Fourier coefficients of modular forms of half-integral weight

Darmon cycles and the Kohnen-Shintani lifting

“…When N is odd square free, W. Kohnen showed the existence of a Hecke equivariant isomorphism (denoted as D-th Shintani liftings) in [22] between • λ g is a non-zero complex number which depends only on the choice of g.…”

“…The exposition is standard and all the material discussed here can be found in the fundamental papers on the subject -for example [39], [42], [22] and [23].…”

Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications

“…In [1, page 65] Kohnen proved that there is a canonically defined subspace S new kC 1 2 .N / S kC 1 2 .N /, and S new kC 1 2 .N / and S new 2k .N / are isomorphic as modules over the Hecke algebra. Later in [2,Theorem 3] he gave a formula for the product a g .m/a g .n/ of two arbitrary Fourier coefficients of a Hecke eigenform g of half-integral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem [1,Theoerm 2] he deduced the formula in [2,Theorem 3].…”

“…We will also construct Shintani and Shimura lifts for these spaces (see Theorems 1.6 and 1.9), and prove in Theorem 1.11 a result analogous to [2,Theorem 3].…”

Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

“…The choice of Schwartz function is crucial to what follows and is inspired by computations in Shintani [33], Kohnen [18] and Waldspurger [37].…”