×

zbMATH — the first resource for mathematics

Definability of restricted theta functions and families of abelian varieties. (English) Zbl 1284.03215
In [Ann. Math. (2) 173, No. 3, 1779–1840 (2011; Zbl 1243.14022)], J. Pila gave an unconditional proof of the André-Oort conjecture for the Shimura variety \(\mathbb C^n\). One of the central ingredients to this proof is that the Weierstrass \(\wp\) function is, in a suitable sense, definable in the o-minimal structure \(\mathbb R_{\mathrm{an}, \mathrm{exp}}\). The present paper proves the analogue of this ingredient for Siegel modular varieties \(\text{Sp}(2g, \mathbb Z) \backslash \mathbb H_g\), \(g \geq 1\). Note however that to prove the André-Oort conjecture in this generality along the same lines, several other ingredients are still missing.
More precisely, fix \(g \geq 1\) and a diagonal matrix \(D = \text{Diag}(d_1, \dots, d_g)\) with \(d_1 \mid d_2 \mid \dots \mid d_g\) non-negative integers. Then all the complex abelian varieties of polarization type \(D\) are of the form \(\mathcal{E}_\tau^D := \mathbb C^g/(\tau \mathbb Z^g + D \mathbb Z^g)\), where \(\tau\) runs over the set \(\mathbb H_g\) of symmetric \((g \times g)\)-matrices with a positive definite imaginary part. Write \(h_\tau^D : \mathbb C^g \to \mathbb P^k (\mathbb C)\) for a suitable holomorphic map inducing an embedding \(\mathcal{E}_\tau^D \to \mathbb P^k (\mathbb C)\) (where \(k\) only depends on \(D\)). Morally, the main result of the paper (Theorem 1.2) is that the map \(\mathbb H_g \times \mathbb C^g \to \mathbb P^k (\mathbb C), (\tau, z) \mapsto h_\tau^D(z)\) is definable in \(\mathbb R_{\mathrm{an}, \mathrm{exp}}\).
Formulated like this, this is not possible for two reasons: \(h_\tau^D\) is periodic for each fixed \(\tau\), and, moreover, \(\tau \mapsto \mathcal{E}_\tau^D\) is periodic in the sense that \(\mathcal{E}_\tau^D\) and \(\mathcal{E}_{\tau'}^D\) are isomorphic as polarized abelian varieties iff \(\tau\) and \(\tau'\) lie in the same orbit of the action of \(G_D\) on \(\mathbb H_g\), where \(G_D\) is a discrete subgroup of \(\text{Sp}(2g, \mathbb Q)\). A more precise formulation of Theorem 1.2 is that the restriction of \((\tau, z) \mapsto h_\tau^D(z)\) to a subset \(U\) of \(\mathbb H_g \times \mathbb C^g\) is definable in \(\mathbb R_{\mathrm{an}, \mathrm{exp}}\), where the projection of \(U\) to \(\mathbb H_g\) contains a fundamental domain of the action of \(G_D\) on \(\mathbb H_g\), and for each \(\tau\) in this projection, the corresponding fiber of \(U\) contains a fundamental domain of the map \(\mathbb C^g \to \mathcal{E}_\tau^D\).
The paper also contains several related results that might be of independent interest. In particular, one of the main ingredients to the proof of Theorem 1.2 is the definability of certain Riemann theta functions \(\vartheta[{a \atop b}](z, \tau)\).

