Relative Manin–Mumford in additive extensions
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Abstract:
In recent papers Masser and Zannier have proved various results of “relative Manin–Mumford” type for various families of abelian varieties, some with field of definition restricted to the algebraic numbers. Typically these imply the finiteness of the set of torsion points on a curve in the family. After Bertrand, Masser, and Zannier discovered some surprising counterexamples for multiplicative extensions of elliptic families, the three authors together with Pillay settled completely the situation for this case over the algebraic numbers. Here we treat the last remaining case of surfaces, that of additive extensions of elliptic families, and even over the field of all complex numbers. In particular analogous counterexamples do not exist. There are finiteness consequences for Pell’s equation over polynomial rings and integration in elementary terms. Our work can be made effective (as opposed to most of that preceding), mainly because we use counting results only for analytic curves.References
- D. Bertrand, Extensions de $D$-modules et groupes de Galois différentiels, $p$-adic analysis (Trento, 1989) Lecture Notes in Math., vol. 1454, Springer, Berlin, 1990, pp. 125–141 (French). MR 1094849, DOI 10.1007/BFb0091136
- D. Bertrand, Special points and Poincaré bi-extensions, arXiv:1104.5178 (2011). With an appendix by Bas Edixhoven.
- Daniel Bertrand, Generalized jacobians and Pellian polynomials, J. Théor. Nombres Bordeaux 27 (2015), no. 2, 439–461 (English, with English and French summaries). MR 3393162, DOI 10.5802/jtnb.909
- Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
- D. Bertrand, D. Masser, A. Pillay, and U. Zannier, Relative Manin-Mumford for semi-Abelian surfaces, Proc. Edinb. Math. Soc. (2) 59 (2016), no. 4, 837–875. MR 3570118, DOI 10.1017/S0013091515000486
- E. Bombieri, D. Masser, and U. Zannier, On unlikely intersections of complex varieties with tori, Acta Arith. 133 (2008), no. 4, 309–323. MR 2457263, DOI 10.4064/aa133-4-2
- E. Bombieri and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), no. 2, 337–357. MR 1016893, DOI 10.1215/S0012-7094-89-05915-2
- Pietro Corvaja, David Masser, and Umberto Zannier, Sharpening ‘Manin-Mumford’ for certain algebraic groups of dimension 2, Enseign. Math. 59 (2013), no. 3-4, 225–269. MR 3189035, DOI 10.4171/LEM/59-3-2
- Sinnou David, Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64 (1997), no. 1, 104–129 (French, with English and French summaries). MR 1450488, DOI 10.1006/jnth.1997.2100
- Philipp Habegger, Special points on fibered powers of elliptic surfaces, J. Reine Angew. Math. 685 (2013), 143–179. MR 3181568, DOI 10.1515/crelle-2012-0007
- G.-H. Halphen, Traité des fonctions elliptiques et de leurs applications, I, Gauthier-Villars, Paris, 1886.
- Marc Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575–603 (French). MR 969244, DOI 10.1007/BF01394276
- Dale Husemöller, Elliptic curves, 2nd ed., Graduate Texts in Mathematics, vol. 111, Springer-Verlag, New York, 2004. With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen. MR 2024529
- Nicholas M. Katz, $p$-adic interpolation of real analytic Eisenstein series, Ann. of Math. (2) 104 (1976), no. 3, 459–571. MR 506271, DOI 10.2307/1970966
- Serge Lang, Introduction to algebraic geometry, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1972. Third printing, with corrections. MR 0344244
- Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960, DOI 10.1007/978-1-4612-4752-4
- Serge Lang, Algebra, 2nd ed., Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. MR 783636
- Qing Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. Translated from the French by Reinie Erné; Oxford Science Publications. MR 1917232
- D. Masser, Rational values of the Riemann zeta function, J. Number Theory 131 (2011), no. 11, 2037–2046. MR 2825110, DOI 10.1016/j.jnt.2011.03.013
- Ju. I. Manin, Rational points on algebraic curves over function fields, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1395–1440 (Russian). MR 0157971
- D. Masser and U. Zannier, Torsion anomalous points and families of elliptic curves, Amer. J. Math. 132 (2010), no. 6, 1677–1691. MR 2766181
- D. Masser and U. Zannier, Torsion points on families of squares of elliptic curves, Math. Ann. 352 (2012), no. 2, 453–484. MR 2874963, DOI 10.1007/s00208-011-0645-4
- David Masser and Umberto Zannier, Bicyclotomic polynomials and impossible intersections, J. Théor. Nombres Bordeaux 25 (2013), no. 3, 635–659 (English, with English and French summaries). MR 3179679, DOI 10.5802/jtnb.851
- D. Masser and U. Zannier, Torsion points on families of products of elliptic curves, Adv. Math. 259 (2014), 116–133. MR 3197654, DOI 10.1016/j.aim.2014.03.016
- David Masser and Umberto Zannier, Torsion points on families of simple abelian surfaces and Pell’s equation over polynomial rings, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 9, 2379–2416. With an appendix by E. V. Flynn. MR 3420511, DOI 10.4171/JEMS/560
- D. Masser and U. Zannier, Torsion points on families of abelian varieties, Pell’s equation, and integration in elementary terms (in preparation).
- Jonathan Pila, Integer points on the dilation of a subanalytic surface, Q. J. Math. 55 (2004), no. 2, 207–223. MR 2068319, DOI 10.1093/qjmath/55.2.207
- Richard Pink, A combination of the conjectures of Mordell-Lang and André-Oort, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 251–282. MR 2166087, DOI 10.1007/0-8176-4417-2_{1}1
- Robert H. Risch, The solution of the problem of integration in finite terms, Bull. Amer. Math. Soc. 76 (1970), 605–608. MR 269635, DOI 10.1090/S0002-9904-1970-12454-5
- Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI 10.1515/crll.1983.342.197
- Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
- Umberto Zannier, Some problems of unlikely intersections in arithmetic and geometry, Annals of Mathematics Studies, vol. 181, Princeton University Press, Princeton, NJ, 2012. With appendixes by David Masser. MR 2918151
- Umberto Zannier, Unlikely intersections and Pell’s equations in polynomials, Trends in contemporary mathematics, Springer INdAM Ser., vol. 8, Springer, Cham, 2014, pp. 151–169. MR 3586397
- Boris Zilber, Exponential sums equations and the Schanuel conjecture, J. London Math. Soc. (2) 65 (2002), no. 1, 27–44. MR 1875133, DOI 10.1112/S0024610701002861
Additional Information
- Harry Schmidt
- Affiliation: Department of Mathematics, University of Basel, Rheinsprung 21, Basel, Switzerland
- MR Author ID: 1092431
- Received by editor(s): October 29, 2015
- Received by editor(s) in revised form: December 4, 2017
- Published electronically: November 13, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6463-6486
- MSC (2010): Primary 11G50, 14H05, 14H52, 14L15, 33C05; Secondary 14J99
- DOI: https://doi.org/10.1090/tran/7612
- MathSciNet review: 3937333