The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny

TL;DR

This paper calculates the probability that a randomly chosen elliptic curve over the p-adic numbers has a rational 3-torsion point or a 3-isogeny, using p-adic volume integrals and modular curve analysis.

Contribution

It provides explicit density formulas for elliptic curves with rational 3-torsion or 3-isogenies over p-adic fields, extending to other primes.

Findings

Derived explicit p-adic volume integrals for densities.

Analyzed modular curves related to 3-torsion and 3-isogenies.

Explored densities for torsion points of primes greater than 3.

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 $\ell$-torsion for $\ell>3$ prime.