On the Lang–Trotter conjecture for two elliptic curves

Abstract

Following Lang and Trotter, we describe a probabilistic model that predicts the distribution of primes p with given Frobenius traces at p for two fixed elliptic curves over $$\mathbb {Q}$$ . In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the $$\ell $$ -adic limits determining the $$\ell $$ -factors of the universal constant, unlike the Lang–Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves, which extends the main results of Fouvry and Murty (Supersingular primes common to two elliptic curves, number theory (Paris, 1992), London Mathematical Society Lecture Note Series, vol 215, Cambridge University Press, Cambridge, 1995) and Akbary et al. (Acta Arith 111(3):239–268, 2004), following the work of David et al. (Math Ann 368(1–2):685–752, 2017).