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

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 -