## Duke Mathematical Journal

### Uniform bounds for the number of rational points on curves of small Mordell–Weil rank

#### Abstract

Let $X$ be a curve of genus $g\geq2$ over a number field $F$ of degree $d=[F:\mathbf{Q}]$. The conjectural existence of a uniform bound $N(g,d)$ on the number $\#X(F)$ of $F$-rational points of $X$ is an outstanding open problem in arithmetic geometry, known by the work of Caporaso, Harris, and Mazur to follow from the Bombieri–Lang conjecture. A related conjecture posits the existence of a uniform bound $N_{\mathrm{tors},\dagger}(g,d)$ on the number of geometric torsion points of the Jacobian $J$ of $X$ which lie on the image of $X$ under an Abel–Jacobi map. For fixed $X$, the finiteness of this quantity is the Manin–Mumford conjecture, which was proved by Raynaud.

We give an explicit uniform bound on $\#X(F)$ when $X$ has Mordell–Weil rank $r\leq g-3$. This generalizes recent work of Stoll on uniform bounds for hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of $F$-rational torsion points of $J$ lying on the image of $X$ under an Abel–Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of $J$ lying on $X$ when the reduction type of $X$ is highly degenerate.

Our methods combine Chabauty–Coleman’s $p$-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.

#### Article information

Source
Duke Math. J., Volume 165, Number 16 (2016), 3189-3240.

Dates
Revised: 19 December 2015
First available in Project Euclid: 14 October 2016

https://projecteuclid.org/euclid.dmj/1476450482

Digital Object Identifier
doi:10.1215/00127094-3673558

Mathematical Reviews number (MathSciNet)
MR3566201

Zentralblatt MATH identifier
06666955

Subjects
Primary: 14G05: Rational points

#### Citation

Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David. Uniform bounds for the number of rational points on curves of small Mordell–Weil rank. Duke Math. J. 165 (2016), no. 16, 3189--3240. doi:10.1215/00127094-3673558. https://projecteuclid.org/euclid.dmj/1476450482

