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 ${\displaystyle (2k+1)/2}$ and character $\chi$ . 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 $\omega _{p}$'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 $\chi ^{2}$ .

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