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Modular Forms of Half Integral Weight

  • Conference paper

Part of the Lecture Notes in Mathematics book series (LNM,volume 320)

Abstract

The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function

$$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty {e^{2\pi in^2 z} } $$

and the Dedekind eta function

$$ \eta \left( z \right) = e^{\pi iz/12} \prod _{n = 1}^\infty (1 - e^{2\pi inz} ) $$

are classical examples of such forms. (For some practical reasons, we take \( e^{2\pi in^2 z} \) instead of the usual \( e^{\pi in^2 z} \) in the definition of θ.) In fact, the function θ satisfies

$$ \theta (\gamma (z)) = j(\gamma ,z)\theta (z)for all\gamma \in \Gamma _0 (4) $$
(1.1)

with

$$ j([\begin{array}{*{20}c} a & b \\ c & d \\ \end{array} ],z) = (\frac{c} {d})\varepsilon _d^{ - 1} (cz + d)^{1/2}. $$
(1.2)

.

Keywords

  • Modular Form
  • Fourier Coefficient
  • Eisenstein Series
  • Cusp Form
  • Dirichlet Series

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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© 1973 Springer-Verlag Berlin Heidelberg

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Shimura, G. (1973). Modular Forms of Half Integral Weight. In: Kuijk, W. (eds) Modular Functions of One Variable I. Lecture Notes in Mathematics, vol 320. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38509-7_3

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  • DOI: https://doi.org/10.1007/978-3-540-38509-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06219-6

  • Online ISBN: 978-3-540-38509-7

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