#### References

• [1] O. Amini, M. Baker, E. Brugallé, and J. Rabinoff, Lifting harmonic morphisms, I: Metrized complexes and Berkovich skeleta, Res. Math. Sci. 2 (2015), Art. 7.
• [2] M. Baker, Specialization of linear systems from curves to graphs, Algebra Number Theory 2 (2008), 613–653.
• [3] M. Baker and X. Faber, Metric properties of the tropical Abel-Jacobi map, J. Algebraic Combin. 33 (2011), 349–381.
• [4] M. Baker, S. Payne, and J. Rabinoff, “On the structure of nonarchimedean analytic curves” in Tropical and Non-Archimedean Geometry, Contemp. Math. 605, Amer. Math. Soc., Providence, 2013, 93–121.
• [5] M. Baker and J. Rabinoff, The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to alegebraic curves, Int. Math. Res. Not. IMRN 2014, no. 16.
• [6] M. Baker and R. Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, Math. Surv. Monogr. 159, Amer. Math. Soc., Providence, 2010.
• [7] J. S. Balakrishnan, R. W. Bradshaw, and K. S. Kedlaya, “Explicit Coleman integration for hyperelliptic curves” in Algorithmic Number Theory, Lecture Notes in Comput. Sci. 6197, Springer, Berlin, 2010, 16–31.
• [8] V. G. Berkovich, Spectral Theory and Analytic Geometry Over Non-Archimedean Fields, Math. Surv. Monogr. 33, Amer. Math. Soc., Providence, 1990.
• [9] V. G. Berkovich, Integration of One-Forms on $p$-Adic Analytic Spaces, Ann. of Math. Stud. 162, Princeton Univ. Press, Princeton, 2007.
• [10] A. Besser and S. Zerbes, Vologodsky integration on semi-stable curves, in preparation.
• [11] M. Bhargava, Most hyperelliptic curves over $\mathbf{Q}$ have no rational points, preprint, arXiv:1308.0395v1 [math.NT].
• [12] M. Bhargava and B. H. Gross, “The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point” in Automorphic Representations and L-Functions, Tata Inst. Fund. Res. Stud. Math. 22, Tata Inst. Fund. Res., Mumbai, 2013.
• [13] M. Bhargava, B. H. Gross, and X. Wang, Pencils of quadrics and the arithmetic of hyperelliptic curves, preprint, arXiv:1310.7692v1 [math.NT].
• [14] M. Bhargava and A. Shankar, The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1, preprint, arXiv:1312.7859v1 [math.NT].
• [15] S. Bosch and W. Lütkebohmert, Stable reduction and uniformization of abelian varieties, II, Invent. Math. 78 (1984), 257–297.
• [16] S. Bosch and W. Lütkebohmert, Degenerating abelian varieties, Topology 30 (1991), 653–698.
• [17] S. Bosch and W. Lütkebohmert, Formal and rigid geometry, I: Rigid spaces, Math. Ann. 295 (1993), 291–317.
• [18] C. Breuil, Intégration sur les variétés $p$-adiques (d’après Coleman, Colmez), Astérisque 266 (2000), 319–350, Séminaire Bourbaki, Vol. 1998/1999, no. 860.
• [19] A. Buium, Geometry of $p$-jets, Duke Math. J. 82 (1996), 349–367.
• [20] L. Caporaso, J. Harris, and B. Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), 1–35.
• [21] C. Chabauty, Sur les points rationnels des courbes algébriques de genre supérieur à l’unité, C. R. Acad. Sci. Paris 212 (1941), 882–885.
• [22] A. Chambert-Loir, Mesures et équidistribution sur les espaces de Berkovich, J. Reine Angew. Math. 595 (2006), 215–235.
• [23] A. Chambert-Loir, “Heights and measures on analytic spaces: A survey of recent results, and some remarks” in Motivic Integration and Its Interactions with Model Theory and non-Archimedean Geometry, Vol. II, London Math. Soc. Lecture Note Ser. 384, Cambridge Univ. Press, Cambridge, 2011, 1–50.
• [24] C. Christensen, Erste Chernform und Chambert–Loir Maße auf dem Quadrat einer Tate-Kurve, Ph.D. dissertation, Universität Tübingen, Tübingen, Germany, 2013, https://publikationen.uni-tuebingen.de/xmlui/handle/10900/52823.
• [25] A. Cohen, M. Temkin, and D. Trushin, Morphisms of Berkovich curves and the different function, preprint, arXiv:1408.2949v2 [math.AG].
• [26] R. F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), 765–770.
• [27] R. F. Coleman, Ramified torsion points on curves, Duke Math. J. 54 (1987), 615–640.
• [28] R. F. Coleman, Reciprocity laws on curves, Compos. Math. 72 (1989), 205–235.
• [29] R. F. Coleman and E. de Shalit, $p$-Adic regulators on curves and special values of $p$-adic $L$-functions, Invent. Math. 93 (1988), 239–266.
• [30] R. F. Coleman and A. Iovita, The Frobenius and monodromy operators for curves and abelian varieties, Duke Math. J. 97 (1999), 171–215.
• [31] P. Colmez, Périodes $p$-adiques des variétés abéliennes, Math. Ann. 292 (1992), 629–644.
• [32] E. de Shalit, Coleman integration versus Schneider integration on semistable curves, Doc. Math. Extra Vol. (2006), 325–334 (electronic).
• [33] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst. Hautes Études Sci. Publ. Math. (1969), 75–109.
• [34] M. Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), 575–603.
• [35] W. Ho, How many rational points does a random curve have? Bull. Amer. Math. Soc. (N.S.) 51 (2014), 27–52.
• [36] E. Katz and D. Zureick-Brown, The Chabauty-Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions, Compos. Math. 149 (2013), 1818–1838.
• [37] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Grad. Texts in Math. 6, Oxford Univ. Press, Oxford, 2002.
• [38] D. Lorenzini and T. J. Tucker, Thue equations and the method of Chabauty-Coleman, Invent. Math. 148 (2002), 47–77.
• [39] W. McCallum and B. Poonen, “The method of Chabauty and Coleman” in Explicit Methods in Number Theory, Panor. Synthèses, 36, Soc. Math. France, Paris, 2012, 99–117.
• [40] G. Mikhalkin and I. Zharkov, “Tropical curves, their Jacobians and theta functions” in Curves and Abelian Varieties, Contemp. Math. 465, Amer. Math. Soc., Providence, 2008, 203–230.
• [41] J. M. Y. Park, Effective Chabauty for symmetric powers of curves, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Mass., 2014.
• [42] J. Pila and U. Zannier, Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19 (2008), 149–162.
• [43] B. Poonen and M. Stoll, Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2) 180 (2014), 1137–1166.
• [44] M. Raynaud, Courbes sur une variété abélienne et points de torsion, Invent. Math. 71 (1983), 207–233.
• [45] A. Shankar and X. Wang, Average size of the $2$-Selmer group of Jacobians of monic even hyperelliptic curves, preprint, arXiv:1307.3531v2 [math.NT].
• [46] S. Siksek, Chabauty for symmetric powers of curves, Algebra Number Theory 3 (2009), 209–236.
• [47] A. Silverberg and Y. G. Zarhin, Semistable reduction and torsion subgroups of abelian varieties, Ann. Inst. Fourier (Grenoble) 45 (1995), 403–420.
• [48] M. Stoll, Independence of rational points on twists of a given curve, Compos. Math. 142 (2006), 1201–1214.
• [49] M. Stoll, Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank, preprint, arXiv:1307.1773v6 [math.NT].
• [50] M. Temkin, Metrization of differential pluriforms on Berkovich analytic spaces, preprint, arXiv:1410.3079v2 [math.AG].
• [51] J. A. Thorne, $E_{6}$ and the arithmetic of a family of non-hyperelliptic curves of genus 3, Forum Math. Pi 3 (2015), e1.
• [52] A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne: Applications à la théorie d’Arakelov, Ph.D. dissertation, University of Rennes, Rennes, France, 2005, http://tel.archives-ouvertes.fr/docs/00/04/87/50/PDF/tel-00010990.pdf.
• [53] E. Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. (2) 147 (1998), 167–179.
• [54] V. Vologodsky, Hodge structure on the fundamental group and its application to $p$-adic integration, Mosc. Math. J. 3 (2003), 205–247, 260.
• [55] Y. G. Zarhin, $p$-Adic abelian integrals and commutative Lie groups, J. Math. Sci. 81 (1996), 2744–2750.
• [56] S. Zhang, Admissible pairing on a curve, Invent. Math. 112 (1993), 171–193.