Torsion Packets on Curves

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Abstract

The torsion packet containing a point P on a curve X over the field K is the set of all points Q in X(Ksep) such that the class of the divisor (P)-(Q) is torsion in the Jacobian J of X. The main result of our paper is the following theorem: If X is a curve of genus at least 2 over a field of characteristic zero, then there are only finitely many torsion packets of size greater than 2 on X, and there are infinitely many nontrivial torsion packets on X if and only if either:

We prove this result by studying the abelian varieties contained in the image of the map from Xn to Jn-1 given by (P1,...,Pn) → (P1 - P2,..., Pn-1-Pn), and then applying a result of Raynaud.

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