Special Values of Zeta Functions, and Eisenstein Series of Half Integral Weight

1980 ◽  
Vol 102 (2) ◽  
pp. 219 ◽  
Author(s):  
Jacob Sturm
Keyword(s):  
Eisenstein Series ◽  
Zeta Functions ◽  
Special Values ◽  
Half Integral Weight ◽  
Integral Weight

scholarly journals Quadratic congruences for Cohen–Eisenstein series

1999 ◽  
Vol 41 (1) ◽  
pp. 141-144
Author(s):  
P. GUERZHOY
Keyword(s):  
Number Theory ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Analytic Number Theory ◽  
Cusp Forms ◽  
Special Values ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Published Paper

The notion of quadratic congruences was introduced in the recently published paper [A. Balog, H. Darmon and K. Ono, Congruences for Fourier coefficients of half-integral weight modular forms and special values of L-functions, in Analytic Number Theory, Vol. 1, Progr. Math.138 (Birkhäuser, Boston, 1996), 105–128.]. In this note we present different, somewhat more conceptual proofs of several results from that paper. Our method allows us to refine the notion and to generalize the results quoted. Here we deal only with the quadratic congruences for Cohen–Eisenstein series. Similar phenomena exist for cusp forms of half-integral weight as well; however, as one would expect, in the case of Eisenstein series the argument is much simpler. In particular, we do not make use of techniques other than p-adic Mazur measure, whereas the consideration of cusp forms of half-integral weight involves a much more sophisticated construction. Moreover, in the case of Cohen–Eisenstein series we are able to obtain a full and exhaustive result. For these reasons we present the argument here.

scholarly journals A remark on the Shimura correspondence

1988 ◽  
Vol 30 (3) ◽  
pp. 285-291 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Fourier Coefficients ◽  
Central Point ◽  
Modular Curve ◽  
Special Values ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Special Case ◽  
Shimura Correspondence

In [4] an identity is given which relates the product of two Fourier coefficients of a Hecke eigenform g of half-integral weight and level 4N with N odd and squarefree to the integral of a Hecke eigenform f of even integral weight associated to g under the Shimura correspondence along a geodesic period on the modular curve X0(N) This formula contains as a special case a refinement of a result of Waldspurger [6] about special values of L-series attached to f at the central point.

scholarly journals Modular forms arising from zeta functions in two variables attached to prehomogeneous vector spaces related to quadratic forms

2004 ◽  
Vol 175 ◽  
pp. 1-37 ◽  
Author(s):  
Takahiko Ueno
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Vector Spaces ◽  
Parabolic Subgroups ◽  
Converse Theorem ◽  
Half Integral Weight ◽  
Prehomogeneous Vector Spaces ◽  
Integral Weight ◽  
Weil Type

AbstractIn this paper, we prove the functional equations for the zeta functions in two variables associated with prehomogeneous vector spaces acted on by maximal parabolic subgroups of orthogonal groups. Moreover, applying the converse theorem of Weil type, we show that elliptic modular forms of integral or half integral weight can be obtained from the zeta functions.

scholarly journals Eisenstein series in the Kohnen plus space for Hilbert modular forms

2016 ◽  
Vol 12 (03) ◽  
pp. 691-723 ◽  
Author(s):  
Ren-He Su
Keyword(s):  
Eisenstein Series ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Hilbert Modular Forms ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
The One ◽  
Totally Real

In 1975, Cohen constructed a kind of one-variable modular forms of half-integral weight, say [Formula: see text], whose [Formula: see text]th Fourier coefficient only occurs when [Formula: see text] is congruent to 0 or 1 modulo 4. The space of modular forms whose Fourier coefficients have the above property is called Kohnen plus space, initially introduced by Kohnen in 1980. Recently, Hiraga and Ikeda generalized the plus space to the spaces for half-integral weight Hilbert modular forms with respect to general totally real number fields. The [Formula: see text]th Fourier coefficients [Formula: see text] of a Hilbert modular form of parallel weight [Formula: see text] lying in the generalized Kohnen plus space does not vanish only if [Formula: see text] is congruent to a square modulo 4. In this paper, we use an adelic way to construct Eisenstein series of parallel half-integral weight belonging to the generalized Kohnen plus spaces and give an explicit form for their Fourier coefficients. These series give a generalization of the one introduced by Cohen. Moreover, we show that the Kohnen plus space is generated by the cusp forms and the Eisenstein series we constructed as a vector space over [Formula: see text].

scholarly journals Hecke operators on half-integral weight Siegel Eisenstein series

2017 ◽  
Vol 13 (09) ◽  
pp. 2335-2372
Author(s):  
Lynne H. Walling
Keyword(s):  
Eisenstein Series ◽  
Hecke Operators ◽  
Linear Combinations ◽  
Half Integral Weight ◽  
Multiplicity One ◽  
Integral Weight

We construct a basis for the space of half-integral weight Siegel Eisenstein series of level [Formula: see text] where [Formula: see text] is odd and square-free. Then we restrict our attention to those Eisenstein series generated from elements of [Formula: see text], commenting on why this restriction is necessary for our methods. We directly apply to these forms all Hecke operators attached to odd primes, and we realize the images explicitly as linear combinations of Siegel Eisenstein series. Using this information, we diagonalize the subspace of Eisenstein series generated from elements of [Formula: see text], obtaining a multiplicity-one result.

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