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Computing torsion points on curves. (English) Zbl 1063.11017
Summary: Let \(X\) be a curve of genus \(g \geq 2\) over a field \(k\) of characteristic zero. Let \(X(A)\) be an Albanese map associated to a point \(P_0\) on \(X\). The Manin-Mumford conjecture, first proved by Raynaud, asserts that the set \(T\) of points in \(X(\overline{k})\) mapping to torsion points on \(A\) is finite. Using a \(p\)-adic approach, we develop an algorithm to compute \(T\), and implement it in the case where \(k=\mathbb Q\), \(g=2\), and \(P_0\) is a Weierstrass point. Improved bounds on \(\#T\) are also proved: for instance, in the context of the previous sentence, if in addition \(X\) has good reduction at a prime \(p \geq 5\), then \(\#T \leq\;2 p^3 + 2 p^2 + 2p + 8\).

MSC:
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H40 Jacobians, Prym varieties
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