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- W4289417743 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." @default.
- W4289417743 created "2022-08-02" @default.
- W4289417743 creator A5052508390 @default.
- W4289417743 date "2018-10-10" @default.
- W4289417743 modified "2023-09-28" @default.
- W4289417743 title "Ramification in Division Fields and Sporadic Points on Modular Curves" @default.
- W4289417743 doi "https://doi.org/10.48550/arxiv.1810.04809" @default.
- W4289417743 hasPublicationYear "2018" @default.
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