Elliptic curves with missing Frobenius traces

Nathan Jones; Kevin Vissuet

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arXiv · 2108.08727

Elliptic curves with missing Frobenius traces

Nathan Jones·Kevin Vissuet

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Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. In 1976, Lang and Trotter conjectured an asymptotic formula for the number $π_{E,r}(X)$ 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 $π_{E,r}(X)$ 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.

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BibTeX RIS

Nathan Jones, Kevin Vissuet. 2021-08-19. Elliptic curves with missing Frobenius traces. https://arxiv.org/abs/2108.08727

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