A Manin-Mumford theorem for the maximal compact subgroup of a universal vectorial extension of a product of elliptic curves

TL;DR

This paper proves a Manin-Mumford type theorem for the intersection of algebraic varieties with the maximal compact subgroup of a universal vectorial extension of elliptic curves, providing effective bounds and answering open questions.

Contribution

It extends the Manin-Mumford theorem to a new setting involving universal vectorial extensions and maximal compact subgroups, with effective bounds.

Findings

Proves a Manin-Mumford type statement for the intersection with maximal compact subgroups.

Provides effective bounds depending on the degree and dimension.

Answers questions posed by Corvaja-Masser-Zannier regarding intersections with real analytic subgroups.

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 a curve with real analytic subgroups of various algebraic groups. They prove finiteness in the situation of a single elliptic curve. 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. This seems new even if restricted to the classical Manin-Mumford statement.