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 2"k", discovered by Goro Shimura (1973). It has the property that the eigenvalue of a Hecke operator "T""n"2 on "F" is equal to the eigenvalue of "T""n" on "f".
Let formula_1 be a holomorphic cusp form with weight formula_2 and character formula_3 . For any prime number "p", let
formula_4
where formula_5's are the eigenvalues of the Hecke operators formula_6 determined by "p".
Using the functional equation of L-function, Shimura showed that
formula_7
is a holomorphic modular function with weight "2k" and character formula_8 .
Shimura's proof uses the Rankin-Selberg convolution of formula_9 with the theta series formula_10 for various Dirichlet characters formula_11 then applies Weil's converse theorem.