Analytic and rational sections of relative semi-abelian varieties

The hyperbolicity statements for subvarieties and complement of hypersurfaces in abelian varieties admit arithmetic analogues, due to Faltings, Ann. Math. 133 (1991) (and for the semiabelian case, Vojta, Invent. Math. 126 (1996); Amer. J. Math. 121 (1999)). In Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 29 (2018) by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud’s theorem (Manin–Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-)abelian scheme $\mathscr{A} \to B$ over an affine algebraic curve $B$. These sections form a group; while the group of the rational sections (the Mordell–Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory.

Keywords

Legendre elliptic, semi-abelian scheme, Diophantine geometry, Nevanlinna theory

2010 Mathematics Subject Classification

Primary 14K99, 32H25. Secondary 14J27.

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Junjiro Noguchi was supported by Grant-in-Aid for Scientific Research (C) 19K03511.

Received 12 May 2020

Accepted 30 July 2020

Published 10 February 2022