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arXiv · 1810.04809

Ramification in Division Fields and Sporadic Points on Modular Curves

Hanson Smith
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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.

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Hanson Smith. 2018-10-11. Ramification in Division Fields and Sporadic Points on Modular Curves. https://arxiv.org/abs/1810.04809

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