On a reduction map for Drinfeld modules

Wojciech Bondarewicz¹ ; Piotr Krasoń¹

¹ Institute of Mathematics University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland

Acta Arithmetica, Tome 195 (2020) no. 2, pp. 109-129.

Résumé

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.

Détail

DOI : 10.4064/aa171025-26-10

Keywords: investigate local to global principle mordell weil group defined ring integers cal mathbf t modules products drinfeld modules widehat varphi phi times dots times phi here finite extension field fractions mathbb assume operatorname rank phi endomorphism rings relevant drinfeld modules generic characteristic simplest possible operatorname end sep phi dots main result following numerical criterion times dots times finitely generated a submodule mordell weil group widehat varphi cal phi cal times dots times phi cal lambda subset a submodule geq operatorname red cal operatorname red cal lambda almost primes cal cal lambda tor build recent results bara czuk concerning dynamical local to global principle mordell weil type groups solvability certain dynamical equations aforementioned mathbf modules