Shimura correspondence

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**n2 on F is equal to the eigenvalue of T**n 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 .

Shimura's proof uses the Rankin-Selberg convolution of f(z) with the theta series \theta_\psi(z)=\sum_{n=-\infty}^\infty \psi(n) n^\nu e^{2i \pi n^2 z} \ ({\scriptstyle\nu = \frac{1-\psi(-1)}{2}}) for various Dirichlet characters \psi then applies Weil's converse theorem.

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