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THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

Part of: Arithmetic and non-Archimedean dynamical systems Arithmetic algebraic geometry

Published online by Cambridge University Press:  24 June 2019

Pietro Corvaja Open the ORCID record for Pietro Corvaja [Opens in a new window] ,
Dragos Ghioca Open the ORCID record for Dragos Ghioca [Opens in a new window] ,
Thomas Scanlon Open the ORCID record for Thomas Scanlon [Opens in a new window]  and
Umberto Zannier Open the ORCID record for Umberto Zannier [Opens in a new window]
Pietro Corvaja
Affiliation:
Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, via delle Scienze, 206, 33100Udine, Italy (pietro.corvaja@uniud.it)
Dragos Ghioca
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada (dghioca@math.ubc.ca)
Thomas Scanlon
Affiliation:
University of California, Berkeley, Mathematics Department, Evans Hall, Berkeley, CA94720-3840, USA (scanlon@math.berkeley.edu)
Umberto Zannier
Affiliation:
Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126Pisa, Italy (u.zannier@sns.it)
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Abstract

Let $K$ be an algebraically closed field of prime characteristic $p$, let $X$ be a semiabelian variety defined over a finite subfield of $K$, let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$, let $V\subset X$ be a subvariety defined over $K$, and let $\unicode[STIX]{x1D6FC}\in X(K)$. The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$-sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$, some rational numbers $c_{i}$ and some non-negative integers $k_{i}$. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$, or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$. We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.

Keywords

dynamical Mordell–Lang problemendomorphisms of semiabelian varieties defined over fields of characteristic p

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 37P55: Arithmetic dynamics on general algebraic varieties
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 2 , March 2021 , pp. 669 - 698
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Copyright
© Cambridge University Press 2019

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Footnotes

The second author has been partially supported by a discovery grant from the National Science and Engineering Board of Canada. The third author has been partially supported by grant DMS-1363372 of the United States National Science Foundation and a Simons Foundation Fellowship.

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