# Shimura correspondence

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In [number theory](/source/Number_theory), the **Shimura correspondence** is a correspondence between [modular forms](/source/Modular_form) *F* of half integral weight *k*+1/2, and modular forms *f* of even weight 2*k*, discovered by txt. It has the property that the eigenvalue of a [Hecke operator](/source/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_{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](/source/Hecke_operator) T(p^2) determined by *p*.

Using the [functional equation](/source/Functional_equation_(L-function)) of [L-function](/source/L-function), [Shimura](/source/Goro_Shimura) showed that

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

is a holomorphic [modular function](/source/Modular_function) with weight *2k* and character \chi^2 .

Shimura's proof uses the [Rankin-Selberg convolution](/source/Rankin%E2%80%93Selberg_method#History) of f(z) with the theta series \theta_\psi(z)=\sum_{n=-\infty}^\infty \psi(n) n^\nu e^{2i \pi n^2 z} \ ({\scriptstyle\nu = \frac{1-\psi(-1)}{2}}) for various Dirichlet characters \psi then applies [Weil's converse theorem](/source/Weil's_converse_theorem).

## See also

- [Theta correspondence](/source/Theta_correspondence)

## References

- Shimura, Goro (1973), "On modular forms of half integral weight", *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **97** (3): 440–481, Second Series, [doi:10.2307/1970831](https://doi.org/10.2307/1970831). [ISSN 0003-486X](https://www.worldcat.org/issn/0003-486X). [JSTOR 1970831](https://www.jstor.org/stable/1970831). MR 0332663

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