A+ CATEGORY SCIENTIFIC UNIT

On a reduction map for Drinfeld modules

Volume 195 / 2020

Wojciech Bondarewicz, Piotr Krasoń Acta Arithmetica 195 (2020), 109-129 MSC: Primary 11G09; Secondary 14G05, 14G25, 11J93. DOI: 10.4064/aa171025-26-10 Published online: 13 May 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.

Authors

  • Wojciech BondarewiczInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
    e-mail
  • Piotr KrasońInstitute of Mathematics
    University of Szczecin
    Wielkopolska 15
    70-451 Szczecin, Poland
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image