Sequences of powers with second differences equal to two and hyperbolicity
HTML articles powered by AMS MathViewer
- by Natalia Garcia-Fritz PDF
- Trans. Amer. Math. Soc. 370 (2018), 3441-3466 Request permission
Abstract:
By explicitly finding the complete set of curves of genus $0$ or $1$ in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound $M>0$ such that there are only finitely many sequences of length $M$ formed by $k$-th rational powers with second differences equal to $2$. Moreover, we prove the unconditional analogue of this result for function fields, with $M$ depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.References
- Ta Thi Hoai An, Hsiu-Lien Huang, and Julie Tzu-Yueh Wang, Generalized Büchi’s problem for algebraic functions and meromorphic functions, Math. Z. 273 (2013), no. 1-2, 95–122. MR 3010153, DOI 10.1007/s00209-012-0997-9
- F. A. Bogomolov, The theory of invariants and its applications to some problems in the algebraic geometry, Algebraic surfaces, C.I.M.E. Summer Sch., vol. 76, Springer, Heidelberg, 2010, pp. 217–245. MR 2757652, DOI 10.1007/978-3-642-11087-0_{4}
- J. Browkin and J. Brzeziński, Some remarks on the $abc$-conjecture, Math. Comp. 62 (1994), no. 206, 931–939. MR 1218341, DOI 10.1090/S0025-5718-1994-1218341-2
- Mireille Deschamps, Courbes de genre géométrique borné sur une surface de type général [d’après F. A. Bogomolov], Séminaire Bourbaki, 30e année (1977/78), Lecture Notes in Math., vol. 710, Springer, Berlin, 1979, pp. Exp. No. 519, pp. 233–247 (French). MR 554224
- David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819
- Natalia Garcia-Fritz, Representation of powers by polynomials and the language of powers, J. Lond. Math. Soc. (2) 87 (2013), no. 2, 347–364. MR 3046275, DOI 10.1112/jlms/jds052
- Natalia Cristina Garcia Fritz, Curves of low genus on surfaces and applications to Diophantine problems, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–Queen’s University (Canada). MR 3564085
- A. Grothendieck and J. A. Dieudonné, Éléments de géométrie algébrique. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 166, Springer-Verlag, Berlin, 1971 (French). MR 3075000
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Serge Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159–205. MR 828820, DOI 10.1090/S0273-0979-1986-15426-1
- Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. MR 1112552, DOI 10.1007/978-3-642-58227-1
- J. Richard Büchi, The collected works of J. Richard Büchi, Springer-Verlag, New York, 1990. Edited and with a preface by Saunders Mac Lane and Dirk Siefkes. MR 1030043
- B. Mazur, Questions of decidability and undecidability in number theory, J. Symbolic Logic 59 (1994), no. 2, 353–371. MR 1276620, DOI 10.2307/2275395
- Michael McQuillan, Diophantine approximations and foliations, Inst. Hautes Études Sci. Publ. Math. 87 (1998), 121–174. MR 1659270, DOI 10.1007/BF02698862
- David Mumford, Lectures on curves on an algebraic surface, Annals of Mathematics Studies, No. 59, Princeton University Press, Princeton, N.J., 1966. With a section by G. M. Bergman. MR 0209285, DOI 10.1515/9781400882069
- Junjiro Noguchi, A higher-dimensional analogue of Mordell’s conjecture over function fields, Math. Ann. 258 (1981/82), no. 2, 207–212. MR 641826, DOI 10.1007/BF01450536
- H. Pasten, T. Pheidas, and X. Vidaux, A survey on Büchi’s problem: new presentations and open problems, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. Issledovaniya po Teorii Chisel. 10, 111–140, 243 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 6, 765–781. MR 2753653, DOI 10.1007/s10958-010-0181-x
- Hector Pasten, Powerful values of polynomials and a conjecture of Vojta, J. Number Theory 133 (2013), no. 9, 2964–2998. MR 3057059, DOI 10.1016/j.jnt.2013.03.001
- Thanases Pheidas and Xavier Vidaux, Extensions of Büchi’s problem: questions of decidability for addition and $k$th powers, Fund. Math. 185 (2005), no. 2, 171–194. MR 2163109, DOI 10.4064/fm185-2-4
- Thanases Pheidas and Xavier Vidaux, The analogue of Büchi’s problem for rational functions, J. London Math. Soc. (2) 74 (2006), no. 3, 545–565. MR 2286432, DOI 10.1112/S0024610706023283
- Thanases Pheidas and Xavier Vidaux, Corrigendum: The analogue of Büchi’s problem for rational functions [MR2286432], J. Lond. Math. Soc. (2) 82 (2010), no. 1, 273–278. MR 2669651, DOI 10.1112/jlms/jdq002
- Paul Vojta, A more general $abc$ conjecture, Internat. Math. Res. Notices 21 (1998), 1103–1116. MR 1663215, DOI 10.1155/S1073792898000658
- Paul Vojta, Diagonal quadratic forms and Hilbert’s tenth problem, Hilbert’s tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999) Contemp. Math., vol. 270, Amer. Math. Soc., Providence, RI, 2000, pp. 261–274. MR 1802018, DOI 10.1090/conm/270/04378
- Paul Vojta, On the $ABC$ conjecture and Diophantine approximation by rational points, Amer. J. Math. 122 (2000), no. 4, 843–872. MR 1771576, DOI 10.1353/ajm.2000.0031
- Paul Vojta, Diophantine approximation and Nevanlinna theory, Arithmetic geometry, Lecture Notes in Math., vol. 2009, Springer, Berlin, 2011, pp. 111–224. MR 2757629, DOI 10.1007/978-3-642-15945-9_{3}
Additional Information
- Natalia Garcia-Fritz
- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street BA6103, Toronto, Ontario, Canada, M5S 2E4
- Address at time of publication: Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avenida Vicuña Mackenna 4860, Santiago, Chile
- Email: natalia.garcia@mat.uc.cl
- Received by editor(s): March 7, 2016
- Received by editor(s) in revised form: August 8, 2016
- Published electronically: December 27, 2017
- Additional Notes: This work was part of the author’s thesis at Queen’s University and was partially supported by a Becas Chile Scholarship
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 3441-3466
- MSC (2010): Primary 11D41, 32Q45, 14G05
- DOI: https://doi.org/10.1090/tran/7040
- MathSciNet review: 3766854