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 2''k'', discovered by . 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

:$\sum^\infty_\Lambda(n)n^=\prod_p(1-\omega_pp^+(\chi_p)^2p^)^\ ,$

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

:$F(z)=\sum^\infty_ \Lambda(n)q^n$

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 $\theta_\psi(z)=\sum_^\infty \psi(n) n^\nu e^ \ ()$ for various Dirichlet characters $\psi$ then applies Weil's converse theorem.

See also

* Theta correspondence

References

*
*{{Citation , last1=Shimura , first1=Goro , title=On modular forms of half integral weight , jstor=1970831 , mr=0332663 , year=1973 , journal=Annals of Mathematics , series=Second Series , issn=0003-486X , volume=97 , pages=440–481 , doi=10.2307/1970831

Modular forms
Langlands program