Torsion hypersurfaces on abelian schemes and Betti coordinates
This paper completes the solution of the Zilber-Pink conjecture for abelian surface schemes over a curve. \par In more detail: Let \(\mathcal C\) be a complex curve, let \(\mathcal A\to\mathcal C\) be a scheme in abelian surfaces, and let \(\sigma:\mathcal C\to\mathcal A\) be a section of \(\mathcal A\to\mathcal C\) whose image is not contained in a proper group subscheme. It is then shown that there are only finitely many \(x\in\mathcal C\) for which \(\sigma(x)\) is torsion in the fiber \(\mathcal A_x\). \par More generally, let \(S\) be an irreducible complex quasi-projective variety, let \(\mathcal A\to S\) be a scheme in abelian surfaces, and let \(\sigma: S\to\mathcal A\) be a section whose image is not contained in a proper group subscheme. The paper shows that there are only finitely many hypersurfaces \(X\) in \(S\) such that \(\sigma(X)\) is torsion. \par When \(\mathcal A\) is isogenous to a product of elliptic curves, the former statement is already known [\textit{D. Masser} and \textit{U. Zannier}, Adv. Math. 259, 116--133 (2014; Zbl 1318.11075)]. Therefore the current paper deals only with the case that \(\mathcal A\) is simple (meaning that the generic fiber is geometrically simple). The proof also relies on the known case in which \(\mathcal A\to\mathcal C\) is simple and \(\mathcal C\) is a curve over a number field [\textit{D. Masser} and \textit{U. Zannier}, J. Eur. Math. Soc. (JEMS) 17, No. 9, 2379--2416 (2015; Zbl 1328.11068)]. \par The paper also resolves a question of Mazur. Let \(S\) be a quasi-projective scheme which is also a finite cover of the moduli space \(M_2\) of curves of genus \(2\) over \(\mathbb C\). This has dimension 3. Let \(\mathcal A\to S\) be the jacobian of the universal curve over \(S\), and let \(\sigma: S\to\mathcal A\) be a non-torsion section. The paper under review shows that there are infinitely many torsion curves in \(S\), and that they are dense in \(S\). In particular, confirming Mazur's expectation, they are not contained in finitely many hypersurfaces. \par Finally, the paper discusses a link with work of Griffiths and Harris on a higher dimensional extension of a theorem of Poncelet.
scientific article; zbMATH DE number 6923793
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