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 . It has the property that the eigenvalue of a Hecke operator T<sub>n<sup>2</sup></sub> on F is equal to the eigenvalue of T<sub>n</sub> 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.