MSC:
03C64 Model theory of ordered structures; o-minimality
03C98 Applications of model theory
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11G18 Arithmetic aspects of modular and Shimura varieties
14G35 Modular and Shimura varieties
14K25 Theta functions and abelian varieties
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] W. L. Baily Jr., On the theory of \(\theta\)-functions, the moduli of abelian varieties, and the moduli of curves , Ann. of Math. (2) 75 (1962), 342-381. · Zbl 0147.39702 · doi:10.2307/1970178
[2] W. L. Baily Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains , Ann. of Math. (2) 84 (1966), 442-528. · Zbl 0154.08602 · doi:10.2307/1970457
[3] C. Birkenhake and H. Lange, Complex Abelian Varieties , 2nd ed., Grundlehren Math. Wiss. 302 , Springer, Berlin, 2004. · Zbl 1056.14063
[4] J.-B. Bost, “Introduction to compact Riemann surfaces, Jacobians, and abelian varieties” in From Number Theory to Physics (Les Houches, 1989) , Springer, Berlin, 1992, 64-211. · Zbl 0815.14018 · doi:10.1007/978-3-662-02838-4_2
[5] A. M. Gabrièlov, Projections of semianalytic sets , Funkcional. Anal. i Priložen 2 (1968), 18-30. · Zbl 0179.08503 · doi:10.1007/BF01075680
[6] J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis , Grundlehren Text Ed., Springer, Berlin, 2001. · Zbl 0998.49001
[7] Jun-ichi Igusa, On the graded ring of theta-constants, II , Amer. J. Math. 88 (1966), 221-236. · Zbl 0146.31704 · doi:10.2307/2373057
[8] J.-i. Igusa, Theta Functions , Grundlehren Math. Wiss. 194 , Springer, New York, 1972.
[9] H. Klingen, Introductory Lectures on Siegel Modular Forms , Cambridge Stud. Adv. Math. 20 , Cambridge Univ. Press, Cambridge, 1990. · Zbl 0693.10023
[10] B. Klingler and A. Yafaev, The André-Oort conjecture , preprint, [math.NT]. 1209.0936 · arxiv.org
[11] D. Mumford, Tata Lectures on Theta, I , Progr. Math. 28 , Birkhäuser, Boston, 1983. · Zbl 0509.14049
[12] Y. Peterzil and S. Starchenko, Uniform definability of the Weierstrass \(\wp\) functions and generalized tori of dimension one , Selecta Math. (N.S.) 10 (2004), 525-550. · Zbl 1071.03022 · doi:10.1007/s00029-005-0393-y
[13] Ya’acov Peterzil and Sergei Starchenko, Complex analytic geometry and analytic-geometric categories , J. Reine Angew. Math. 626 (2009), 39-74. · Zbl 1168.03029 · doi:10.1515/CRELLE.2009.002
[14] Y. Peterzil and S. Starchenko, Mild manifolds and a non-standard Riemann existence theorem , Selecta Math. (N.S.) 14 (2009), 275-298. · Zbl 1173.03035 · doi:10.1007/s00029-008-0064-x
[15] Ya’acov Peterzil and Sergei Starchenko, “Tame complex analysis and o-minimality” in Proceedings of the International Congress of Mathematicians, Vol. II , Hindustan Book Agency, 2010, 58-81. · Zbl 1246.03061 · ebooks.worldscinet.com
[16] J. Pila, Rational points of definable sets and results of André-Oort-Manin-Mumford type , Int. Math. Res. Not. IMRN 13 (2009), 2476-2507. · Zbl 1243.14021
[17] J. Pila, O-minimality and the André-Oort conjecture for \(\mathbb{C}^{n}\) , Ann. of Math. (2) 173 (2011), 1779-1840. · Zbl 1243.14022 · doi:10.4007/annals.2011.173.3.11
[18] J. Pila and A. J. Wilkie, The rational points of a definable set , Duke Math. J. 133 (2006), 591-616. · Zbl 1217.11066 · doi:10.1215/S0012-7094-06-13336-7 · euclid:dmj/1150201203 · eprints.ma.man.ac.uk
[19] C. L. Siegel, Lectures on Quadratic Forms , notes by K. G. Ramanathan, Tata Inst. Fund. Res. Lectures Math. 7 , Tata Inst. Fund. Res., Bombay, 1967. · Zbl 0248.10019
[20] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, II , Springer, Berlin, 1988. · Zbl 0668.10033
[21] L. van den Dries, Tame Topology and O-Minimal Structures , London Math. Soc. Lecture Note Ser. 248 , Cambridge Univ. Press, Cambridge, 1998. · Zbl 0953.03045
[22] L. van den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation , Ann. of Math. (2) 140 (1994), 183-205. · Zbl 0837.12006 · doi:10.2307/2118545
[23] L. van den Dries and C. Miller, On the real exponential field with restricted analytic functions , Israel J. Math. 85 (1994), 19-56. · Zbl 0823.03017 · doi:10.1007/BF02758635
[24] Lou van den Dries and Christopher Miller, Geometric categories and o-minimal structures , Duke Math. J. 84 (1996), 497-540. · Zbl 0889.03025 · doi:10.1215/S0012-7094-96-08416-1
[25] G. van der Geer, “Siegel modular forms and their applications” in The 1-2-3 of Modular Forms (Nordfjordeid, Norway, 2004) , Universitext, Springer, Berlin, 2008, 181-245. · Zbl 1259.11051 · doi:10.1007/978-3-540-74119-0_3
[26] G. M. Ziegler, Lectures on Polytopes , Grad. Texts in Math. 152 , Springer, New York, 1995.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.