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 f be a holomorphic cusp form with weight (2k+1)/2 and character \chi . For any prime number p, let

\sum^\infty_{n=1}\Lambda(n)n^{-s}=\prod_p(1-\omega_pp^{-s}+(\chi_p)^2p^{2k-1-2s})^{-1}\ ,

where \omega_p's are the eigenvalues of the Hecke operators T(p^2) determined by p.

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

F(z)=\sum^\infty_{n=1} \Lambda(n)q^n

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