Modular forms of half integral weight and the integral of certain theta-functions

Shinji Niwa

doi:10.1017/S0027763000016445

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Recently G. Shimura [1] constructed modular forms of integral weight from the forms of half integral weight. His construction is rather indirect. Indeed, he proved that the Dirichlet series, obtained from a form of half integral weight, multiplied by a certain L-function, corresponds to a modular form of an integral weight by means of the characterization of modular forms due to Weil.

References

[1] Shimura, G., On modular forms of half integral weight, Ann. of Math., 97 (1973), 440–481.Google Scholar

[2] Shintani, T., On construction of holomorphic cusp forms of half-integral weight, to appear.Google Scholar

[3] Doi, K. and Naganuma, H., On the functional equation of certain Dirichlet series, Inventiones Math., 9 (1969), 1–14.Google Scholar

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