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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.

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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