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A conjecture on arithmetic fundamental groups

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The conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth\(\overline {\mathbb{F}_\ell ((t))} \) is finite. We indicate how to prove this for rank 2, using results of Drinfeld. We also show that the conjecture implies that certain deformation rings of Galois representations are complete intersection rings.

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This material is based upon work supported by the National Science Foundation under Grant No. 9970049.

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de Jong, A.J. A conjecture on arithmetic fundamental groups. Isr. J. Math. 121, 61–84 (2001).

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