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 \(p \le X\) 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 \(\mathbb {Q}\) that arise as specializations of an elliptic curve \(\mathbb {E}\) over \(\mathbb {Q}(t)\) , all but finitely many of whose specializations have a missing Frobenius trace.
