Galois theory of Moore-Carlitz-Drinfeld modules

Abhyankar, Shreeram S. ; Sundaram, Ganapathy S. (1997) Galois theory of Moore-Carlitz-Drinfeld modules Comptes Rendus de l'Academie des Sciences - Series I: Mathematics, 325 (4). pp. 349-353. ISSN 0764-4442

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In 1896, E. H. Moore showed that the Galois group of the generic vectorial (= additive) q-polynomial of q-degree m is GL(m,q), where m > 0 is any integer and q > 1 is any power of any prime p. We show that, for any integer n > 0, the Galois group of the n-th iterate of the said polynomial is the generalized general linear group GL(m, q, n) consisting of all m by m matrices with invertible determinant over the local ring GF(q)[T]/Tn. For m=1, this was proved by Carlitz in 1938 as part of his explicit class field theory over finite fields. The case of m=1 was further enhanced by Drinfeld in 1974.

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