The Betti map associated to a section of an abelian scheme
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The Betti map associated to a section of an abelian scheme Y. André1 · P. Corvaja2 · U. Zannier3
Received: 22 February 2017 / Accepted: 5 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Given a point ξ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of ξ . When (A, ξ ) varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often ξ takes a torsion value (for instance, Manin’s theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when ξ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira–Spencer map associated to (A, ξ ) (assuming A without fixed part, and Zξ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension ≤ 3, and study in detail the case of jacobians of families of hyperelliptic curves. Our
B P. Corvaja
[email protected] Y. André [email protected] U. Zannier [email protected]
1
Institut Mathématique de Jussieu - Paris Rive Gauche, 4, Place Jussieu, 75005 Paris, France
2
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Via delle Scienze, 206, Udine, Italy
3
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
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Y. André et al.
main application, obtained in collaboration with Z. Gao, states that if A → S is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space Ag has dimension at least g, then the Betti map of any non-torsion section ξ is generically a submersion, so that ξ −1 Ators is dense in S(C). Mathematics Subject Classification 11G · 11J · 14K · 14M Contents 1 Introduction I: motivation . . . . . . . . . . . . . . . . . . . . . . . . 2 Introduction II: main results . . . . . . . . . . . . . . . . . . . . . . . 3 The Betti map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 From Betti to Kodaira–Spencer . . . . . . . . . . . . . . . . . . . . . 5 Generic rank of the Betti map . . . . . . . . . . . . . . . . . . . . . . 6 Abelian schemes with End S A = Z and dμ A ≥ g . Main theorem . . . 7 Case study I: g ≤ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Case study II: complex hyperelliptic case . . . . . . . . . . . . . . . . 9 Case study III: real hyperelliptic case . . . . . . . . . . . . . . . . . . 10 Appendix by Z. Gao: an application of the pure Ax-Schanuel Theorem References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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