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2001 Computing Torsion Points on Curves
Bjorn Poonen
Experiment. Math. 10(3): 449-466 (2001).


Let $X$ be a curve of genus $g \ge 2$ over a field $k$ of characteristic zero. Let $X\box0 A$ 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 $X(\kbar)$ 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 $k=\funnyQ$, $g=2$, 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 $p \ge 5$, then $\#T \le\ 2 p^3 + 2 p^2 + 2p + 8$.


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Bjorn Poonen. "Computing Torsion Points on Curves." Experiment. Math. 10 (3) 449 - 466, 2001.


Published: 2001
First available in Project Euclid: 25 November 2003

zbMATH: 1063.11017
MathSciNet: MR1917430

Primary: 11G30
Secondary: 11G10 , 14H25 , 14K15 , 14K20

Keywords: $p$-adic integrals , cuspidal torsion packet , Greenberg functor , Greenberg transform, , hyperelliptic torsion packet , Kummer surface , Manin--Mumford conjecture , torsion packet , torsion point

Rights: Copyright © 2001 A K Peters, Ltd.


Vol.10 • No. 3 • 2001
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