×

Arithmetic variation of fibers in families of curves. I: Hurwitz monodromy criteria for rational points on all members of the family. (English) Zbl 0699.14033

Datum given entirely by group theory, called a Nielsen class, determines families of covers of the projective line \(\mathbb P^ 1\) that are complete with respect to a simple moduli problem. Many of the properties of this family are determined by corresponding properties of representations of the Hurwitz monodromy group (a quotient of the Artin braid group) on the Nielsen class and related sets.
This paper considers the arithmetic of the points lying over the ramification locus in each fiber of the family. The main result (theorem 3.14) computes generically the degrees of the rational divisors with support contained in this locus of ramified points using Hurwitz monodromy action. This leads to a criterion, using ramification data, for the existence of rational points on a cover of \(\mathbb P^ 1\). This criterion is especially effective when the genus of the cover is 0 or 1.
Several examples illustrate this and allied phenomena: (1) for \(g=0,\) the arithmetic constancy of the family in relation with the Lewis-Schinzel criterion for families of conics to have sections (§3.1); and (2) for \(g=1\) the possible application of theorem 3.14 to the construction of elliptic curves of high rank over \(\mathbb Q\) (§3.7).
Reviewer: M. Fried

MSC:

14E22 Ramification problems in algebraic geometry
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
12F10 Separable extensions, Galois theory
14H30 Coverings of curves, fundamental group
PDFBibTeX XMLCite
Full Text: EuDML