Rational points in periodic analytic sets and the Manin-Mumford conjecture
Let \(A\) be an abelian variety and \(X\) an algebraic subvariety of \(A\), both defined over a number field. Denote by \(A_{\mathrm{tor}}\) the group of torsion points of \(A\). Then the Zariski closure of \(X\cap A_{\mathrm{tor}}\) is contained in a finite union of translates of abelian subvarieties by torsion points. In particular, if \(X\) is not contained in any translate of an abelian subvariety of dimension \(>0\) by a torsion point, then \(X\cap A_{\mathrm{tor}}\) is finite. These statements are due to \textit{M. Raynaud} [Invent. Math. 74, 207--233 (1983; Zbl 0564.14020)]. In the special case when \(X\) is a curve of genus \(\geq 2\) and \(A\) its Jacobian, they solve the original Manin--Mumford Conjecture. The authors give a list of different proofs of Raynaud's Theorem and produce a new completely different proof of it. They combine an upper bound for the number of rational points on a transcendental subvariety of a real torus, due to the work of \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, No. 2, 337--357 (1989; Zbl 0718.11048)], \textit{J. Pila} [Q. J. Math. 55, 207--223 (2004; Zbl 1111.32004)], \textit{J.~Pila} and \textit{A.J.~Wilkie} [Duke Math. J. 133, No. 3, 591--616 (2006; Zbl 1217.11066)], with a lower bound for the degree of torsion points due to \textit{D. W. Masser} [Compos. Math. 53, 153--170 (1984; Zbl 0551.14015)] . The proof involves a number of auxiliary results of independent interest, including a study of semi algebraic curves in \({\mathbb {C}}^g\) lying on the inverse image of \(X\) under the analytic uniformization \({\mathbb {C}}^g\rightarrow A\) of \(A\).
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