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Abhyankar's arithmetic conjecture for the affine line

From papers

Let pp be a prime, let qq be a power of pp, and let GG be a finite group. Write p(G)p(G) for the subgroup generated by the pp-Sylow subgroups of GG; thus GG is cyclic-by-quasi-pp when G/p(G)G/p(G) is cyclic. Abhyankar's arithmetic conjecture. If G/p(G)G/p(G) is cyclic, then there exists a Galois extension L/Fq(T)L/\mathbb{F}_q(T), not necessarily geometric, ramified only over the infinite prime, such that

Gal⁡(L/Fq(T))=G.\operatorname{Gal}(L/\mathbb{F}_q(T))=G.

This is an arithmetic analogue over a finite field of Abhyankar's conjecture for the affine line. The paper presents evidence for it, but the assertion is not established in general.

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Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Abhyankar's arithmetic conjecture for the affine line

    Let qq be a power of a prime pp, and let GG be a finite cyclic-by-quasi-pp group, meaning that G/p(G)G/p(G) is cyclic. Abhyankar's arithmetic conjecture. There exists a Galois extension K/Fq(T)K/\mathbb{F}_q(T) ramified only over ∞\infty such that

    Gal⁡(K/Fq(T))=G.\operatorname{Gal}(K/\mathbb{F}_q(T))=G.

    This is the arithmetic analogue of Abhyankar's geometric conjecture, replacing an algebraically closed constant field by a finite field. The claim is included as the arithmetic focus of the paper; the supplied text does not state its resolution, so its database status is left open.

    source: Alexei Entin and Noam Pirani, “Abhyankar's Affine Arithmetic Conjecture for the Symmetric and Alternating Groups”, arXiv:2205.03879 (2023).

Sources & referencesView supporting material

Primary source

Lior Bary-Soroker, Alexei Entin and Arno Fehm, “The minimal ramification problem for rational function fields over finite fields”, arXiv:2106.09126 (2022).

finite-fieldsfunction-fieldsgalois-theoryramification

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