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A Chabauty–Coleman bound for surfaces

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Abstract

Building on work by Chabauty from 1941, Coleman proved in 1985 an explicit bound for the number of rational points of a curve \(C\) of genus \(g\ge 2\) defined over a number field \(F\), with Jacobian of rank at most \(g-1\). Namely, in the case \(F=\mathbb{Q}\), if \(p>2g\) is a prime of good reduction, then the number of rational points of \(C\) is at most the number of \(\mathbb{F}_{p}\)-points plus a contribution coming from the canonical class of \(C\). We prove a result analogous to Coleman’s bound in the case of a hyperbolic surface \(X\) over a number field, embedded in an abelian variety \(A\) of rank at most one, under suitable conditions on the reduction type at the auxiliary prime. This provides the first extension of Coleman’s explicit bound beyond the case of curves. The main innovation in our approach is a new method to study the intersection of a \(p\)-adic analytic subgroup with a subvariety of \(A\) by means of overdetermined systems of differential equations in positive characteristic.

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References

  1. Aubry, Y., Perret, M.: On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields. Finite Fields Appl. 10(3), 412–431 (2004)

    MathSciNet  MATH  Google Scholar 

  2. Balakrishnan, J., Dogra, N.: Quadratic Chabauty and rational points, I: \(p\)-adic heights. Duke Math. J. 167(11), 1981–2038 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Balakrishnan, J., Dogra, N.: An effective Chabauty-Kim theorem. Compos. Math. 155, 1057–1075 (2019)

    MathSciNet  MATH  Google Scholar 

  4. Balakrishnan, J., Dogra, N.: Quadratic Chabauty and rational points II: generalised height functions on Selmer varieties. Int. Math. Res. Not. (2020). https://doi.org/10.1093/imrn/rnz362

    Article  MATH  Google Scholar 

  5. Balakrishnan, J., Besser, A., Müller, J.: Computing integral points on hyperelliptic curves using quadratic Chabauty. Math. Comput. 86(305), 1403–1434 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Balakrishnan, J., Dogra, N., Müller, J., Tuitman, J., Vonk, J.: Explicit Chabauty–Kim for the split Cartan modular curve of level 13. Ann. Math. 189(3), 885–944 (2019)

    MathSciNet  MATH  Google Scholar 

  7. Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact Complex Surfaces, 2nd edn. Ergebnisse der Mathematik und Ihrer Grenzgebiete, vol. 4. Springer, Berlin (2004)

    MATH  Google Scholar 

  8. Berthelot, P., Grothendieck, A., Illusie, L. (eds.): Theorie des Intersections et Theoreme de Riemann-Roch (SGA6). 1966/67. Lecture Notes in Math., vol. 225. Springer, Berlin (1971)

    Google Scholar 

  9. Bhargava, M., Shankar, A.: Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0. Ann. Math. 181, 587–621 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Bhargava, M., Skinner, C.: A positive proportion of elliptic curves over Q have rank one. J. Ramanujan Math. Soc. 29(2), 221–242 (2014)

    MathSciNet  MATH  Google Scholar 

  11. Bourbaki, N.: Lie groups and Lie Algebras. Elements of Mathematics (Berlin). Springer, Berlin (1998). Chapters 1-3

    MATH  Google Scholar 

  12. Brody, R.: Compact manifolds and hyperbolicity. Trans. Am. Math. Soc. 235, 213–219 (1978)

    MathSciNet  MATH  Google Scholar 

  13. Chabauty, C.: Sur les points rationnels des courbes algébriques de genre supérieur à l’unité. C. R. Acad. Sci. Paris 212, 882–885 (1941)

    MathSciNet  MATH  Google Scholar 

  14. Chavdarov, N.: The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. Duke Math. J. 87(1), 151–180 (1997)

    MathSciNet  MATH  Google Scholar 

  15. Coleman, R.: Effective Chabauty. Duke Math. J. 52(3), 765–770 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Debarre, O.: Degrees of curves in Abelian varieties. Bull. Soc. Math. Fr. 122(3), 343–361 (1994)

    MathSciNet  MATH  Google Scholar 

  17. Deligne, P.: La conjecture de Weil: I. Publ. Math. IHES 43, 273–307 (1974)

    MathSciNet  MATH  Google Scholar 

  18. Deschamps, M.: 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 Mathematics, vol. 710. Springer, Berlin (1978)

    Google Scholar 

  19. Edixhoven, B., Lido, G.: Geometric quadratic Chabauty. Preprint, v3 in arXiv:1910.10752

  20. Faltings, G.: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. [Finiteness theorems for Abelian varieties over number fields]. Invent. Math. 73(3), 349–366 (1983). (German)

