Riv. Mat. Univ. Parma, Vol. 7, No. 1, 2016

Evelina Viada[1]

Explicit height bounds and the effective Mordell-Lang Conjecture

Pages: 101-131
Received: 3 December 2015   
Accepted: 5 February 2016.
Mathematics Subject Classification (2010): Primary 11G50, 11G05, Secondary 14G05.
Keywords: Heights, Rational Points, Anomalous Intersections, Mordell-Lang Conjecture.
Author address:
[1] : Department of mathematics, ETH Zurich, Rämistrasse 101, CH-8092 Zurich, Swithzerland.

Abstract: We first give an overview of some landmark theorems and recent conjectures in Diophantine Geometry. In the elliptic case, we prove some new bounds for torsion anomalous points and we clarify the implications of several height bounds on the effective Mordell-Lang Conjecture. In addition, we explicitly bound the Néron-Tate height of the rational points of a new family of curves of increasing genus, proving the effective Mordell Conjecture for these curves.

Work supported by the FNS Fonds National Suisse


[AD99] F. Amoroso and S. David, Le problème de Lehmer en dimension supérieure, J. Reine Angew. Math. 513 (1999), 145-179. MR1713323
[AD03] F. Amoroso and S. David, Minoration de la hauteur normalisée dans un tore, J. Inst. Math. Jussieu 2 (2003), no. 3, 335-381. MR1990219
[AV09] F. Amoroso and E. Viada, Small points on subvarieties of a torus, Duke Math. J. 150 (2009), no. 3, 407-442. MRMR2582101
[AV12] F. Amoroso and E. Viada, Small points on rational subvarieties of tori, Comment. Math. Helv. 87 (2012), no. 2, 355-383. MR2914852
[BS04] M. Baker and J. Silverman, A lower bound for the canonical height on abelian varieties over abelian extensions, Math. Res. Lett. 11 (2004), no. 2-3, 377-396. MR2067482
[Bog80] F. A. Bogomolov, Points of finite order on abelian varieties (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 4, 782804. MR0587337
[BG06] E. Bombieri and W. Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge 2006. MR2216774
[BMZ99] E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Internat. Math. Res. Notices 1999, no. 20, 1119-1140. MR1728021
[BMZ07] E. Bombieri, D. Masser and U. Zannier, Anomalous subvarieties-structure theorems and applications, Int. Math. Res. Not. IMRN 2007, no. 19, Art. ID rnm057, 33 pp. MR2359537
[BZ96] E. Bombieri and U. Zannier, Heights of algebraic points on subvarieties of abelian varieties, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 4, 779-792 (1997). MR1469574
[Car09] M. Carrizosa, Petits points et multiplication complexe, Int. Math. Res. Not. IMRN 2009, no. 16, 3016-3097. MR2533796
[CVV12] S. Checcoli, F. Veneziano and E. Viada, A sharp Bogomolov-type bound, New York J. Math. 18 (2012), 891-910. MR2991428
[CVV14] S. Checcoli, F. Veneziano and E. Viada, On torsion anomalous intersections, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25 (2014), no. 1, 1-36. MR3180478
[CVV15] S. Checcoli, F. Veneziano and E. Viada, On the explicit torsion anomalous conjecture, Trans. Amer. Math. Soc., to appear.
[CVV16] S. Checcoli, F. Veneziano and E. Viada, The explicit Mordell Conjecture for families of curves, arXiv:1602.04097, preprint 2016.
[CV14] S. Checcoli and E. Viada, On the torsion anomalous conjecture in CM abelian varieties, Pacific J. Math. 271 (2014), no. 2, 321-345. MR3267532
[CV16] S. Checcoli and E. Viada, An explicit Chaubauty Theorem, preprint 2016.
[DH00] S. David and M. Hindry, Minoration de la hauteur de Néron-Tate sur les variétés abéliennes de type C. M., J. Reine Angew. Math. 529 (2000), 1-74. MR1799933
[DP07] S. David and P. Philippon, Minorations des hauteurs normalisées des sous-variétés des puissances des courbes elliptiques, Int. Math. Res. Pap. IMRP 2007, no. 3, Art. ID rpm006, 113 pp. MR2355454
[Dob79] E. Dobrowolski, On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith. 34 (1979), 391-401. MR0543210
[Fal83] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349-366. MR0718935
