Elliptic curves with missing Frobenius traces

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In 1976, Lang and Trotter conjectured an asymptotic formula for the number $\pi_{E,r}(X)$ of primes $p \leq X$ of good reduction for which the Frobenius trace at $p$ associated to $E$ is equal to a given fixed integer $r$. We investigate elliptic curves $E$ over $\mathbb{Q}$ that have a missing Frobenius trace, i.e. for which the counting function $\pi_{E,r}(X)$ remains bounded as $X \rightarrow \infty$, for some $r \in \mathbb{Z}$. In particular, we classify all elliptic curves $E$ over $\mathbb{Q}(t)$ that have a missing Frobenius trace.