# Counting number fields in fibers (with an Appendix by Jean Gillibert)

## Abstract

Let *X* be a projective curve over \({\mathbb Q}\) and \({t\in {\mathbb Q}(X)}\) a non-constant rational function of degree \({n\ge 2}\). For every \({\tau \in {\mathbb Z}}\) pick \({P_\tau \in X(\bar{\mathbb Q})}\) such that \({t(P_\tau )=\tau }\). Dvornicich and Zannier proved that, for large *N*, the field \({\mathbb Q}(P_1, \ldots , P_N)\) is of degree at least \(e^{cN/\log N}\) over \({\mathbb Q}\), where \({c>0}\) depends only on *X* and *t*. In this paper we extend this result, replacing \({\mathbb Q}\) by an arbitrary number field.

## Notes

### Acknowledgements

A substantial part of this article was written during my stay the Max-Planck-Institut für Mathematik in Bonn. I thank this institute for the financial support and stimulating working conditions.

This article belongs to a joint project with Jean Gillibert. I thank him for allowing me to publish this part of this project as a separate article, for adding a beautiful appendix, and for many stimulating discussions.

## References

- 1.Beckmann, S.: On extensions of number fields obtained by specializing branched coverings. J. Reine Angew. Math.
**419**, 27–53 (1991)MathSciNetzbMATHGoogle Scholar - 2.Bilu, Yu., Gillibert, J.: Chevalley–Weil Theorem and Subgroups of Class Groups, a manuscriptGoogle Scholar
- 3.Bilu, Yu.F., Luca, F.: Divisibility of class numbers: enumerative approach. J. Reine Angew. Math.
**578**, 79–91 (2005)Google Scholar - 4.Bilu, Yu., Luca, F.: Diversity in parametric families of number fields. In: Elsholtz, C., Grabner, P. (eds.) Number Theory-Diophantine Problems, Uniform Distribution and Applications. Springer (
**to appear**)Google Scholar - 5.Conrad, B.: Inertia groups and fibers. J. Reine Angew. Math.
**522**, 1–26 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Davenport, H., Lewis, D., Schinzel, A.: Polynomials of certain special types. Acta Arith.
**9**, 107–116 (1964)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Dvornicich, R., Zannier, U.: Fields containing values of algebraic functions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)
**21**, 421–443 (1994)MathSciNetzbMATHGoogle Scholar - 8.Grothendieck, A., et al.: Revêtements étales et groupe fondamental (SGA 1), Lecture Notes in Mathematics vol. 224. Springer, Berlin (1971)Google Scholar
- 9.Grothendieck, A., Murre, J.P.: The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, Lecture Notes in Mathematics. vol. 208. Springer, Berlin (1971)Google Scholar
- 10.Lang, S.: Algebra (Revised Third Edition), GTM 211. Springer, Berin (2002)Google Scholar
- 11.Serre, J.-P.: Lectures on the Mordell–Weil Theorem, 3rd edn. Vieweg & Sohn, Braunschweig (1997)CrossRefzbMATHGoogle Scholar
- 12.Walker, R.J.: Algebraic Curves. Springer, Berlin (1950)zbMATHGoogle Scholar