scholarly journals Twisted Maaß–Koecher series and spinor zeta functions

1999 ◽  
Vol 155 ◽  
pp. 153-160 ◽  
Author(s):  
Stefan Breulmann ◽  
Winfried Kohnen
Keyword(s):  
Fourier Coefficients ◽  
Zeta Functions ◽  
Positive Definite ◽  
Integral Weight ◽  
Genus 2 ◽  
Hecke Eigenform ◽  
Positive Integers

AbstractIt is shown that a Siegel-Hecke eigenform of integral weight k and genus 2 is uniquely determined by its Fourier coefficients indexed by nT where T runs over all half-integral positive definite primitive matrices of size 2 and n over all squarefree positive integers. The proof uses analytic arguments involving Koecher-Maaß series and spinor zeta functions.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

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 Estimates for fourier coefficients of siegel cusp forms of degree two, II

1992 ◽  
Vol 128 ◽  
pp. 171-176 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Positive Definite ◽  
Cusp Forms ◽  
Integral Weight ◽  
Siegel Cusp Form

Let F be a Siegel cusp form of integral weight k on Γ2: = Sp2(Z) and denote by a(T) (T a positive definite symmetric half-integral (2,2)-matrix) its Fourier coefficients. In [2] Kitaoka proved that(1)(the result is actually stated only under the assumption that k is even). In our previous paper [3] it was shown that one can attain(2)

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

2017 ◽  
Vol 15 (1) ◽  
pp. 304-316
Author(s):  
SoYoung Choi ◽  
Chang Heon Kim
Keyword(s):  
Fourier Coefficients ◽  
Hecke Algebra ◽  
Cusp Forms ◽  
Arithmetic Groups ◽  
Multiplicity One ◽  
Integral Weight ◽  
Hecke Eigenform

Abstract For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace $S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\subset S_{\kappa+\frac{1}{2}}(N),\,\,{\text{and}}\,\,S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\,\,{\text{and}}\,\,S_{2k}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product $a_{g}(m)\overline{a_{g}(n)}$ 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 $S_{2k}^{+}(p)\,\,{\text{and}}\,\,\mathbb{S}_{k+\frac{1}{2}}(p).$ We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].

ARITHMETIC TRACES OF NON-HOLOMORPHIC MODULAR INVARIANTS

2010 ◽  
Vol 06 (01) ◽  
pp. 69-87 ◽  
Author(s):  
ALISON MILLER ◽  
AARON PIXTON
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Algebraic Numbers ◽  
Heegner Points ◽  
Positive Weight ◽  
Poincare Series ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Modular Invariants

We extend results of Bringmann and Ono that relate certain generalized traces of Maass–Poincaré series to Fourier coefficients of modular forms of half-integral weight. By specializing to cases in which these traces are usual traces of algebraic numbers, we generalize results of Zagier describing arithmetic traces associated to modular forms. We define correspondences [Formula: see text] and [Formula: see text]. We show that if f is a modular form of non-positive weight 2 - 2 λ and odd level N, holomorphic away from the cusp at infinity, then the traces of values at Heegner points of a certain iterated non-holomorphic derivative of f are equal to Fourier coefficients of the half-integral weight modular forms [Formula: see text].

INDEX-DEPENDENT DIVISORS OF COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1841-1853 ◽  
Author(s):  
B. K. MORIYA ◽  
C. J. SMYTH
Keyword(s):  
Prime Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Survey Methods ◽  
Integer Sequences ◽  
Positive Integers

We evaluate [Formula: see text] for a certain family of integer sequences, which include the Fourier coefficients of some modular forms. In particular, we compute [Formula: see text] for all positive integers n for Ramanujan's τ-function. As a consequence, we obtain many congruences — for instance that τ(1000m) is always divisible by 64000. We also determine, for a given prime number p, the set of n for which τ(pn-1) is divisible by n. Further, we give a description of the set {n ∈ ℕ : n divides τ(n)}. We also survey methods for computing τ(n). Finally, we find the least n for which τ(n) is prime, complementing a result of D. H. Lehmer, who found the least n for which |τ(n)| is prime.

On linear relations for special L-values over certain totally real number fields

2021 ◽  
pp. 1-18
Author(s):  
Seiji Kuga
Keyword(s):  
Real Number ◽  
Fourier Coefficients ◽  
Number Fields ◽  
Arithmetic Functions ◽  
Hilbert Modular Form ◽  
Linear Relations ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hilbert Modular ◽  
Totally Real

In this paper, we give linear relations between the Fourier coefficients of a special Hilbert modular form of half integral weight and some arithmetic functions. As a result, we have linear relations for the special [Formula: see text]-values over certain totally real number fields.

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