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## Open Mathematics

2017 | 15 | 1 | 304-316
Tytuł artykułu

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

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EN
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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Kategorie tematyczne
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Tom
Numer
Strony
304-316
Opis fizyczny
Daty
wydano
2017-01-01
otrzymano
2016-08-02
zaakceptowano
2017-01-03
online
2017-03-30
Twórcy
autor
• Department of Mathematics Education and RINS, Gyeongsang National University, 501 Jinjudae-ro, Jinju, 660-701,, mathsoyoung72@gmail.com
autor
• Department of Mathematics, Sungkyunkwan University, Suwon, 440-746,, chhkim@skku.edu
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