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A MORDELL–LANG-TYPE PROBLEM FOR $\mathrm{GL}_{m}$

Part of: Polynomials and matrices Diophantine equations Arithmetic and non-Archimedean dynamical systems

Published online by Cambridge University Press:  04 October 2024

JASON BELL Open the ORCID record for JASON BELL [Opens in a new window] ,
DRAGOS GHIOCA Open the ORCID record for DRAGOS GHIOCA [Opens in a new window]  and
YIFENG HUANG Open the ORCID record for YIFENG HUANG [Opens in a new window]
JASON BELL
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada e-mail: jpbell@uwaterloo.ca
DRAGOS GHIOCA*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
YIFENG HUANG
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada e-mail: huangyf@math.ubc.ca
*
e-mail: dghioca@math.ubc.ca
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Abstract

We prove a nonabelian variant of the classical Mordell–Lang conjecture in the context of finite- dimensional central simple algebras. We obtain the following result as a particular case of a more general statement. Let K be an algebraically closed field of characteristic zero, let $B_1,\dots ,B_r\in \mathrm {GL}_m(K)$ be matrices with multiplicatively independent eigenvalues and let V be a closed subvariety of $\mathrm {GL}_m(K)$ not passing through zero. Then there exist only finitely many elements of $\mathrm {GL}_m(K)$ of the form $B_1^{n_1}\cdots B_r^{n_r}$ (as we vary $n_1,\dots ,n_r$ in $\mathbb {Z}$) lying on the subvariety V.

Keywords

finite-dimensional central simple algebrasMordell–Lang conjecture

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11C99: None of the above, but in this section 37P55: Arithmetic dynamics on general algebraic varieties

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 3 , June 2025 , pp. 433 - 444
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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