Papers › Classification of torsion of elliptic curves over quartic fields
Classification of torsion of elliptic curves over quartic fields
Maarten Derickx, Filip Najman
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Let E be an elliptic curve over a quartic field K. By the Mordell-Weil theorem, E(K) is a finitely generated group. We determine all the possibilities for the torsion group E(K)ₜₒᵣ where K ranges over all quartic fields K and E ranges over all elliptic curves over K. We show that there are no sporadic torsion groups, or in other words, that all torsion groups either do not appear or they appear for infinitely many non-isomorphic elliptic curves E. Proving this requires showing that numerous modular curves X₁(m,n) have no non-cuspidal degree 4 points. We deal with almost all the curves using one of 3 methods: a method for the rank 0 cases requiring no computation; the Hecke sieve, a local method requiring computer-assisted computations; and the global method, an argument for the positive rank cases also requiring no computation. We deal with the handful of remaining cases using ad hoc methods.
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