Arithmetic of 3 And 4 Branch Point Covers
The method of choice nowadays for achieving a group G as a Galois group of a regular extension of ℚ(x) goes under the heading of rigidity. It works essentially, only, to produce Galois extensions of ℚ(x) ramified over 3 points. The three rigidity conditions ((0.1) below) imply that G is generated in a very special way by two elements. Generalization of rigidity that considers extensions with any number r of branch points has been around even longer than rigidity (§ 5.1). Of the three conditions, the generalization of the transitivity condition, 0.1 c), requires only the addition of an action of the Hurwitz monodromy group H r (a quotient of the Artin braid group). But it also adds a 4th condition that in many situations amounts to asking for a ℚ-point on the Hurwitz space associated the data for the generators of G. Theorem 1 below —our main theorem— is that in the case r = 4 this is equivalent to finding ℚ-point on a curve derived from a quotient of the upper half plane by a subgroup of PSL 2(Z).
Unable to display preview. Download preview PDF.
- [DDH]S. Diaz, R. Donagi and D. Harbater.-Every curve is a Hurwitz space, preprint.Google Scholar
- [DFr]P. Debes and M. Fried.-Arithmetic variation of fibers in families of curves Pari I: Hurwitz monodromy criteria for rational points on all members of the family; preprint.Google Scholar
- [Fr,3]M. Fried.-Rigidity and applications of the classification of simple groups to monodromy Pari I-Super rational connectivity with examples; Pari II-Applications of connectivity; Davenport and Hilbert-Siegel problems.Google Scholar
- [FrT]M. Fried and J.G. Thompson.-The Hurwitz monodromy group H 4 and modular curves, preprint.Google Scholar
- [Gro]A. Grothendieck.-Géométrie jormelle et geometrie algébrique, Séminaire Bourbaki t. 11, 182 (1958).Google Scholar
- [Groa]A. Grothendieck.-Géométrie jormelle et geometrie algébrique, Séminaire Bourbaki t. 11, 182 (1959).Google Scholar
- [Gu]R.C. Gunning.-Lectures on Riemann Surfaces, Princeton Math. Notes (1966).Google Scholar
- [Hi]D. Hilbert.-Uber die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten, J. Reine Angew. Math. 110 (1892), (Ges. Abh. II, 264–286).Google Scholar
- [Ma,1]H. Matzat.-Konstructive Galoistheorie, Lecture Notes in Math-Springer Verlag 1284 (1986).Google Scholar
- [Ma,2]H. Matzat.-Rationality Criteria for Galois Extensions, preprint.Google Scholar