On a reduction map for Drinfeld modules

Abstract

In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the 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 the ${\mathrm{rank}}(\phi)_{i})=d_{i}$ and endomorphism rings of the involved Drinfeld modules of generic characteristic are the simplest possible, i.e. ${\mathrm{End}}({\phi}_{i})=A$ for $ i=1,\dots , t.$ Our main result is the following numeric 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 we assume $d_{i}\geq e_{i}$ and $P\in N$ such that $r_{\cal W}(P)\in r_{\cal W}({\Lambda}) $ for almost all primes ${\cal W}$ of ${\cal O}_{K},$ then $P\in {\Lambda}+N_{tor}.$ We also build on the recent results of S.Bara{\'n}czuk \cite{b17} concerning the dynamical local to global principle in Mordell-Weil type groups and the solvability of certain dynamical equations to the aforementioned $t$-modules.