Duke Mathematical Journal

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

Gebhard Böckle and Chandrashekhar Khare

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There is a well-known conjecture of Serre that any continuous, irreducible representation ρ̲:GQGL2(F̲) which is odd arises from a newform. Here GQ is the absolute Galois group of Q, and F̲ is an algebraic closure of the finite field F of of ℓ elements. We formulate such a conjecture for n-dimensional mod ℓ representations of π1(X) for any positive integer n and where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p (p), and we prove this conjecture in a large number of cases. In fact, a proof of all cases of the conjecture for >2 follows from a result announced by Gaitsgory in [G]. The methods are different.

Article information

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

First available in Project Euclid: 27 September 2005

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Zentralblatt MATH identifier

Primary: 11F80: Galois representations 11F70: Representation-theoretic methods; automorphic representations over local and global fields 14H30: Coverings, fundamental group [See also 14E20, 14F35] 11R34: Galois cohomology [See also 12Gxx, 19A31]


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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