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$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields

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arxiv 2401.04510 v1 pith:V42OIGAR submitted 2024-01-09 math.NT math.AGmath.KT

François Brunault , Rob de Jeu , Hang Liu , Fernando Rodriguez Villegas This is my paper · ORCID
classification math.NTmath.AGmath.KT
keywords curvesfieldscubicelementsellipticfamiliesnon-abelianquartic
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We give two constructions of families of elliptic curves over cubic or quartic fields with three, respectively four, `integral' elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we prove a new criterion that does not ignore any torsion. We also verify Beilinson's conjecture numerically for just over 90 of the curves.

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  1. Bernoulli determinants and cuspidal subgroups

    math.NT 2026-07 accept novelty 6.0 of 10

    The order of the rational cuspidal class group of X_1(N) is given by an explicit product over even Dirichlet characters involving generalized Bernoulli numbers B_{2,χ}, valid for all N ≥ 5.

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