CONGRUENCES RELATED TO MODULAR FORMS

2010 ◽  
Vol 06 (06) ◽  
pp. 1367-1390 ◽  
Author(s):  
H. A. VERRILL
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Congruence Relation ◽  
Mellin Transform ◽  
Congruence Subgroup ◽  
Modular Function ◽  
Euler Product ◽  
Mod P

Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL 2(Z), and t a weight 0 modular function for Γ. Assume that near t = 0, we can write f = ∑n≥0bn tn, bn ∈ Z. Let ℓ(z) be a weight k + 2 modular form with q-expansion ∑γnqn, such that the Mellin transform of ℓ can be expressed as an Euler product. Then we show that if [Formula: see text] for some integers ai, di, then the congruence relation bmpr -γpbmpr-1 + εppk+1bmpr-2 ≡ 0 ( mod pr) holds. We give a number of examples of this phenomena.

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

scholarly journals On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

scholarly journals On the divisibility of r2(n)

1977 ◽  
Vol 18 (1) ◽  
pp. 109-111 ◽  
Author(s):  
E. J. Scourfield
Keyword(s):  
Modular Form ◽  
Algebraic Number ◽  
Modular Forms ◽  
Congruence Subgroup ◽  
Algebraic Number Field ◽  
The Past ◽  
Divisibility Property ◽  
Integral Weight ◽  
Image Position ◽  
Positive Integers

During the past few years, some papers of P. Deligne and J.-P. Serre (see [2], [9], [10] and other references cited there) have included an investigation of certain properties of coefficients of modular forms, and in particular Serre [10] (see also [11]) obtained the divisibility property (1) below. Letbe a modular form of integral weight k ≧ 1 on a congruence subgroup of SL2(Z), and suppose that each cn belongs to the ring RK of integers of an algebraic number field K finite over Q. For c ∈ RK and m ≧ 1 an integer, write c ≡ 0 (mod m) if c ∈ m RK and c ≢ 0 (mod m) otherwise. Then Serre showed that there exists α > 0 such thatas x → ∞, where throughout this note N(n ≦ x: P) denotes the number of positive integers n ≦ x with the property P.

scholarly journals Restriction of Siegel modular forms to modular curves

2002 ◽  
Vol 65 (2) ◽  
pp. 239-252 ◽  
Author(s):  
Cris Poor ◽  
David S. Yuen
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Modular Curves ◽  
Linear Relations

We study homomorphisms form the ring of Siegel modular forms of a given degree to the ring of elliptic modular forms for a congruence subgroup. These homomorphisms essentially arise from the restriction of Siegel modular forms to modular curves. These homomorphisms give rise to linear relations among the Fourier coefficients of a Siegel modular form. We use this technique to prove that dim .

scholarly journals On Ramanujan and Dirichlet series with Euler products

1984 ◽  
Vol 25 (2) ◽  
pp. 203-206 ◽  
Author(s):  
S. Raghavan
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Hecke Operators ◽  
Euler Product ◽  
Arithmetical Functions ◽  
In Trans ◽  
Euler Products ◽  
Unpublished Manuscripts ◽  
Profound Insight

In his unpublished manuscripts (referred to by Birch [1] as Fragment V, pp. 247–249), Ramanujan [3] gave a whole list of assertions about various (transforms of) modular forms possessing naturally associated Euler products, in more or less the spirit of his extremely beautiful paper entitled “On certain arithmetical functions” (in Trans. Camb. Phil. Soc. 22 (1916)). It is simply amazing how Ramanujan could write down (with an ostensibly profound insight) a basis of eigenfunctions (of Hecke operators) whose associated Dirichlet series have Euler products, anticipating by two decades the famous work of Hecke and Petersson. That he had further realized, in the event of a modular form f not corresponding to an Euler product, the possibility of restoring the Euler product property to a suitable linear combination of modular forms of the same type as f, is evidently fantastic.

