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A Modular André–Oort Statement with Derivatives

  • Haden Spence (a1)

In unpublished notes, Pila discussed some theory surrounding the modular function j and its derivatives. A focal point of these notes was the statement of two conjectures regarding j, j′ and j″: a Zilber–Pink-type statement incorporating j, j′ and j″, which was an extension of an apparently weaker conjecture of André–Oort type. In this paper, I first cover some background regarding j, j′ and j″, mostly covering the work already done by Pila. Then I use a seemingly novel adaptation of the o-minimal Pila–Zannier strategy to prove a weakened version of Pila's ‘Modular André–Oort with Derivatives’ conjecture. Under the assumption of a certain number-theoretic conjecture, the central theorem of the paper implies Pila's conjecture in full generality, as well as a more precise statement along the same lines.

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1.André, Y., Finitude des couples d'invariants modulaires singuliers sur une courbe algébrique plane non modulaire, J. Reine Angew. Math. 505 (1998), 203208.
2.Bertolin, C., Priodes de 1-motifs et transcendance, J. Number Theory 97(2) (2002), 204221.
3.Bombieri, E. and Gubler, W., Heights in diophantine geometry, New Mathematical Monographs, Volume 4 ( Cambridge University Press, Cambridge, 2006).
4.Diaz, G., Transcendance et indépendance algébrique: liens entre les points de vue elliptique et modulaire, Ramanujan J. 4(2) (2000), 157199.
5.Habegger, P., Weakly bounded height on modular curves, Acta Math. Vietnam 35(1) (2010), 4369.
6.Habegger, P. and Pila, J., Some unlikely intersections beyond André–Oort, Compos. Math. 148(1) (2012), 127.
7.Klingler, B. and Yafaev, A., The André–Oort conjecture, Ann. Math. (2) 180(3) (2014), 867925.
8.Masser, D., Elliptic functions and transcendence, Lecture Notes in Mathematics, Volume 437 (Springer-Verlag, Berlin–New York, 1975).
9.Mertens, M. H. and Rolen, L., On class invariants for non-holomorphic modular functions and a question of Bruinier and Ono, Res. Number Theory 1 (2015), Art. 4, 13.
10.Orr, M., Height bounds and the Siegel property, arXiv:1609.01315 (2016).
11.Pellarin, F., Sur une majoration explicite pour un degré d'isogénie liant deux courbes elliptiques, Acta Arith. 100(3) (2001), 203243.
12.Peterzil, Y. and Starchenko, S., Uniform definability of the Weierstrass ℘ functions and generalized tori of dimension one, Sel. Math. (N.S.) 10(4) (2004), 525550.
13.Pila, J., Rational points of definable sets and results of André–Oort–Manin–Mumford type, Int. Math. Res. Not. IMRN 13 (2009), 24762507.
14.Pila, J., O-minimality and the André–Oort conjecture for ℂn, Ann. Math. (2) 173(3) (2011), 17791840.
15.Pila, J., Modular Ax–Lindemann–Weierstrass with derivatives, Notre Dame J. Form. Log. 54(3–4) (2013), 553565.
16.Pila, J. and Wilkie, A. J., The rational points of a definable set, Duke Math. J. 133(3) (2006), 591616.
17.Scanlon, T., Automatic uniformity, Int. Math. Res. Not. 62 (2004), 33173326.
18.Siegel, C., Über die Klassenzahl quadratischer Zahlkörper, Acta Arith. 1(1) (1935), 8386.
19.Silverman, J. H., Heights and elliptic curves, pp. 253265 ( Springer, New York, 1986).
20Spence, H., André–Oort for a nonholomorphic modular function, Journal de Théorie des Nombres de Bordeaux (to appear), Preprint available at arXiv:1607.03769.
21.van den Dries, L., Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, Volume 248 ( Cambridge University Press, Cambridge, 1998).
22.van den Dries, L. and Miller, C., On the real exponential field with restricted analytic functions, Israel J. Math. 85(1–3) (1994), 1956.
23.Zagier, D., Elliptic modular forms and their applications, in The 1-2-3 of modular forms, pp. 1103 ( Universitext, Springer, Berlin, 2008).
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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