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Quadratic Chabauty and p-adic Gross--Zagier

Sachi Hashimoto

Transactions of the American Mathematical Society, 2023-05, Vol.376 (5), p.3725 [Periódico revisado por pares]

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Título:
Quadratic Chabauty and p-adic Gross--Zagier
Autor: Sachi Hashimoto
É parte de: Transactions of the American Mathematical Society, 2023-05, Vol.376 (5), p.3725
Descrição: 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.
Idioma: Inglês

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