Elliptic curves with missing Frobenius traces

Publication date: 19 August 2021

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 $X ightarrow infty$, for some $r in mathbb{Z}$. In particular, we classify all elliptic curves $E$ over $mathbb{Q}(t)$ that have a missing Frobenius trace.

Has companion code repository: https://github.com/ncjones-uic/missingfrobeniustraces

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Galois representations (11F80)

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