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

SoYoung Choi ; Chang Heon Kim

Open Mathematics, Tome 15 (2017), p. 304-316 / Harvested from The Polish Digital Mathematics Library

Résumé

For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace [...] S κ + 1 2 n e w ( N ) ⊂ S κ + 1 2 ( N ) , and S κ + 1 2 n e w ( N ) and S 2 k n e w ( N ) $S_{\kappa +\frac{1}{2}}^{\mathrm {n}\mathrm {e}\mathrm {w}}(N)\subset S_{\kappa +\frac{1}{2}}(N),\,\,{\text{and}}\,\,S_{\kappa +\frac{1}{2}}^{\mathrm {n}\mathrm {e}\mathrm {w}}(N)\,\,{\text{and}}\,\,S_{2k}^{\mathrm {n}\mathrm {e}\mathrm {w}}(N)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product [...] a g ( m ) a g ( n ) ¯ $a_{g}(m)\overline{a_{g}(n)}$ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral 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 he deduced the formula in [2, Theorem 3]. In this paper we will prove that there is a Hecke equivariant isomorphism between the spaces [...] S 2 k + ( p ) and S k + 1 2 ( p ) . $S_{2k}^{+}(p)\,\,{\text{and}}\,\,\mathbb {S}_{k+\frac{1}{2}}(p).$ We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].

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Publié le : 2017-01-01

Zbl 06704086

EUDML-ID : urn:eudml:doc:287970