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arXiv · 2210.10152

Galois representations ramified at one prime and with suitably large image

Anwesh Ray
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Abstract

Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$ which are unramified outside $\{p, \infty\}$ with large image. More precisely, the Galois representations constructed have image containing the kernel of the mod-$p^t$ reduction map $SL_n(\mathbb{Z}_p)\rightarrow SL_n(\mathbb{Z}/p^t\mathbb{Z})$, where $t:=8(n^2-n)\left(3+\lfloor log_p(2^n+1)\rfloor\right)+8$. The results are proven via a purely Galois theoretic lifting construction. When $p\equiv 1\mod{4}$, our results are conditional since in this case, we assume a very weak version of Vandiver's conjecture.

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Anwesh Ray. 2022-10-18. Galois representations ramified at one prime and with suitably large image. https://doi.org/10.1090/tran%2F8999

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