Galois representations are surjective for almost all Drinfeld modules
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909262230323200 |
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| author | Ray, Anwesh |
| author_facet | Ray, Anwesh |
| contents | This article advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let $F$ be the rational function field over a finite field. I establish that for Drinfeld modules of rank $r \geq 2$, the $T$-adic Galois representation: $\widehatρ_{ϕ, T}: Gal(F^{sep}/F) \rightarrow GL_r(\mathbb{F}_q[[T]])$ is surjective for a density $1$ set of such modules. The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's uniformization theory and sieve methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Galois representations are surjective for almost all Drinfeld modules Ray, Anwesh Number Theory Algebraic Geometry 11F80, 11G09, 11R45 This article advances the results of Duke on the average surjectivity of Galois representations for elliptic curves to the context of Drinfeld modules over function fields. Let $F$ be the rational function field over a finite field. I establish that for Drinfeld modules of rank $r \geq 2$, the $T$-adic Galois representation: $\widehatρ_{ϕ, T}: Gal(F^{sep}/F) \rightarrow GL_r(\mathbb{F}_q[[T]])$ is surjective for a density $1$ set of such modules. The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's uniformization theory and sieve methods. |
| title | Galois representations are surjective for almost all Drinfeld modules |
| topic | Number Theory Algebraic Geometry 11F80, 11G09, 11R45 |
| url | https://arxiv.org/abs/2407.14264 |