# 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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## Abstract

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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 ) ${S}_{\kappa +\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(N\right)\subset {S}_{\kappa +\frac{1}{2}}\left(N\right),\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}_{\kappa +\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(N\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{S}_{2k}^{\mathrm{n}\mathrm{e}\mathrm{w}}\left(N\right)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product [...] a g ( m ) a g ( n ) ¯ ${a}_{g}\left(m\right)\overline{{a}_{g}\left(n\right)}$ 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 ) . ${S}_{2k}^{+}\left(p\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\text{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{𝕊}_{k+\frac{1}{2}}\left(p\right).$ We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].

## How to cite

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SoYoung 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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