Duke Mathematical Journal

Mod ℓ representations of arithmetic fundamental groups, I: An analog of Serre's conjecture for function fields

Abstract

There is a well-known conjecture of Serre that any continuous, irreducible representation $\overline{\rho}:G_\mathbf{Q}\rightarrow {\rm GL}_2(\overline{\mathbf{F}}_\ell)$ which is odd arises from a newform. Here $G_\mathbf{Q}$ is the absolute Galois group of $\mathbf{Q}$, and $\overline{\mathbf{F}}_\ell$ is an algebraic closure of the finite field $\mathbf{F}_\ell$ of $\ell$ of ℓ elements. We formulate such a conjecture for $n$-dimensional mod ℓ representations of $\pi_1(X)$ for any positive integer $n$ and where $X$ is a geometrically irreducible, smooth curve over a finite field $k$ of characteristic $p$ ($p \neq \ell$), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for $\ell>2$ follows from a result announced by Gaitsgory in [G]. The methods are different.

Article information

Source
Duke Math. J., Volume 129, Number 2 (2005), 337-369.

Dates
First available in Project Euclid: 27 September 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1127831441

Digital Object Identifier
doi:10.1215/S0012-7094-05-12925-8

Mathematical Reviews number (MathSciNet)
MR2165545

Zentralblatt MATH identifier
1078.11036

Citation

Böckle, Gebhard; Khare, Chandrashekhar. Mod ℓ representations of arithmetic fundamental groups, I: An analog of Serre's conjecture for function fields. Duke Math. J. 129 (2005), no. 2, 337--369. doi:10.1215/S0012-7094-05-12925-8. https://projecteuclid.org/euclid.dmj/1127831441

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