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_n² on F is equal to the eigenvalue of T_n on f.

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

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

where ${\displaystyle \omega _{p}}$'s are the eigenvalues of the Hecke operators ${\displaystyle T(p^{2})}$ determined by p.

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

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

is a holomorphic modular function with weight 2k and character ${\displaystyle \chi ^{2}}$ .

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