A MANIN-MUMFORD THEOREM FOR THE MAXIMAL COMPACT SUBGROUP OF A UNIVERSAL VECTORIAL EXTENSION OF A PRODUCT OF ELLIPTIC CURVES

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Abstract

We study the intersection of an algebraic variety with the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves. For this intersection we show a Manin- Mumford type statement. This answers some questions posed by Corvaja-Masser-Zannier which arose in connection with their investigation of the intersection of an algebraic curve with the maximal compact subgroup of various algebraic groups. In particular they proved that these intersections are finite for universal vectorial extensions of elliptic curves. Using Khovanskii’s zero-estimates combined with a stratification result of Gabrielov-Vorobjov and recent work of the authors, we obtain effective bounds for this intersection that only depend on the degree of the algebraic variety and the dimension of the group. As a corollary, we obtain new uniform results of Manin-Mumford type for additive extensions of certain abelian varieties.

Bibliographical metadata

Original languageEnglish
Article numberrnz207
JournalInternational Mathematics Research Notices
Volume0
Early online date20 Nov 2019
DOIs
Publication statusPublished - 20 Nov 2019

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