Shimura correspondence - Alchetron, The Free Social Encyclopedia

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 ( 2 k + 1 ) / 2 and character χ . For any prime number p, let

∑ n = 1 ∞ Λ ( n ) n − s = ∏ p ( 1 − ω p p − s + ( χ p ) 2 p 2 k − 1 − 2 s ) − 1 ,

where ω 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 ) = ∑ n = 1 ∞ Λ ( n ) q n

is a holomorphic modular function with weight 2k and character χ 2 .

References

Shimura correspondence Wikipedia

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