The density of elliptic curves over ℚ_p with a rational 3-torsion point or a rational 3-isogeny

2025/02/12 by Gajović, Stevan, Radičević, Lazar, Verzobio, Matteo
#11G07 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.08583

Abstract

We determine the probability that a random Weierstrass equation with coefficients in the p-adic integers defines an elliptic curve with a non-trivial 3-torsion point, or with a degree 3 isogeny, defined over the field of p-adic numbers. We determine these densities by calculating the corresponding p-adic volume integrals and analyzing certain modular curves. Additionally, we explore the case of ℓ-torsion for ℓ>3 prime.

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