Density of Elliptic Curves over Number Fields with Prescribed Torsion Subgroups

2022/09/07 by Im, Bo-Hae, Kim, Hansol · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2209.02889

Abstract

Let K be a number field. For positive integers m and n such that m| n, we let \mathscrS_m,n be the set of elliptic curves E/K defined over K such that E(K)_tors⊇ \mathscrT≅ ℤ/mℤ× ℤ/nℤ. We prove that if the genus of the modular curve X₁(m,n) is 0, then `almost all' E∈ \mathscrS_m,n satisfy that E(K)_tors= \mathscrT, i.e., not larger than \mathscrT. In particular, if m=n=1, this result generalizes Duke's theorem over ℚ to arbitrary number fields K for the trivial torsion subgroup.

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