scholarly journals Arithmetic properties of Fourier coefficients of meromorphic modular forms

2021 ◽  
Vol 15 (9) ◽  
pp. 2381-2401
Author(s):  
Steffen Löbrich ◽  
Markus Schwagenscheidt
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Arithmetic Properties

scholarly journals Sequences, modular forms and cellular integrals

2018 ◽  
Vol 168 (2) ◽  
pp. 379-404 ◽  
Author(s):  
DERMOT McCARTHY ◽  
ROBERT OSBURN ◽  
ARMIN STRAUB
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Arithmetic Properties ◽  
General Conjecture

AbstractIt is well-known that the Apéry sequences which arise in the irrationality proofs for ζ(2) and ζ(3) satisfy many intriguing arithmetic properties and are related to the pth Fourier coefficients of modular forms. In this paper, we prove that the connection to modular forms persists for sequences associated to Brown's cellular integrals and state a general conjecture concerning supercongruences.

scholarly journals Petersson inner products of weight-one modular forms

2019 ◽  
Vol 2019 (749) ◽  
pp. 133-159
Author(s):  
Maryna Viazovska
Keyword(s):  
Quadratic Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Theta Series ◽  
Binary Quadratic Form ◽  
Positive Definite ◽  
Arithmetic Properties ◽  
Factorization Formula ◽  
Weight One

Abstract In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

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].

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

scholarly journals On mod p singular modular forms

2016 ◽  
Vol 28 (6) ◽  
Author(s):  
Siegfried Böcherer ◽  
Toshiyuki Kikuta
Keyword(s):  
Modular Form ◽  
Number Field ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Siegel Modular Form ◽  
Mod P

AbstractWe show that a Siegel modular form with integral Fourier coefficients in a number field

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.

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