Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T07:51:40.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Six unlikely intersection problems in search of effectivity

Published online by Cambridge University Press:  28 July 2016

P. HABEGGER ,
G. JONES  and
D. MASSER
P. HABEGGER
Affiliation:
Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland. e-mail: [email protected]
G. JONES
Affiliation:
School of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL. e-mail: [email protected]
D. MASSER
Affiliation:
Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate four properties related to an elliptic curve Et in Legendre form with parameter t: the curve Et has complex multiplication, E−t has complex multiplication, a point on Et with abscissa 2 is of finite order, and t is a root of unity. Combining all pairs of properties leads to six problems on unlikely intersections. Using a variety of techniques we solve these problems with varying degrees of effectivity (and for three of them we even present the list of all possible t).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 3 , May 2017 , pp. 447 - 477
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] André, Y. Finitude des couples d'invariants modulaires singuliers sur une courbe algébrique plane non modulaire. J. Reine Angew. Math. 505 (1998), 203–208.CrossRefGoogle Scholar
[2] André, Y. Shimura varieties, subvarieties, and CM points. Six lectures at the University of Hsinchu (Taiwan) (with an appendix by C.-L. Chai) http://math.cts.nthu.edu.tw/Mathematics/lecnotes/andre2001all.ps (2001).Google Scholar
[3] Badzyan, A. I. The Euler–Kronecker constant. Mathematical Notes 87 (2010), 31–42.Google Scholar
[4] Bilu, Y., Masser, D. and Zannier, U. An effective “theorem of André” for CM-points on a plane curve. Math. Proc. Camb. Phil. Soc. 154 (2013), no. 1, 145–152.Google Scholar
[5] Bombieri, E. and Gubler, W. Heights in Diophantine geometry. New Mathematical Monographs, vol. 4. (Cambridge University Press, Cambridge, 2006).Google Scholar
[6] Boxall, G. J. and Jones, G. O. Algebraic values of certain analytic functions. Int. Math. Res. Not. 2015 (2015), no. 4, 1141–1158.Google Scholar
[7] Cox, D. A. Primes of the form x 2 + ny 2 (John Wiley & Sons, 1989).Google Scholar
[8] David, S. Points de petite hauteur sur les courbes elliptiques. J. Number Theory 64 (1997), no. 1, 104–129.Google Scholar
[9] David, S. and Hirata–Kohno, N. Linear forms in elliptic logarithms. J. Reine Angew. Math. 628 (2009), 37–89.Google Scholar
[10] Dwork, B. p-adic cycles. Inst. Hautes Études Sci. Publ. Math. (1969), no. 37, 27–115.Google Scholar
[11] Habegger, P. Weakly bounded height on modular curves. Acta Math. Vietnam. 35 (2010), no. 1, 43–69.Google Scholar
[12] Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers (Oxford University Press, 2005).Google Scholar
[13] Hua, L. K. Introduction to Number Theory. (Springer-Verlag, Berlin-New York, 1982), translated from the Chinese by Peter Shiu.Google Scholar
[14] Husemöller, D. Elliptic curves, second ed. Graduate Texts in Mathematics, vol. 111 (Springer-Verlag, New York, 2004) With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen.Google Scholar
[15] Ihara, Y. On the Euler-Kronecker constants of global fields and primes with small norms. Algebraic geometry and number theory. Progr. Math. vol. 253 (Birkhäuser Boston, 2006), pp. 407–451.Google Scholar
[16] Kühne, L. An effective result of André–Oort type. Ann. of Math. (2) 176 (2012), no. 1, 651–671.Google Scholar
[17] Kühne, L. An effective result of André–Oort type II. Acta Arith. 161 (2013), no. 1, 1–19.CrossRefGoogle Scholar
[18] Lang, S. Elliptic Functions (Springer, 1987).Google Scholar
[19] Lang, S. Complex analysis, fourth ed. Graduate Texts in Mathematics, vol. 103 (Springer-Verlag, New York, 1999).Google Scholar
[20] Loher, T. and Masser, D. Uniformly counting points of bounded height. Acta Arith. 111 (2004), no. 3, 277–297.Google Scholar
[21] Masser, D. Rational values of the Riemann zeta function. J. Number Theory 131 (2011), no. 11, 2037–2046.CrossRefGoogle Scholar
[22] Masser, D. and Zannier, U. Torsion anomalous points and families of elliptic curves. Amer. J. Math. 132 (2010), no. 6, 1677–1691.Google Scholar
[23] Masser, D. and Zannier, U. Torsion points on families of squares of elliptic curves. Math. Ann. 352 (2012), no. 2, 453–484.Google Scholar
[24] Nakkajima, Y. and Taguchi, Y. A generalisation of the Chowla-Selberg formula. J. Reine Angew. Math. 419 (1991), 119–124.Google Scholar
[25] Parish, J. L. Rational torsion in complex-multiplication elliptic curves. J. Number Theory 33 (1989), no. 2, 257–265.Google Scholar
[26] Paulin, R. An explicit André–Oort type result for $\mathbb{P}^1(\mathbb{C}) \times \mathbb{G}_m(\mathbb{C})$ based on logarithmic forms. Publ. Math. Debrecen 88 (2016), no. 1–2, 21–33.Google Scholar
[27] Paulin, R. An explicit André–Oort type result for $\Bbb{P}^1(\Bbb{C})\times\Bbb{G}_m(\Bbb{C})$ . Math. Proc. Camb. Phil. Soc. 159 (2015), no. 1, 153–163.Google Scholar
[28] Pila, J. Rational points of definable sets and results of André–Oort–Manin–Mumford type. Int. Math. Res. Not. IMRN (2009), no. 13, 2476–2507.Google Scholar
[29] Pila, J. O-minimality and the André–Oort conjecture for ℂ n . Ann. of Math. (2011), no. 173, 1779–1840.Google Scholar
[30] Pink, R. A common generalisation of the conjectures of André–Oort, Manin–Mumford and Mordell–Lang. Preprint (2005), 13pp.Google Scholar
[31] Poonen, B. Spans of hecke points on modular curves. Mathematical Research Letters 8 (2001), 767–770.Google Scholar
[32] Rosser, J. B. and Schoenfeld, L. Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94.Google Scholar
[33] Silverman, J. H. The Arithmetic of Elliptic Curves (Springer, 1986).Google Scholar
[34] Stoll, M. Simultaneous torsion in the Legendre family. Exp. Math., accepted for publication.Google Scholar
[35] Titchmarsh, E. C. The Theory of Functions (Oxford University Press, 1939).Google Scholar
[36] Waldschmidt, M. Diophantine approximation on linear algebraic groups. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 326 (Springer-Verlag, Berlin, 2000). Transcendence properties of the exponential function in several variables.Google Scholar
[37] Weinberger, P. J. Exponents of the class groups of complex quadratic fields. Acta Arith. 22 (1973), 117–124.Google Scholar
[38] Wüstholz, G. A note on the conjectures of André–Oort and Pink with an appendix by Lars Kühne. Bull. Inst. Math. Acad. Sin. (N.S.) 9 (2014), no. 4, 735–779. With an appendix by Lars Kühne.Google Scholar
[39] Zannier, U. Some problems of unlikely intersections in arithmetic and geometry. Ann. of Math. Stud. vol. 181 (Princeton University Press, Princeton, NJ, 2012). With appendixes by David Masser.Google Scholar
[40] Zilber, B. Exponential sums equations and the Schanuel conjecture. J. London Math. Soc. (2) 65 (2002), no. 1, 27–44.Google Scholar