Quadratic Chabauty and 𝑝-adic Gross–Zagier

Hashimoto, Sachi

doi:10.1090/tran/8862

Quadratic Chabauty and $p$-adic Gross–Zagier
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by Sachi Hashimoto;

Trans. Amer. Math. Soc. 376 (2023), 3725-3760

DOI: https://doi.org/10.1090/tran/8862

Published electronically: February 28, 2023

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Abstract:

Let $X$ be a quotient of the modular curve $X_0(N)$ whose Jacobian $J_X$ is a simple factor of $J_0(N)^{new}$ over $\mathbf {Q}$. Let $f$ be the newform of level $N$ and weight $2$ associated with $J_X$; assume $f$ has analytic rank $1$. We give analytic methods for determining the rational points of $X$ using quadratic Chabauty by computing two $p$-adic Gross–Zagier formulas for $f$. Quadratic Chabauty requires a supply of rational points on the curve or its Jacobian; this new method eliminates this requirement. To achieve this, we give an algorithm to compute the special value of the anticyclotomic $p$-adic $L$-function of $f$ constructed by Bertolini, Darmon, and Prasanna [Duke Math. J. 162 (2013), pp. 1033–1148], which lies outside of the range of interpolation.

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