On the algebra of modular forms on a congruence subgroup

1979 ◽  
Vol 86 (3) ◽  
pp. 461-466 ◽  
Author(s):  
A. J. Scholl
Keyword(s):  
Modular Form ◽  
Finite Index ◽  
Modular Forms ◽  
Modular Group ◽  
Fourier Expansion ◽  
Congruence Subgroup ◽  
Complex Numbers ◽  
Graded Algebra

Let A be a subring of the complex numbers containing 1, and Γ a subgroup of the modular group of finite index. We say that a modular form on Γ is A-integral if the coefficients of its Fourier expansion at infinity lie in A. We denote by Mk(Γ,A) the A-module of holomorphic A-integral modular forms of weight k, and by M(Γ, A) the graded algebra of A-integral modular forms on Γ.

scholarly journals Fourier Coefficients of Vector-valued Modular Forms of Dimension 2

2014 ◽  
Vol 57 (3) ◽  
pp. 485-494 ◽  
Author(s):  
Cameron Franc ◽  
Geoffrey Mason
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Dimension 2 ◽  
Vector Valued

AbstractWe prove the following theorem. Suppose that F = ( f1, f2) is a 2-dimensional, vectorvalued modular form on SL2(ℤ) whose component functions f1, f2 have rational Fourier coefficients with bounded denominators. Then f1, f2 are classical modular forms on a congruence subgroup of the modular group.

scholarly journals Quotients of L-functions

2000 ◽  
Vol 160 ◽  
pp. 143-159
Author(s):  
Bernhard E. Heim
Keyword(s):  
Modular Form ◽  
Dirichlet Series ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Euler Product ◽  
Jacobi Forms ◽  
New Variant

AbstractIn this paper a certain type of Dirichlet series, attached to a pair of Jacobi forms and Siegel modular forms is studied. It is shown that this series can be analyzed by a new variant of the Rankin-Selberg method. We prove that for eigenforms the Dirichlet series have an Euler product and we calculate all the local L-factors. Globally this Euler product is essentially the quotient of the standard L-functions of the involved Jacobi- and Siegel modular form.

Weights of the Mod p Kernels of Theta Operators

2018 ◽  
Vol 70 (2) ◽  
pp. 241-264
Author(s):  
Siegfried Böcherer ◽  
Toshiyuki Kikuta ◽  
Sho Takemori
Keyword(s):  
Modular Form ◽  
Modular Forms ◽  
Siegel Modular Forms ◽  
Siegel Modular Form ◽  
Mod P

AbstractLet Θ[j] be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime p, we give the weights of elements of mod p kernel of Θ[j], where the mod p kernel of Θ[j] is the set of all Siegel modular forms F such that Θ[j](F) is congruent to zero modulo p. In order to construct examples of the mod p kernel of Θ[j] fromany Siegel modular form, we introduce new operators A(j)(M) and show the modularity of F|A(j)(M) when F is a Siegel modular form. Finally, we give some examples of the mod p kernel of Θ[j] and the filtrations of some of them.

ON THE FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF-INTEGRAL WEIGHT

2013 ◽  
Vol 09 (08) ◽  
pp. 1879-1883 ◽  
Author(s):  
YOUNG JU CHOIE ◽  
WINFRIED KOHNEN
Keyword(s):  
Modular Form ◽  
Growth Condition ◽  
Cusp Form ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Congruence Subgroup ◽  
Half Integral Weight ◽  
Fundamental Discriminant ◽  
Integral Weight ◽  
Elliptic Modular Form

It is known that if the Fourier coefficients a(n)(n ≥ 1) of an elliptic modular form of even integral weight k ≥ 2 on the Hecke congruence subgroup Γ0(N)(N ∈ N) satisfy the bound a(n) ≪f nc for all n ≥ 1, where c > 0 is any number strictly less than k - 1, then f must be cuspidal. Here we investigate the case of half-integral weight modular forms. The main objective of this note is to show that to deduce that f is a cusp form, it is sufficient to impose a suitable growth condition only on the Fourier coefficients a(|D|) where D is a fundamental discriminant with (-1)kD > 0.

Sign in / Sign up

Export Citation Format

Share Document