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

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

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

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

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 ${\theta }_{\psi }(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\psi (n)n^{\nu }{e}^{2i\pi n^{2}z}(\nu =\frac{1-\psi (-1)}{2})$ for various Dirichlet characters $\psi$ then applies Weil's converse theorem.

See also

• Theta correspondence