Ramification in Division Fields and Sporadic Points on Modular Curves

TL;DR

This paper classifies valuations of torsion points on supersingular elliptic curves over number fields and uses this to analyze ramification and sporadic points on modular curves, extending results to composite levels.

Contribution

It provides a complete classification of torsion point valuations and establishes bounds on ramification needed for points of certain orders, linking to properties of sporadic points on modular curves.

Findings

Valuations of $p^n$-torsion points are classified by division polynomial coefficients.

Minimum ramification necessary for elliptic curves to have points of order $p^n$ is determined.

Sporadic points on $X_1(p^n)$ relate to supersingular curves with canonical subgroups.

Abstract

Consider an elliptic curve $E$ over a number field $K$. Suppose that $E$ has supersingular reduction at some prime $\mathfrak{p}$ of $K$ lying above the rational prime $p$. We completely classify the valuations of the $p^n$-torsion points of $E$ by the valuation of a coefficient of the $p^{\text{th}}$ division polynomial. We apply this description to find the minimum necessary ramification at $\mathfrak{p}$ in order for $E$ to have a point of exact order $p^n$. Using this bound we show that sporadic points on the modular curve $X_1(p^n)$ cannot correspond to supersingular elliptic curves without a canonical subgroup. We generalize our methods to $X_1(N)$ with $N$ composite.