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

$\sum^\infty_{n=1}\Lambda(n)n^{-s}=\prod_p(1-\omega_pp^{-s}+(\chi_p)^2p^{2k-1-2s})^{-1}\ ,$

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

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

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

References

• Bump, D. (2001), "Shimura correspondence", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Shimura, Goro (1973), "On modular forms of half integral weight", Annals of Mathematics. Second Series 97: 440–481, ISSN 0003-486X, JSTOR 1970831, MR 0332663