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

Open Mathematics (2017)

- Volume: 15, Issue: 1, page 304-316
- ISSN: 2391-5455

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topSoYoung Choi, and Chang Heon Kim. "Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications." Open Mathematics 15.1 (2017): 304-316. <http://eudml.org/doc/287970>.

@article{SoYoungChoi2017,

abstract = {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].},

author = {SoYoung Choi, Chang Heon Kim},

journal = {Open Mathematics},

keywords = {Modular forms; Shintani lifts; Shimura lifts; modular forms},

language = {eng},

number = {1},

pages = {304-316},

title = {Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications},

url = {http://eudml.org/doc/287970},

volume = {15},

year = {2017},

}

TY - JOUR

AU - SoYoung Choi

AU - Chang Heon Kim

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

JO - Open Mathematics

PY - 2017

VL - 15

IS - 1

SP - 304

EP - 316

AB - 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].

LA - eng

KW - Modular forms; Shintani lifts; Shimura lifts; modular forms

UR - http://eudml.org/doc/287970

ER -

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