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 .