The Manin-Mumford Conjecture: A Brief Survey

Abstract

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)∩J(K)tors is finite. Although the conjecture was proved by Raynaud 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)∩J(K)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. 1991 Mathematics Subject Classification 11G10, 11G30, 11G35, 14G25, 14H25, 14H40.