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CATEGORICITY OF MODULAR AND SHIMURA CURVES

  • Christopher Daw (a1) and Adam Harris (a2)
Abstract

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the model-theoretic statement of a certain ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$ -sentence having a unique model of cardinality $\aleph _{1}$ is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this ${\mathcal{L}}_{\unicode[STIX]{x1D714}_{1},\unicode[STIX]{x1D714}}$ -sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

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1. Baily, W. L. and Borel, A., Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442528.
2. Bays, M., Hart, B., Hyttinen, T., Kesälä, M. and Kirby, J., Quasiminimal structures and excellence, Bull. LMS 46(1) (2014), 155163.
3. Bays, M. and Zilber, B., Covers of multiplicative groups of algebraically closed fields of arbitrary characteristic, Bull. Lond. Math. Soc. 43(4) (2011), 689702.
4. Cadoret, A., An open adelic image theorem for abelian schemes, Int. Math. Res. Not. IMRN, 2014, to appear.
5. Daw, C. and Orr, M., Heights of pre-special points of Shimura varieties, 2015, available on the first author’s web page.
6. Keisler, H. J., On the quantifier ‘there exist uncountably many x, Ann. Math. Log. 1 (1970), 191.
7. Kirby, J., On quasiminimal excellent classes, J. Symbolic Logic 75(2) (2010), 551564.
8. Klingler, B. and Yafaev, A., The André–Oort conjecture, Ann. of Math. (3) 180 (2006), 867925.
9. Lang, S., Elliptic functions: GTM 112, 2nd edn (Springer, New York, 1987).
10. Marker, D., Model Theory: An Introduction (Springer, New York, 2002).
11. Milne, J. S., Introduction to Shimura varieties, in Harmonic Analysis, the Trace Formula and Shimura Varieties, (ed. Arthur, J. and Kottwitz, R.), Lectures at the Summer School held at the Fields Institute, June 2–June 27, 2003, (The AMS and the Clay Mathematics Institute, Cambridge, MA, 2005).
12. Milne, J. S., Modular functions and modular forms, 2012, available at www.jmilne.org/math/CourseNotes/mf.
13. Gavrilovich, M., A remark on transitivity of Galois action on the set of uniquely divisible abelian extensions in …, K-Theory 38(2) (2008), 135152.
14. Ohta, M., On l-adic representations of Galois groups obtained from certain two dimensional abelian varieties, J. Fac. Sci. Univ. Tokyo 21 (1974), 299308.
15. Pillay, A., Model theory, 2002, Lecture notes available on author’s web page.
16. Pink, R., A combination of the conjectures of Mordell–Lang and André–Oort, in Geometric Methods in Algebra and Number Theory,(ed. Bogomolov, F. and Tschinkel, Y.), Progress in Mathematics, Volume 253, pp. 251282 (Birkhäuser, 2005).
17. Ribet, K. A., On l-adic representations attached to modular forms, Invent. Math. 28 (1975), 245275.
18. Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259331.
19. Ullmo, E. and Yafaev, A., Mumford–Tate and generalized Shafarevich conjectures, Ann. Sci. Québec 37(2) (2013), 255284.
20. Zilber, B., Covers of the multiplicative group of an algebraically closed field of characteristic zero, J. Lond. Math. Soc. (2) 74(1) (2006), 4158.
21. Zilber, B., Model theory, geometry and arithmetic of the universal cover of a semi-abelian variety, in Model Theory and Applications, Quad. Mat., Volume 11, pp. 427458. (Aracne, Rome, 2002).
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Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
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