Papers › Elliptic curves with missing Frobenius traces

Elliptic curves with missing Frobenius traces

19 Aug 2021arXiv:2108.08727links table onlyarchive 2025-07-28

Nathan Jones, Kevin Vissuet

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Let E be an elliptic curve defined over ℚ. In 1976, Lang and Trotter conjectured an asymptotic formula for the number π_(E,r)(X) of primes p ≤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 ℚ that have a missing Frobenius trace, i.e. for which the counting function π_(E,r)(X) remains bounded as X →∞, for some r ∈ℤ. In particular, we classify all elliptic curves E over ℚ(t) that have a missing Frobenius trace.

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