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The Manin-Mumford conjecture: a brief survey. (English) Zbl 1073.14525
Summary: The Manin-Mumford conjecture asserts that if \(K\) is a field of characteristic zero, \(C\) a smooth proper geometrically irreducible curve over \(K\), and \(J\) the Jacobian of \(C\), then for any embedding of \(C\) in \(J\), the set \(C(K) \cap J(K)_{\text{tors}}\) is finite. Although the conjecture was proved by M. Raynaud [Invent. Math. 71, 207–233 (1983; Zbl 0564.14020)] in 1983, and several other proofs have appeared since, a number of natural questions remain open, notably concerning bounds on the size of the intersection and the complete determination of \(C(K) \cap J(K)_{\text{tors}}\) for special families of curves \(C\). The first half of this survey paper presents the Manin-Mumford conjecture and related general results, while the second describes recent work mostly dealing with the above questions.

MSC:
14G25 Global ground fields in algebraic geometry
11G10 Abelian varieties of dimension \(> 1\)
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H40 Jacobians, Prym varieties
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