scholarly journals Supplement to the decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators

1991 ◽  
Vol 31 (1) ◽  
pp. 307-309 ◽  
Author(s):  
Masaru Ueda
Keyword(s):  
Trace Formula ◽  
Hecke Operators ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Integral Weight

A TRACE FORMULA FOR HECKE OPERATORS ON VECTOR-VALUED MODULAR FORMS

2016 ◽  
Vol 59 (1) ◽  
pp. 143-165
Author(s):  
NICOLE RAULF ◽  
OLIVER STEIN
Keyword(s):  
Trace Formula ◽  
Modular Forms ◽  
Hecke Operators ◽  
Weil Representation ◽  
Cusp Forms ◽  
Dimension Formula ◽  
Integral Weight ◽  
Vector Valued

AbstractWe present a ready to compute trace formula for Hecke operators on vector-valued modular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

scholarly journals CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

2010 ◽  
Vol 06 (05) ◽  
pp. 1117-1137 ◽  
Author(s):  
T. SHEMANSKE ◽  
S. TRENEER ◽  
L. WALLING
Keyword(s):  
Hecke Operators ◽  
Cusp Forms ◽  
Hecke Eigenforms ◽  
Linearly Independent ◽  
Integral Weight ◽  
Trivial Character ◽  
Free Level ◽  
Fixed Space ◽  
Space Of Cusp Forms

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.

Zeros of $L$-functions attached to cusp forms of half-integral weight

2018 ◽  
Vol 147 (1) ◽  
pp. 131-143
Author(s):  
Jaban Meher ◽  
Sudhir Pujahari ◽  
Karam Deo Shankhadhar
Keyword(s):  
Cusp Forms ◽  
Half Integral Weight ◽  
Integral Weight

scholarly journals Newforms of Half-integral Weight: The Minus Space Counterpart

2019 ◽  
Vol 72 (2) ◽  
pp. 326-372 ◽  
Author(s):  
Ehud Moshe Baruch ◽  
Soma Purkait
Keyword(s):  
Hecke Algebras ◽  
Double Cover ◽  
Hecke Operators ◽  
Characterization Result ◽  
Generators And Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
New Space

AbstractWe study genuine local Hecke algebras of the Iwahori type of the double cover of $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ and translate the generators and relations to classical operators on the space $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$, $M$ odd and square-free. In [9] Manickam, Ramakrishnan, and Vasudevan defined the new space of $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$ that maps Hecke isomorphically onto the space of newforms of $S_{2k}(\unicode[STIX]{x1D6E4}_{0}(2M))$. We characterize this newspace as a common $-1$-eigenspace of a certain pair of conjugate operators that come from local Hecke algebras. We use the classical Hecke operators and relations that we obtain to give a new proof of the results in [9] and to prove our characterization result.

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