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Shimura correspondence

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In number theory, the Shimura correspondence is a correspondence between modular forms F of half integral weight k+1/2, and modular forms f of even weight 2k, discovered by Goro Shimura (1973). It has the property that the eigenvalue of a Hecke operator Tn2 on F is equal to the eigenvalue of Tn on f.

Let be a holomorphic cusp form with weight and character . For any prime number p, let

where 's are the eigenvalues of the Hecke operators determined by p.

Using the functional equation of L-function, Shimura showed that

is a holomorphic modular function with weight 2k and character .

Shimura's proof uses the Rankin-Selberg convolution of with the theta series for various Dirichlet characters then applies Weil's converse theorem.

References

  • Bump, D. (2001) [1994], "Shimura correspondence", Encyclopedia of Mathematics, EMS Press
  • Shimura, Goro (1973), "On modular forms of half integral weight", Annals of Mathematics, Second Series, 97: 440–481, doi:10.2307/1970831, ISSN 0003-486X, JSTOR 1970831, MR 0332663