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

    $\sum _{n=1}^{\infty }\Lambda (n)n^{-s}=\prod _{p}(1-\omega _{p}p^{-s}+(\chi _{p})^{2}p^{2k-1-2s})^{-1}\ ,$

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

    $F(z)=\sum _{n=1}^{\infty }\Lambda (n)q^{n}$

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.

