Mathematical Research Letters

Volume 26 (2019)

Number 5

A variant of the Mordell–Lang conjecture

Pages: 1383 – 1392

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n5.a7

Authors

Dragos Ghioca (Department of Mathematics, University of British Columbia, Vancouver, BC, Canada)

Fei Hu (Department of Mathematics, University of British Columbia, Vancouver, BC, Canada; and Pacific Institute for the Mathematical Sciences, Vancouver, BC, Canada)

Thomas Scanlon (Department of Mathematics, University of California at Berkeley)

Umberto Zannier (Scuola Normale Superiore, Pisa, Italy)

Abstract

The Mordell–Lang conjecture (proven by Faltings, Vojta and McQuillan) states that the intersection of a subvariety $V$ of a semi-abelian variety $G$ defined over an algebraically closed field $\mathbb{k}$ of characteristic $0$ with a finite rank subgroup $\Gamma \leq G(\mathbb{k})$ is a finite union of cosets of subgroups of $\Gamma$. We explore a variant of this conjecture when $G = \mathbb{G}_a \times A$ for an abelian variety $A$ defined over $\mathbb{k}$.

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1fundingThe first author has been partially supported by a Discovery Grant from the National Science and Engineering Board of Canada. The second author was partially supported by a UBC-PIMS Postdoctoral Fellowship. The third author has been partially supported by grant DMS-1363372 of the United States National Science Foundation and a Simons Foundation Fellowship.

Received 27 April 2018

Accepted 8 June 2019

Published 27 November 2019