×

Quantitative Chevalley-Weil theorem for curves. (English) Zbl 1288.11064

Let \(K\) be a number field and \(\bar{K}\) the algebraic closure of \(K\). Let \(\phi : \tilde{V}\rightarrow V\) be a finite étale covering of normal projective varieties, defined over \(K\). The Chevalley-Weil theorem asserts that there exists an integer \(T> 0\) such that for any \(P\in V(K)\) and \(\tilde{P} \in \tilde{V}(\bar{K})\) with \(\phi(\tilde{P}) = P\), the relative discriminant of \(K(\tilde{P})/K\) divides \(T\). This theorem is useful in diophantine analysis since it provides a reduction of a diophantine problem on the variety \(V\) to that on the variety \(\tilde{V}\) which could be easier. Furthermore, it is used implicitly in the proofs of some finiteness theorems of Mordell-Weil. In this paper, a quantitative version of this theorem in dimension \(1\) is presented which improves and generalizes previous results of the first author [Ph. D. Thesis, Beer Sheva (1993)] and of K. Draziotis and D. Poulakis [Rocky Mt. J. Math. 39, No. 1, 49–70 (2009; Zbl 1222.14063)] and [Houston J. Math. 38, No. 1, 29–39 (2012; Zbl 1239.14018)]. The method of the proof is different of that of Draziotis and Poulakis and goes back to the first author’s thesis.

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G05 Rational points
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Bilu, Yu.: Effective Analysis of Integral Points on Algebraic Curves, Ph. D. Thesis, Beer Sheva (1993)
[2] Bilu, Yu.: Quantitative Siegel’s theorem for Galois coverings. Compost. Math. 106(2), 125-158 (1997) · Zbl 1044.11593 · doi:10.1023/A:1000172615719
[3] Bilu, Yu., Borichev, A.: Remarks on Eisenstein. J. Aust. Math. Soc. arXiv:1112.2290 (to appear) · Zbl 1326.11064
[4] Bilu, Yu., Strambi, M.: Quantitative Riemann existence theorem over a number field. Acta Arith. 145, 319-339 (2010) · Zbl 1222.11082 · doi:10.4064/aa145-4-2
[5] Chevalley, C., Weil, A.: Un théorème d’arithmétique sur les courbes algébriques. C. R. Acad. Sci. Paris 195, 570-572 (1932) · JFM 58.0182.04
[6] Dedekind, R.: Gesammelte mathematische Werke. Bände I-III. Herausgegeben von Robert Fricke, Emmy Noether und Öystein Ore. Chelsea Publishing Co., New York (1968) · Zbl 0001.38501
[7] Draziotis, K., Poulakis, D.: Explicit Chevalley-Weil theorem for affine plane curves. Rocky Mt. J. Math. 39, 49-70 (2009) · Zbl 1222.14063 · doi:10.1216/RMJ-2009-39-1-49
[8] Draziotis, K., Poulakis, D.: An effective version of Chevalley-Weil theorem for projective plane curves. Houston J. Math. 38, 29-39 (2012) · Zbl 1239.14018
[9] Dwork, B., Robba, P.: On natural radii of \[p\]-adic convergence. Trans. Am. Math. Soc. 256, 199-213 (1979) · Zbl 0426.12013
[10] Dwork, B.M., van der Poorten, A.J.: The Eisenstein constant. Duke Math. J. 65(1), 23-43 (1992) · Zbl 0770.11051 · doi:10.1215/S0012-7094-92-06502-1
[11] Hindry, M., Silverman, J.H.: Diophantine geometry: an introduction. In: Graduate Texts in Math, vol. 201. Springer, Berlin (2000) · Zbl 0948.11023
[12] Krick, T., Pardo, L.M., Sombra, M.: Sharp estimates for the arithmetic Nullstellensatz. Duke Math. J. 109, 521-598 (2001) · Zbl 1010.11035
[13] Lang, S.: Fundamentals of Diophantine Geometry. Springer, New York (1983) · Zbl 0528.14013 · doi:10.1007/978-1-4757-1810-2
[14] Rosser, J.B., Schoenfeld, L.: Lowell Approximate formulas for some functions of prime numbers. Illinois J. Math. 6, 64-94 (1962) · Zbl 0122.05001
[15] Schmidt, W.M.: Eisenstein’s theorem on power series expansions of algebraic functions. Acta Arith. 56(2), 161-179 (1990) · Zbl 0659.12003
[16] Schmidt, W.M.: Construction and estimates of bases in function fields. J. Number Theory 39, 181-224 (1991) · Zbl 0764.11046 · doi:10.1016/0022-314X(91)90044-C
[17] Serre, J.-P.: Quelques applications du théorème de densité de Chebotarev. Publ. Math. IHES 54, 323-401 (1981)
[18] Serre, J.-P.: Lectures on Mordell-Weil Theorem, 3rd edn. Vieweg, Braunschweig (1997) · Zbl 0863.14013 · doi:10.1007/978-3-663-10632-6
[19] Silverman, J.H.: Lower bounds for height functions. Duke Math. J. 51, 395-403 (1984) · Zbl 0579.14035 · doi:10.1215/S0012-7094-84-05118-4
[20] Weil, A.: Arithmétique et géométrie sur les variétés algébriques. Act. Sc. et Ind. 206, 3-16 (1935) · JFM 61.1078.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.