Diophantine geometry of the torsion of a Drinfeld module.

*(English)*Zbl 1055.11037It 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.

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.

Reviewer: Carlo Toffalori (Camerino)

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##### References:

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