[Fal91] G. Faltings, The general case of S. Lang's Conjecture, In: "Barsotti Symposium in Algebraic Geometry" (Abano Terme, 1991), Perspect. Math., 15, Academic Press, San Diego, CA 1994, 175-182. MR1307396
[Mas85] D. Masser, Small values of heights on families of abelian varieties, In: "Diophantine approximation and transcendence theory" (Bonn, 1985), Lecture Notes in Math., vol. 1290, Springer-Verlag, Berlin 1987, 109148. MR0927559
[Gal10] A. Galateau, Une minoration du minimum essentiel sur les variétés abéliennes, Comment. Math. Helv. 85 (2010), no. 4, 775-812. MR2718139
[Hab08] P. Habegger, Intersecting subvarieties of \(\mathbb{G}^n_m\) with algebraic subgroups, Math. Ann. 342 (2008), no. 2, 449-466. MR2425150
[Hab09] P. Habegger, Intersecting subvarieties of abelian varieties with algebraic subgroups of complementary dimension, Invent. Math. 176 (2009), no. 2, 405-447. MR2495768
[HP16] P. Habegger and J. Pila, O-Minimality and certain atypical intersections, Ann. Sci. Éc. Norm. Supér. 49 (2016), n. 4.
[Hin88] M. Hindry, Autour d'une conjecture de Serge Lang, Invent. Math. 94 (1988), 575-603. MR0969244
[Hru01] E. Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43-115. MR1854232
[Lau84] M. Laurent, Equations diophantiennes exponentielles, Invent. Math. 78 (1984), 299-327. MR0767195
[Leh33] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math (2) 34 (1933), 461-479. MR1503118
[Mau08] G. Maurin, Courbes algébriques et équations multiplicatives, Math. Ann. 341 (2008), no. 4, 789-824. MR2407327
[McC94] W. McCallum, On the method of Coleman and Chabauty, Math. Ann. 299 (1994), no. 3, 565-596. MR1282232
[MP10] W. McCallum and B. Poonen, The method of Chabauty and Coleman, Panor. Synthèses, 36, Soc. Math. France, Paris 2012, 99117. MR3098132
[Mcq95] M. McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995), no. 1, 143-159. MR1323985
[Phi91] P. Philippon, Sur des hauteurs alternatives. I, Math. Ann. 289 (1991), no. 2, 255-283. MR1092175
[Phi95] P. Philippon, Sur des hauteurs alternatives. III, J. Math. Pures Appl. (9) 74 (1995), no. 4, 345-365. MR1341770
[PZ08] 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), no. 2, 149-162. MR2411018
[Ray83] M. Raynaud, Sous-varié té d'une variété abélienne et points de torsion, In: "Arithmetic and geometry", vol. I, Birkhäuser Boston, Boston 1983, 327-352. MR0717600
[Rém00] G. Rémond, Décompte dans une conjecture de Lang, Invent. Math. 142 (2000), no. 3, 513-545. MR1804159
[Rém09] G. Rémond, Intersection de sous-groupes et de sous-variétés. III, Comment. Math. Helv. 84 (2009), no. 4, 835-863. MR2534482
[Ser89] J.-P. Serre, Lectures on the Mordell-Weil theorem, Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. MR1002324
[Sil07] J. H. Silverman, The arithmetic of dynamical system, Graduate Texts in Mathematics, 241, Springer, New York 2007. MR2316407
[Ulm98] E. Ullmo, Positivité et discrétion des points algébriques des courbes, Ann. of Math. 147 (1998), no.1, 167-179. MR1609514
[Via03] E. Viada, The intersection of a curve with algebraic subgroups in a product of elliptic curves, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 1, 47-75. MR1990974
[Via08] E. Viada, The intersection of a curve with a union of translated codimension-two subgroups in a power of an elliptic curve, Algebra Number Theory 2 (2008), no. 3, 249-298. MR2407116
[Via09] E. Viada, Nondense subsets of varieties in a power of an elliptic curve, Int. Math. Res. Not. IMRN 2009, no. 7, 1214-1246. MR2495303
[Via10] E. Viada, Lower bounds for the normalized height and non-dense subsets of subvarieties of abelian varieties, Int. J. Number Theory 6 (2010), no. 3, 471-499. MR2652892
[Voj96] P. Vojta, Integral points on subvarieties of semiabelian varieties. I, Invent. Math. 126 (1996), no. 1, 133-181. MR1408559
[Zha98] S.-W. Zhang, Equidistribution of small points on abelian varieties, Ann. of Math. (2) 147 (1998), no. 1, 159-165. MR1609518
[Zil02] B. Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. 65 (2002), no. 1, 27-44. MR1875133

Home Riv.Mat.Univ.Parma