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An Introduction to the Theory of Harmonic Maass Forms

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L-Functions and Automorphic Forms

Part of the book series: Contributions in Mathematical and Computational Sciences ((CMCS,volume 10))

Abstract

In this note we give a short introduction to the theory of harmonic Maass forms. We start by introducing modular forms and Maass forms and then present the notion of (vector valued) harmonic Maass forms as developed by Bruinier and Funke in [4]. We end by giving two recent applications of this theory.

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References

  1. Alfes, C.: CM values and Fourier coefficients of harmonic Maass forms. Dissertation. TU prints (2015)

    Google Scholar 

  2. Alfes, C., Griffin, M., Ono, K., Rolen, L.: Weierstrass mock modular forms and elliptic curves. Res. Number Theory 1(1), 1–31 (2015)

    Google Scholar 

  3. Bringmann, K., Folsom, A., Ono, K., Rolen, L.: Harmonic Maass Forms and Mock Modular Forms: Theory and Applications, vol. 64. American Mathematical Society, Colloquium Publications (2018)

    Google Scholar 

  4. Bruinier, J.H., Funke, J.: On two geometric theta lifts. Duke Math. J. 125(1), 45–90 (2004)

    Google Scholar 

  5. Bruinier, J.H., Funke, J.: Traces of CM values of modular functions. J. Reine Angew. Math. 594, 1–33 (2006)

    Google Scholar 

  6. Bruinier, J.H., Ono, K.: Heegner divisors, L-functions and harmonic weak Maass forms. Ann. Math. (2) 172(3), 2135–2181 (2010)

    Google Scholar 

  7. Bruinier, J.H., Ono, K.: Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms. Adv. Math. 246, 198–219 (2013)

    Google Scholar 

  8. Bruinier, J.H., Ono, K., Rhoades, R.C.: Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues. Math. Ann. 342(3), 673–693 (2008)

    Google Scholar 

  9. Bruinier, J.H., van der Geer, G., Harder, G., Zagier, D.: The 1-2-3 of Modular Forms. In: Ranestad, K. (ed.) Lectures from the Summer School on Modular Forms and their Applications held in Nordfjordeid, June 2004. Universitext. Springer, Berlin (2008)

    Google Scholar 

  10. Diamond, F., Shurman, J.: A First Course in Modular Forms. Graduate Texts in Mathematics, vol. 228. Springer, New York (2005)

    Google Scholar 

  11. Iwaniec, H.: Spectral Methods of Automorphic Forms. Graduate Studies in Mathematics, vol. 53, 2nd edn. American Mathematical Society, Providence, RI (2002)

    Google Scholar 

  12. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics, vol. 97, 2nd edn. Springer, New York (1993)

    Google Scholar 

  13. Koecher, M., Krieg, A.: Elliptische Funktionen und Modulformen, Revised edn. Springer, Berlin (2007)

    Google Scholar 

  14. Larson, E., Rolen, L.: Integrality properties of the CM-values of certain weak Maass forms. Forum Math. 27, 961–972 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series. CBMS Regional Conference Series in Mathematics, vol. 102. Conference Board of the Mathematical Sciences, Washington, DC (2004)

    Google Scholar 

  16. Ono, K.: Unearthing the visions of a master: Harmonic Maass forms and number theory. In: Current Developments in Mathematics, 2008, pp. 347–454. International Press, Somerville, MA (2009)

    Google Scholar 

  17. Zagier, D.B.: Traces of singular moduli. In: Motives, Polylogarithms and Hodge theory, Part I (Irvine, CA, 1998). International Press Lecture Series, vol. 3, pp. 211–244. International Press, Somerville, MA (2002)

    Google Scholar 

  18. Zwegers, S.P.: Mock θ-functions and real analytic modular forms. In: q-Series with Applications to Combinatorics, Number Theory, and Physics (Urbana, IL, 2000). Contemporary Mathematics, vol. 291, pp. 269–277. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

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Acknowledgements

The author thanks the referee, Eric Hofmann and Markus Schwagenscheidt for comments on an earlier version of this paper.

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Authors and Affiliations

  1. Fakultät für Eleklrotechnik, Informatik und Mathematik, Institut für Mathematik, Warburger Str. 100, 33098, Paderborn, Germany

    Claudia Alfes-Neumann

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  1. Claudia Alfes-Neumann
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Correspondence to Claudia Alfes-Neumann .

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Editors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Jan Hendrik Bruinier

  2. Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

    Winfried Kohnen

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Alfes-Neumann, C. (2017). An Introduction to the Theory of Harmonic Maass Forms. In: Bruinier, J., Kohnen, W. (eds) L-Functions and Automorphic Forms. Contributions in Mathematical and Computational Sciences, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-69712-3_17

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