Fourier Coefficients of Modular Forms of Half-Integral Weight in Arithmetic Progressions

2020 ◽  
Author(s):  
Corentin Darreye
Keyword(s):  
Gaussian Distribution ◽  
Modular Forms ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Mixed Distribution ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

Abstract We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry–Ganguly–Kowalski–Michel and Kowalski–Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution that comes from the asymptotic mixed distribution of Salié sums.

On the Ramanujan–Petersson conjecture for modular forms of half-integral weight

2019 ◽  
Vol 31 (3) ◽  
pp. 703-711
Author(s):  
Sanoli Gun ◽  
Winfried Kohnen
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Integral Weight

Abstract We investigate the (still unknown) Ramanujan–Petersson conjecture about the growth of the Fourier coefficients of cusp forms of half-integral weight and prove that it is optimal, at least for newforms in the plus space.

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.

A SHORT NOTE ON FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT MODULAR FORMS

2010 ◽  
Vol 06 (06) ◽  
pp. 1255-1259 ◽  
Author(s):  
WINFRIED KOHNEN
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Short Note ◽  
Cusp Forms ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

We give an unconditional proof of a result on sign changes of Fourier coefficients of cusp forms of half-integral weight that before was proved only under the hypothesis of Chowla's conjecture.

scholarly journals THE SIGN OF FOURIER COEFFICIENTS OF HALF-INTEGRAL WEIGHT CUSP FORMS

2012 ◽  
Vol 08 (03) ◽  
pp. 749-762 ◽  
Author(s):  
THOMAS A. HULSE ◽  
E. MEHMET KIRAL ◽  
CHAN IEONG KUAN ◽  
LI-MEI LIM
Keyword(s):  
Cusp Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Square Root ◽  
Critical Strip ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Finite Set ◽  
Invent Math

From a result of Waldspurger [W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math.64 (1981) 175–198], it is known that the normalized Fourier coefficients a(m) of a half-integral weight holomorphic cusp eigenform 𝔣 are, up to a finite set of factors, one of [Formula: see text] when m is square-free and f is the integral weight cusp form related to 𝔣 by the Shimura correspondence [G. Shimura, On modular forms of half-integral weight, Ann. of Math.97 (1973) 440–481]. In this paper we address a question posed by Kohnen: which square root is a(m)? In particular, if we look at the set of a(m) with m square-free, do these Fourier coefficients change sign infinitely often? By partially analytically continuing a related Dirichlet series, we are able to show that this is so.

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].

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

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.

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