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From Fourier Analysis and Number Theory to Radon Transforms and Geometry pp 475–490Cite as

A Transcendence Criterion for CM on Some Families of Calabi–Yau Manifolds

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Part of the book series: Developments in Mathematics ((DEVM,volume 28))

Abstract

In this chapter, we give some examples of the validity of a special case of a recent conjecture of Green et al. (Ann. Math. Studies, no. 183, Princeton University Press, 2012). This special case is an analogue of a celebrated theorem of Schneider (Math. Annalen 113:1–13, 1937) on the transcendence of values of the elliptic modular function and its generalization in Cohen (Rocky Mountain J.Math. 26:987–1001, 1996) and Shiga and Wolfart (J. Reine Angew. Math. 463:1–25, 1995). Related techniques apply to all the examples of CMCY families in the work of Rohde (Lecture Notes in Mathematics 1975, Springer, Berlin, 2009), and this is the subject of a paper in preparation by the author (Tretkoff, Transcendence and CM on Borcea-Voisin towers of Calabi-Yau manifolds).

Mathematics Subject Classification(2010):11J81 (Primary), 14C30 (Secondary)

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Acknowledgements

The author is supported by NSF grant number DMS–0800311 and NSA grant 1100362.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77842–3368, USA

    Paula Tretkoff & Marvin D. Tretkoff

  2. CNRS, UMR 8524, Université de Lille 1, Cité Scientifique, 59655, Villeneuve d’Ascq CEDEX, France

    Paula Tretkoff

Authors
  1. Paula Tretkoff
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  2. Marvin D. Tretkoff
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Corresponding author

Correspondence to Paula Tretkoff .

Editor information

Editors and Affiliations

  1. , Einstein Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, 91904, Israel

    Hershel M. Farkas

  2. , Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, 08544, New Jersey, USA

    Robert C. Gunning

  3. , Department of Mathematics, Temple University, Wachman Hall 608, Philadelphia, 19122, Pennsylvania, USA

    Marvin I. Knopp

  4. , Department of Mathematics, University of Michigan, Church St. 530, Ann Arbor, 48109, Michigan, USA

    B. A. Taylor

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Dedicated to the memory of Leon Ehrenpreis, my Ph.D. advisor and friend for 50 years (Marvin Tretkoff)

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Tretkoff, P., Tretkoff, M.D. (2013). A Transcendence Criterion for CM on Some Families of Calabi–Yau Manifolds. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_23

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