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 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 $$\chi^2$$.

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.