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.