Drinfeld modules with maximal Galois action
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917910016950272 |
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| author | Zywina, David |
| author_facet | Zywina, David |
| contents | With a fixed prime power $q>1$, define the ring of polynomials $A=\mathbb{F}_q[t]$ and its fraction field $F=\mathbb{F}_q(t)$. For each pair $a=(a_1,a_2) \in A^2$ with $a_2$ nonzero, let $ϕ(a)\colon A\to F\{τ\}$ be the Drinfeld $A$-module of rank $2$ satisfying $t\mapsto t+a_1τ+a_2τ^2$. The Galois action on the torsion of $ϕ(a)$ gives rise to a Galois representation $ρ_{ϕ(a)}\colon \operatorname{Gal}(F^{\operatorname{sep}}/F)\to \operatorname{GL}_2(\widehat{A})$, where $\widehat{A}$ is the profinite completion of $A$. We show that the image of $ρ_{ϕ(a)}$ is large for random $a$. More precisely, for all $a\in A^2$ away from a set of density $0$, we prove that the index $[\operatorname{GL}_2(\widehat{A}):ρ_{ϕ(a)}(\operatorname{Gal}(F^{\operatorname{sep}}/F))]$ divides $q-1$ when $q>2$ and divides $4$ when $q=2$. We also show that the representation $ρ_{ϕ(a)}$ is surjective for a positive density set of $a\in A^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01030 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Drinfeld modules with maximal Galois action Zywina, David Number Theory 11G09 (Primary) 11F80, 11R58 (Secondary) With a fixed prime power $q>1$, define the ring of polynomials $A=\mathbb{F}_q[t]$ and its fraction field $F=\mathbb{F}_q(t)$. For each pair $a=(a_1,a_2) \in A^2$ with $a_2$ nonzero, let $ϕ(a)\colon A\to F\{τ\}$ be the Drinfeld $A$-module of rank $2$ satisfying $t\mapsto t+a_1τ+a_2τ^2$. The Galois action on the torsion of $ϕ(a)$ gives rise to a Galois representation $ρ_{ϕ(a)}\colon \operatorname{Gal}(F^{\operatorname{sep}}/F)\to \operatorname{GL}_2(\widehat{A})$, where $\widehat{A}$ is the profinite completion of $A$. We show that the image of $ρ_{ϕ(a)}$ is large for random $a$. More precisely, for all $a\in A^2$ away from a set of density $0$, we prove that the index $[\operatorname{GL}_2(\widehat{A}):ρ_{ϕ(a)}(\operatorname{Gal}(F^{\operatorname{sep}}/F))]$ divides $q-1$ when $q>2$ and divides $4$ when $q=2$. We also show that the representation $ρ_{ϕ(a)}$ is surjective for a positive density set of $a\in A^2$. |
| title | Drinfeld modules with maximal Galois action |
| topic | Number Theory 11G09 (Primary) 11F80, 11R58 (Secondary) |
| url | https://arxiv.org/abs/2502.01030 |