Torsion packets on curves (Q2731699)

Let \(X\) be a smooth, proper and irreducible curve of genus \(g \geq 2\) over an algebraically closed field of characteristic zero. A torsion packet on \(X\) is an equivalence class under the relation on \(X\) by defining \(P \sim Q\) if and only if the divisors \(mP\) and \(mQ\) are linearly equivalent for some positive integer \(m\). In other words, the equivalence class containing \(P \in X\) is the preimage of the torsion points of the Jacobian of \(X\), \(J(X) \), by the Abel map \(X\to J(X)\) given by the point \(P\). The Manin-Mumford conjecture, proved by \textit{M. Raynaud} [Invent. Math. 71, 207-233 (1983; Zbl 0564.14020)], states that the every torsion packet on \(X\) is finite. The main result of this article is concerned with the number and size of torsion packets on a curve. The authors prove that there are at most finitely many torsion packets of size greater than 2 on \(X\), and there are infinitely many non-trivial torsion packets on \(X\) if and only if either \(g=2\) or \(g=3\) and \(X\) is both hyperelliptic and bielliptic. Moreover, they deduce that there is a constant \(M\), depending on \(X\), such that every torsion packet on \(X\) has size at most \(M\). The authors ask the open questions: Does the above constant \(M\) depend only on the genus of \(X\)? Does there exist a constant \(M(g,s)\) for fixed numbers \(g\geq 2\) and \(s\geq 3\) such that for all curves \(X\) of genus \(g\), the number of torsion packets on \(X\) of size at least \(s\) is bounded by \(M(g,s)\)? In particular, they show that for every \(n\geq 1\), there exists a curve \(X\) of genus \(g\geq 2\) such that \(X\) has at least \(n\) torsion packets each of size at least \(n\).