogenous and both without complex multiplication over $\overline{\mathbb{Q}}$. Motivated by the Lang-Trotter conjecture, 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 GRH) and unconditional upper bounds. 查看全文>>