    MathSciNet  MATH  Google Scholar 

  21. Faltings, G.: Diophantine approximation on Abelian varieties. Ann. Math. 133, 549–576 (1991)

    MathSciNet  MATH  Google Scholar 

  22. Faltings, G.: The general case of S. Lang’s conjecture. In: Barsotti Symposium in Algebraic Geometry, Abano Terme, 1991. Perspect. Math., vol. 15, pp. 175–182. Academic Press, San Diego (1994)

    Google Scholar 

  23. Flynn, E.: A flexible method for applying Chabauty’s theorem. Compos. Math. 105(1), 79–94 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Fulton, W.: Intersection Theory, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2. Springer, Berlin (1998)

    MATH  Google Scholar 

  25. Garcia-Fritz, N.: Curves of low genus on surfaces and applications to Diophantine problems. PhD Thesis, Queen’s University (2015)

  26. Garcia-Fritz, N.: Sequences of powers with second differences equal to two and hyperbolicity. Trans. Am. Math. Soc. 370(5), 3441–3466 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Garcia-Fritz, N.: Quadratic sequences of powers and Mohanty’s conjecture. Int. J. Number Theory 14(02), 479–507 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Green, M.: Holomorphic maps to complex tori. Am. J. Math. 100(3), 615–620 (1978)

    MathSciNet  MATH  Google Scholar 

  29. Grothendieck, A., Raynaud, M.: Revêtement étales et groupe fondamental (SGA1). Lecture Note in Math., vol. 224. Springer, Berlin (1971)

    Google Scholar 

  30. Gunther, J., Morrow, J.: Irrational points on random hyperelliptic curves. Preprint, v3 in arXiv:1709.02041

  31. Hartshorne, R.: Algebraic Geometry. Graduate Texts in Math., vol. 52. Springer, Berlin (2013)

    MATH  Google Scholar 

  32. Hindry, M., Silveman, J.: Diophantine Geometry, an Introduction. Graduate Texts in Mathematics, vol. 201. Springer, New York (2000)

    Google Scholar 

  33. Hironaka, H.: On the arithmetic genera and the effective genera of algebraic curves. Mem. Coll. Sci., Univ. Kyoto, Ser. A: Math. 30(2), 177–195 (1957)

    MathSciNet  MATH  Google Scholar 

  34. Hu, P.-C., Yang, C.-C.: Meromorphic Functions over Non-Archimedean Fields. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  35. Katsura, T., Ueno, K.: On elliptic surfaces in characteristic \(p\). Math. Ann. 272(3), 291–330 (1985)

    MathSciNet  MATH  Google Scholar 

  36. Katz, D.: On the number of minimal prime ideals in the completion of a local domain. Rocky Mt. J. Math. 16(3), 575–578 (1986)

    MathSciNet  MATH  Google Scholar 

  37. Katz, E., Zureick-Brown, D.: The Chabauty-Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions. Compos. Math. 149(11), 1818–1838 (2013)

    MathSciNet  MATH  Google Scholar 

  38. Katz, E., Rabinoff, J., Zureick-Brown, D.: Uniform bounds for the number of rational points on curves of small Mordell-Weil rank. Duke Math. J. 165(16), 3189–3240 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Kim, M.: The motivic fundamental group of \(\mathbf{P}^{1}\setminus \{0,1,\infty \}\) and the theorem of Siegel. Invent. Math. 161, 629–656 (2005)

    MathSciNet  Google Scholar 

  40. Klassen, M.J.: Algebraic points of low degree on curves of low rank. Thesis, University of Arizona (1993)

  41. Kouvidakis, A.: Divisors on symmetric products of curves. Trans. Am. Math. Soc. 337(1), 117–128 (1993)

    MathSciNet  MATH  Google Scholar 

  42. Kunz, E.: Kähler Differentials. Advanced Lectures in Mathematics. Vieweg, Braunschweig (1986)

    MATH  Google Scholar 

  43. Kunz, E.: Introduction to Plane Algebraic Curves. Birkhäuser Boston, Boston (2005)

    MATH  Google Scholar 

  44. Lorenzini, D., Tucker, T.: Thue equations and the method of Chabauty-Coleman. Invent. Math. 148, 47–77 (2002)

    MathSciNet  MATH  Google Scholar 

  45. Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1987). (M. Reid, Trans.)

