Conjecture on maximal compact quotients of affine-line fundamental groups

At least 32 years old · documented by

Let K=Fq(T)K={\bf F}_q(T), let G/KG/K be an isotropic, semisimple, simply-connected algebraic group, and let ZZ be its center. Let AKfin{\bf A}^{\mathrm{fin}}_K denote the finite adeles of KK. The maximal compact quotient conjecture. Every maximal compact subgroup of

G(AKfin)/Z(Fq)G({\bf A}^{\mathrm{fin}}_K)/Z({\bf F}_q)

occur as a continuous quotient of the fundamental group of the affine line over CC. The conjecture proposes a broad generalization of the preceding constructions and results on Galois groups of coverings of the affine line; the source presents it as a conjecture following work of Madhav Nori and the paper's result, but supplies no resolution evidence.

References

Primary source

Kirti Joshi, “A family of étale coverings of the affine line”, arXiv:alg-geom/9309002 (1993).

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