Shimura correspondence explained

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 ( 2 k +1 )/ 2 and character \chi . For any prime number p, let\sum ^{inf ty} _{n =1} Λ (n)n ^{-s} = \prod _{p(1-\omega} ^{- s} ^{pp} + (\chi ^{2 p} _{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 ^{inf ty} _{n =1} Λ (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} ^{inf ty} ^{n =-inf ty} \psi(n) n ^{\nu} e ^{2 i \pi n ^{2} z} ({\scriptstyle\nu = (1- \psi( -1 ))/(2) }) for various Dirichlet characters \psi then applies Weil's converse theorem.

See also

• Theta correspondence