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Vector-Valued Modular Forms with an Unnatural Boundary

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Book cover From Fourier Analysis and Number Theory to Radon Transforms and Geometry

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

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Abstract

We characterize all logarithmic, holomorphic vector-valued modular forms which can be analytically continued to a region strictly larger than the upper half-plane.

Marvin Isadore Knopp, born January 4, 1933, passed away unexpectedly on December 24, 2011.

Mathematics Subject Classification (2010): 11F12, 11F99

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Notes

  1. 1.

    Superscript t means transpose of vectors and matrices.

  2. 2.

    A minor variant of the usual Jordan canonical form. See [KM3] for details.

References

  1. Bol, G., Invarianten linearer Differentialgleichungen, Abh. Math. Sem. Univ. Hamburg 16 Nos. 3–4 (1949), 1–28.

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  2. Knopp, M., and Mason, G., On vector-valued modular forms and their Fourier coefficients, Acta Arith. 110.2 (2003), 117–124.

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  3. Knopp, M. and Mason, G., Vector-valued modular forms and Poincaré series, Ill. J. Math. 48 No. 4 (2004), 1345–1366.

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  4. Knopp, M. and Mason, G., Logarithmic vector-valued modular forms, to appear in Acta Arith. 147.3(2011), 264–282.

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Acknowledgements

Supported by the NSF and NSA.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Cruz, California, USA

    Geoffrey Mason

Authors
  1. Marvin Knopp
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  2. Geoffrey Mason
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Corresponding author

Correspondence to Geoffrey Mason .

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.

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Knopp, M., Mason, G. (2013). Vector-Valued Modular Forms with an Unnatural Boundary. 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_16

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