    Google Scholar 

  46. Matsusaka, T.: On a characterization of a Jacobian variety. Mem. Coll. Sci., Univ. Kyoto, Ser. A: Math. 32(1), 1–19 (1959)

    MathSciNet  MATH  Google Scholar 

  47. McCallum, W., Poonen, B.: The method of Chabauty and Coleman. Explicit methods in number theory; rational points and Diophantine equations. In: Panoramas et Synthèses, vol. 36, pp. 99–117. Société Math. De France (2012)

    Google Scholar 

  48. Morikawa, H.: Cycles and endomorphisms of Abelian varieties. Nagoya Math. J. 7, 95–102 (1954)

    MathSciNet  MATH  Google Scholar 

  49. Mumford, D.: Appendix to “The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface” by O. Zariski. Ann. Math. 76(3), 560–615 (1962)

    MathSciNet  Google Scholar 

  50. Mumford, D.: Abelian Varieties. Tata Inst. Fund. Res. Stud. Math., vol. 5. Tata, Bombay (1970)

    MATH  Google Scholar 

  51. Murty, V.K., Patankar, V.: Splitting of Abelian varieties. Int. Math. Res. Not. 12 (2008)

  52. Nakai, Y.: Notes on invariant differentials on Abelian varieties. J. Math. Kyoto Univ. 3(1), 127–135 (1963)

    MathSciNet  MATH  Google Scholar 

  53. Nakai, Y.: On the theory of differentials on algebraic varieties. J. Sci. Hiroshima Univ. Ser. A-I 27, 7–34 (1963)

    MathSciNet  MATH  Google Scholar 

  54. Park, J.: Effective Chabauty for symmetric powers of curves. Preprint, v1 in arXiv:1504.05544

  55. Płoski, A.: Introduction to the local theory of plane algebraic curves. In: Krasiński, T., Spodzieja, S. (eds.) Analytic and Algebraic Geometry, pp. 115–134. Łódź University Press, Łódź (2013)

    MATH  Google Scholar 

  56. Rémond, G.: Décompte dans une conjecture de Lang. Invent. Math. 142(3), 513–545 (2000)

    MathSciNet  MATH  Google Scholar 

  57. Rémond, G.: Sur les sous-variétés des tores. Compos. Math. 134(3), 337–366 (2002)

    MATH  Google Scholar 

  58. Siksek, S.: Chabauty for symmetric powers of curves. Algebra Number Theory 3(2), 209–236 (2009)

    MathSciNet  MATH  Google Scholar 

  59. Stoll, M.: Independence of rational points on twists of a given curve. Compos. Math. 142, 1201–1214 (2006)

    MathSciNet  MATH  Google Scholar 

  60. Stoll, M.: Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank. J. Eur. Math. Soc. 21(3), 923–956 (2019)

    MathSciNet  MATH  Google Scholar 

  61. The Stacks Project: Schemes smooth over fields (Tag 04QM). https://stacks.math.columbia.edu/tag/04QM

  62. Vemulapalli, S., Wang, D.: Uniform bounds for the number of rational points on symmetric squares of curves with low Mordell-Weil rank. Preprint, v3 in arXiv:1708.07057

  63. Vojta, P.: Siegel’s theorem in the compact case. Ann. Math. 133(3), 509–548 (1991)

    MathSciNet  MATH  Google Scholar 

  64. Vojta, P.: Diagonal quadratic forms and Hilbert’s tenth problem. Hilbert’s tenth problem. In: Relations with Arithmetic and Algebraic Geometry, Ghent, 1999. Contemp. Math., vol. 270, pp. 261–274. Am. Math. Soc., Providence (2000)

    MATH  Google Scholar 

  65. Yau, S.-T.: Intrinsic measures of compact complex manifolds. Math. Ann. 212, 317–329 (1975)

    MathSciNet  MATH  Google Scholar 

  66. Zariski, O., Samuel, P.: Commutative Algebra, Volume II. Graduate Texts in Mathematics, vol. 29. Springer, Berlin (1960)

    MATH  Google Scholar 

  67. Zywina, D.: The splitting of reductions of an Abelian variety. Int. Math. Res. Not. 2014(18), 5042–5083 (2014)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

We would like to thank Natalia Garcia-Fritz for answering numerous questions regarding branches of curves and \(\omega \)-integrality. Initially, our results were proved for surfaces contained in abelian threefolds, and we thank Peter Sarnak for asking a question that led us to address the general case. We heartily thank the referees for carefully reading this article and for several useful comments on an earlier version of it. Specially, we thank one of the referees who found a mistake in an earlier version of Lemma 3.15.

Funding

J.C. was supported by ANID Doctorado Nacional 21190304 and H.P. was supported by ANID (ex CONICYT) FONDECYT Regular grant 1190442 from Chile.

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  1. Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, 4860 Av. Vicuña Mackenna, Macul, RM, Chile

    Jerson Caro & Hector Pasten

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Caro, J., Pasten, H. A Chabauty–Coleman bound for surfaces. Invent. math. 234, 1197–1250 (2023). https://doi.org/10.1007/s00222-023-01217-1

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