On a reduction map for Drinfeld modules
Volume 195 / 2020
Abstract
We investigate a local-to-global principle for the Mordell–Weil group defined over a ring of integers ${\cal O}_K$ of $\mathbf t$-modules that are products of Drinfeld modules, ${\widehat \varphi }={\phi }_{1}^{e_1}\times \dots \times {\phi }_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\mathbb F}_{q}[t].$ We assume that $\operatorname{rank} (\phi _{i})=d_{i}$ and the endomorphism rings of the relevant Drinfeld modules of generic characteristic are simplest possible, i.e. $\operatorname{End} _{K^{\rm sep }}({\phi }_{i})=A$ for $ i=1,\dots , t$. Our main result is the following numerical criterion. Let ${N}={N}_{1}^{e_1}\times \dots \times {N}_{t}^{e_t}$ be a finitely generated $A$-submodule of the Mordell–Weil group ${\widehat \varphi }({\cal O}_{K})={\phi }_{1}({\cal O}_{K})^{e_{1}}\times \dots \times {\phi }_{t}({\cal O}_{K})^{{e}_{t}},$ and let ${\Lambda }\subset N$ be an $A$-submodule. If $d_{i}\geq e_{i}$ and $P\in N$ with $\operatorname{red} _{\cal W}(P)\in \operatorname{red} _{\cal W}({\Lambda }) $ for almost all primes ${\cal W}$ of ${\cal O}_{K},$ then $P\in {\Lambda }+N_{{\rm tor}}.$ We also build on the recent results of S. Barańczuk (2017) concerning the dynamical local-to-global principle in Mordell–Weil type groups and the solvability of certain dynamical equations for the aforementioned ${\mathbf t}$-modules.
