×

zbMATH — the first resource for mathematics

Diophantine geometry of the torsion of a Drinfeld module. (English) Zbl 1055.11037
It has been conjectured by Denis that certain qualitative Diophantine results valid for subgroups of semi-Abelian varieties might be extended to Drinfeld modules. In particular, according to Denis, this should be the case of the Manin-Mumford conjecture saying that an irreducible subvariety of a semi-Abelian variety containing a Zariski dense set of torsion points is a translate of a semialgebraic group, unless replacing \` \` semi-Abelian variety” by \` \` power of the additive group considered as an \({\mathbb F}_p[t]\)-module via a Drinfeld module”; actually Denis permits finite extensions of \({\mathbb F}_p [t]\) but requires that the Drinfeld module have generic characteristic.
The paper under review proves a version of this conjecture. Let us sketch its setting. For \(q\) a power of some prime, a smooth absolutely irreducible projective curve \(C\) over \({\mathbb F}_q\) is fixed, as well as a closed point \(\infty\), and the ring \({\mathbb A}\) of regular functions on \(C - \{ \infty \}\) is considered. Let \(G_a\) be the additive group scheme \(\text{Spec\,} {\mathbb Z}[X]\). An \({\mathbb A}\)-field is an \({\mathbb F}_q\)-morphism \(\iota\) from \({\mathbb A}\) to a field \(K\), and a Drinfeld module (over \(\iota\)) is an \({\mathbb F}_q\)-algebra homomorphism \(\varphi\) from \({\mathbb A}\) to \(\text{End}_K \, G_a\) such that \(\varphi\) and \(\iota\) have the same differential and \(\varphi({\mathbb A})\) is not contained in \(K\) (as canonically embedded into \(\text{End}_K \, G_a\)). The characteristic of \(\varphi\) is the kernel of \(\iota\), and \(\varphi\) is said to have generic characteristic if this ideal is \((0)\). For \(N\) a positive integer, \(K^N\) can be regarded as an \({\mathbb A}\)-module via \(\varphi\). The \(\varphi\)-torsion group of \(\varphi_{\text{tor}} (K^N)\) of \(K^N\) consists of the elements \(x \in K^N\) for which \(\varphi(b)(x) = 0\) for some non-zero \(b \in {\mathbb A}\).
As said, the main theorem of the paper is concerned with the Denis conjecture and says that, if \(\varphi\) is a Drinfeld module of generic characteristic, \(X \subseteq G_{a/K^{\text{alg}}}^N\) is an irreducible subvariety and \(X(K^{alg}) \cap \varphi_{\text{tor}}(K^{\text{alg}})^N\) is Zariski dense in \(X\), then \(X\) is a translate of an algebraic \({\mathbb A}\)-module. This result is approached along the lines of Hrushovski’s proof of the Manin-Mumford conjecture; in particular, the model theory of difference fields of positive characteristic (as developed by Chatzidakis, Hrushovki and Peterzil) is used. First the theory of reductions of Drinfeld modules is involved to find difference equations for a large submodule of the torsion. Then these equations are examined with respect to the dichotomy theorem for difference fields, to see that they define modular groups (which already provides the conclusion of the main theorem under the stronger assumption that \(X(K)\) contains a Zariski dense set of points from the submodule found before). Finally, it is proved how to work with two primes of \({\mathbb A}\) to cover all the torsion points.

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
03C60 Model-theoretic algebra
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boxall, J., Sous-variétés algébriques de variétés semi-abéliennes sur un corps fini, Number theory, 215, (1995), Cambridge Univ. Press Cambridge · Zbl 0840.14029
[2] Chatzidakis, Z.; Hrushovski, E., The model theory of difference fields, Trans. amer. math. soc., 351, 2997-3071, (1999) · Zbl 0922.03054
[3] Chatzidakis, Z.; Hrushovski, E.; Peterzil, Y., Model theory of difference fields. II. periodic ideals and the trichotomy in all characteristics, Proc. London math. soc. (3), 85, 257-311, (2002) · Zbl 1025.03026
[4] L. Denis, Géométrie diophantienne sur les modules de Drinfeld, inThe Arithmetic of Function FieldsD. Goss et al., Eds., pp. 285-302, 1992.
[5] Hrushovski, E., The manin – mumford conjecture and the model theory of difference fields, Ann. pure appl. logic, 112, 43-115, (2001) · Zbl 0987.03036
[6] Goss, D., Basic structures of function field arithmetic, Ergebniße der Mathematik und ihrer grenzgebiete, (1996), Springer Berlin · Zbl 0874.11004
[7] Scanlon, T., p-adic distance from torsion points of semi-abelian varieties, J. reine angew. math., 499, 225-236, (1998) · Zbl 0932.11041
[8] Scanlon, T.; Voloch, J.F., Difference subgroups of commutative algebraic groups over finite fields, Manuscripta math., 99, 329-339, (1999) · Zbl 0974.12013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.