Elliptic curves with missing Frobenius traces
Nathan Jones, Kevin Vissuet

TL;DR
This paper classifies elliptic curves over rational function fields that have a missing Frobenius trace, meaning their Frobenius trace count remains bounded, contrasting with the expected asymptotic growth predicted by Lang and Trotter.
Contribution
It provides a complete classification of elliptic curves over $\
Findings
Identifies all elliptic curves over $\
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Abstract
Let be an elliptic curve defined over . In 1976, Lang and Trotter conjectured an asymptotic formula for the number of primes of good reduction for which the Frobenius trace at associated to is equal to a given fixed integer . We investigate elliptic curves over that have a missing Frobenius trace, i.e. for which the counting function remains bounded as , for some . In particular, we classify all elliptic curves over that have a missing Frobenius trace.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Cryptography and Residue Arithmetic · Analytic Number Theory Research
