Details

Pietro Corvaja ^[a]

On the local-to-global principle for value sets

Pages: 47-72
Received: 6 July 2021
Accepted in revised form: 12 October 2021
Mathematics Subject Classification: 11D99, 11R09, 11G35.
Keywords: Local global principle, Diophantine equations.
Authors address:
[a]: Dipartimento di Scienze Matematiche, Informatiche e Fisiche, Università di Udine, Via delle Scienze 206, 33100 Udine, Italy.

Abstract: We consider the following problem: given a morphism \( \mathcal Y\to \mathcal X \) of algebraic curves over a number field \(k\), describe the rational points \(x\in \mathcal X (k)\) lifting locally at every place to some rational point on \(\mathcal Y\), but admitting no rational pre-image. In particular, we provide examples where there exist infinitely many such points.

References

[1]: P. Buser, Cayley graphs and planar isospectral domains, in ''Geometry and analysis on manifolds'' (Katata/Kyoto 1987), Lecture Notes in Math., 1339, Springer, Berlin, 1988. MR0961473
[2]: P. Corvaja, Integral points on algebraic varieties, An introduction to Diophantine geometry, Inst. Math. Sci. Lect. Notes, 3, Hindustan Book Agency, New Delhi, 2016. MR3497029
[3]: P. Corvaja and U. Zannier, Applications of Diophantine approximation to integral points and transcendence, Cambridge Tracts in Math., 212, Cambridge Univ. Press, Cambridge, 2018. MR3793125
[4]: R. Dvornicich and U. Zannier, Local-global divisibility of rational points in some commutative algebraic groups, Bull. Soc. Math. France 129 (2001), 317-338. MR1881198
[5]: R. Dvornicich and U. Zannier, An analogue for elliptic curves of the Grunwald-Wang example, C. R. Math. Acad. Sci. Paris 338 (2004), 47-50. MR2038083
[6]: W. L. Edge, Bring's curve, J. London Math. Soc. (2) 18 (1978), 539-545. MR0518240
[7]: M. Fried, Arithmetical properties of value sets of polynomials, Acta Arith. 15 (1968/69), 91-115. MR0244150
[8]: M. Fried, On a conjecture of Schur, Michigan Math. J. 17 (1970), 41-45. MR0257033
[9]: F. Gassmann, Bemerkungen zur Vorstehenden Arbeit von Hurwitz: Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppen, Math. Z. 25 (1926), 661-675. ZbMath | DOI
[10]: D. Harari and J. F. Voloch, The Brauer-Manin obstruction for integral points on curves, Math. Proc. Cambridge Philos. Soc. 149 (2010), 413-421. MR2726726
[11]: K. Kunen and W. Rudin, Lacunarity and the Bohr topology, Math. Proc. Cambridge Philos. Soc. 126 (1999), 117-137. MR1681658
[12]: S. Levy (editor), The eightfold way, The beauty of Klein's quartic curve, Math. Sci. Res. Inst. Publ., 35, Cambridge Univ. Press, Cambridge, 1999. MR1722410
[13]: P. Müller, Kronecker conjugacy of polynomials, Trans. Amer. Math. Soc. 350 (1998), 1823-1850. MR1458331
[14]: R. Perlis, On the equation \(\zeta_{\mathbf{k}}(s)=\zeta_{\mathbf{k}'}(s)\), J. Number Theory 9 (1977), 342-360. MR0447188
[15]: J.-P. Serre, Topics in Galois theory, Res. Notes Math., 1, Johnes & Bartlett Publisher, Boston, 1992. MR1162313
[16]: J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), 429-440. MR1997347
[17]: M. Stoll, Finite descent obstructions and rational points on curves, Algebra and Number Theory 1 (2007), 349-391. MR2368954
[18]: T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 169-186. MR0782558
[19]: M. Weber, Kepler's small stellated dodecahedron as a Riemann surface, Pacific J. Math. 220 (2005), 167-182. MR2195068