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 Tn2 on F is equal to the eigenvalue of Tn on f. Let [math]\displaystyle{ f }[/math] be a holomorphic cusp form with weight [math]\displaystyle{ (2k+1)/2 }[/math] and character [math]\displaystyle{ \chi }[/math] . For any prime number p, let
- [math]\displaystyle{ \sum^\infty_{n=1}\Lambda(n)n^{-s}=\prod_p(1-\omega_pp^{-s}+(\chi_p)^2p^{2k-1-2s})^{-1}\ , }[/math]
where [math]\displaystyle{ \omega_p }[/math]'s are the eigenvalues of the Hecke operators [math]\displaystyle{ T(p^2) }[/math] determined by p.
Using the functional equation of L-function, Shimura showed that
- [math]\displaystyle{ F(z)=\sum^\infty_{n=1} \Lambda(n)q^n }[/math]
is a holomorphic modular function with weight 2k and character [math]\displaystyle{ \chi^2 }[/math] .
Shimura's proof uses the Rankin-Selberg convolution of [math]\displaystyle{ f(z) }[/math] with the theta series [math]\displaystyle{ \theta_\psi(z)=\sum_{n=-\infty}^\infty \psi(n) n^\nu e^{2i \pi n^2 z} \ ({\scriptstyle\nu = \frac{1-\psi(-1)}{2}}) }[/math] for various Dirichlet characters [math]\displaystyle{ \psi }[/math] then applies Weil's converse theorem.
See also
References
- Hazewinkel, Michiel, ed. (2001), "Shimura correspondence", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=S/s130280
- Shimura, Goro (1973), "On modular forms of half integral weight", Annals of Mathematics, Second Series 97: 440–481, doi:10.2307/1970831, ISSN 0003-486X
 |  |
