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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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