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 Tn2 on F is equal to the eigenvalue of Tn on f.
Let be a holomorphic cusp form with weight and character . For any prime number p, let
where 's are the eigenvalues of the Hecke operators determined by p.
is a holomorphic modular function with weight 2k and character .
- Bump, D. (2001) , "Shimura correspondence", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, 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