The density of elliptic curves over $\mathbb{Q}_p$ with a rational 3-torsion point or a rational 3-isogeny
Stevan Gajović, Lazar Radičević, Matteo Verzobio
TL;DR
This work computes exact $p$-adic densities for elliptic curves over $\mathbb{Q}_p$ to have a non-trivial $\mathbb{Q}_p$-rational $3$-torsion point or a $\mathbb{Q}_p$-rational degree-$3$ isogeny, expressing them as rational functions in $p$ depending on $p\bmod 3$ (with a special $p=3$ case). The authors couple modular-curve parametrizations of level-3 structures (via the Tate normal form and the Hessian/Hesse framework) with Igusa $p$-adic integration to treat both good and bad reduction uniformly. They extend the analysis to $\ell$-torsion for $\ell>3$, obtain asymptotic results as $p\to\infty$, and develop sieve-based methods to bound counts of curves over $\mathbb{Q}$ with rational $3$-torsion, including twists and short-interval settings. The results provide concrete densities that illuminate how often $3$-torsion phenomena occur locally, with implications for arithmetic statistics and cryptographic considerations around anomalous curves. The techniques—precise modular-curve parametrizations, $p$-adic integration, and covering-map analyses—offer a flexible framework for local-probability calculations in broader torsion and isogeny contexts.
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.