Full Modular Group Research Articles

Reduced modular symmetries of threshold corrections and gauge coupling unification

We revisit the question of gauge coupling unification at the string scale in orbifold compactifications of the heterotic string for the supersymmetric Standard Model. In the presence of discrete Wilson lines threshold corrections with modular symmetry that is a subgroup of the full modular group arise. We find that reduced modular symmetries not previously reported are possible. We conjecture that the effects of such threshold corrections can be simulated using sums of terms built from Dedekind eta functions to obtain the appropriate modular symmetry. For the cases of the ℤ8-I orbifold and the ℤ3 × ℤ6 orbifold it is easily possible to obtain gauge coupling unification at the “observed” scale with Kahler moduli T of approximately one.

Moments of L-functions attached to the twist of modular form by Dirichlet characters

Let f(z) be a holomorphic cusp form of weight κ with respect to the full modular group SL2(ℤ). Let L(s, f) be the automorphic L-function associated with f(z) and χ be a Dirichlet character modulo q. In this paper, the authors prove that unconditionally for \(k = \tfrac{1} {n}\) with n ∈ ℕ, $$M_k \left( {q,f} \right) = \sum\limits_{\begin{array}{*{20}c} {\chi (\bmod q)} \\ {\chi \ne \chi _0 } \\ \end{array} } {\left| {L\left( {\frac{1} {2},f \otimes \chi } \right)} \right|^{2k} < < _k \varphi \left( q \right)(\log q)^{k^2 } ,}$$ and the result also holds for any real number 0 < k < 1 under the GRH for L(s, f ⊗ χ). The authors also prove that under the GRH for L(s, f ⊗ χ), $$M_k \left( {q,f} \right) > > _k \varphi (q)(log q)^{k^2 }$$ for any real number k > 0 and any large prime q.

Vector-valued Siegel modular forms of weight detk ⊗ Sym(8)

We determine the structures of modules of vector-valued Siegel modular forms of weight det k ⊗ Sym (8) with respect to the full Siegel modular group of degree two.

Uniform estimates for sums of coefficients of symmetric square L-function

Let ϕ(z) denote a holomorphic or Maass cusp form for the full modular group Γ=SL(2,Z). And let λSym2ϕ(n) be the n-th coefficient of symmetric square L-function associated with ϕ(z). We establish the uniform upper bound for the summatory function∑n≤xλSym2ϕ(n), which improves the results of Ichihara [4], Lü [10], Sankaranarayanan [13] and Tang [14].

Construction of vector-valued modular integrals and vector-valued mock modular forms

It is known that, given a vector-valued modular form of negative weight, its Fourier coefficients can be calculated based on the principal part of the form. In this paper we start with an arbitrary principal part and complete the Fourier expansion using the calculation. We show that the so-obtained function is a vector-valued modular integral of negative weight on the full modular group. Next, we construct the supplementary function associated to a vector-valued modular cusp form of positive weight. The constructions are inspired by the construction of Eichler integrals by Knopp. We conclude with a comparison of these forms and their integrals to vector-valued weak harmonic Maass forms.

COMPUTING WITH SUBGROUPS OF THE MODULAR GROUP

AbstractWe give several algorithms for finitely generated subgroups of the modular group PSL2(ℤ) given by sets of generators. First, we present an algorithm to check whether a finitely generated subgroup H has finite index in the full modular group. Then we discuss how to parametrise the right cosets of H in PSL2(ℤ), whether the index is finite or not. Further, we explain how an element in H can be written as a word in a given set of generators of H.

Differential equations for septic theta functions

We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends an elementary technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein’s quartic relation induces symmetric representations for low-weight Eisenstein series in terms of weight one modular forms of level seven.

On modular solutions of fractional weights for the Kaneko–Zagier equation for $$\varGamma ^*_0(2) \,\hbox {and}\, \varGamma ^*_0(3)$$ Γ 0 ∗ ( 2 ) and Γ 0 ∗ ( 3 )

For the full modular group, Kaneko gave a modular form solution of fractional weights for a holomorphic modular differential equation of second order. In this paper, we give modular form solutions of fractional weights for a modular differential equation on the Fricke group of levels 2 and 3.

Weakly holomorphic modular forms for some moonshine groups

In an article in the Pure and Applied Mathematics Quarterly in 2008, Duke and Jenkins investigated a certain natural basis of the space of weakly holomorphic modular forms for the full modular group SL 2(Z). We show here that their results can be generalized to certain moonshine groups, also allowing characters that are real on the underlying subgroup Γ0(N).

A note on the characterizations of Jacobi cusp forms and cusp forms of Maass Spezialschar

In this paper, we give characterizations of Jacobi cusp forms of weight \(k\) and index \(1\) on a congruence subgroup \(\Gamma _0(N)\) and cusp forms of weight \(k\) on the full Siegel modular group \(\hbox {Sp}_{4}(\mathbb Z)\) in Maass Spezialschar for \(k\ge 2\) even and \(N\ge 1\) odd and squarefree.

The spaces of new forms for harmonic weak Maalss forms and their level raising properties modulo $$\ell $$ ℓ

Bringmann et al. (Math. Ann., 355(3):1085–1121, 2013) and Guerzhoy (Proc. Am. Math. Soc., 136:3051–3059, 2008) studied the multiplicity two Hecke theory for weakly holomorphic modular forms on the full modular group. In this note, we consider a generalization of these results to congruence subgroups \(\Gamma _1(N)\) with a new form theory. Using this, we define new forms for harmonic weak Maass forms and study a level raising property modulo \(\ell \) for them.

On the trace and norm maps from [formula omitted] to [formula omitted

Let f be a Drinfeld modular form for Γ0(p). From such a form, one can obtain two forms for the full modular group GL2(A): by taking the trace or the norm from Γ0(p) to GL2(A). In this paper we show some connections between the arithmetic modulo p of the coefficients of the u-series expansion of f and those of a form closely related to its trace, and of the coefficients of f and those of its norm.

Sums of absolute values of cusp form coefficients and their application

Let Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let λf(n), λg(n) and λh(n) be the nth normalized Fourier coefficients of three holomorphic Hecke cusp forms f(z),g(z),h(z)∈Sk(Γ), Sk1(Γ) and Sk2(Γ) respectively. In this paper we consider several averages of absolute values of Fourier coefficients of holomorphic Hecke cusp forms. In particular, we have (under suitable conditions)∑n⩽x|λf(n)λg(n)|≪x(logx)−2θ3,∑n⩽x|λf(n)λg(n)λh(n)|≪x(logx)−θ3, where and throughout this paperθ=1−83π=0.1512…. As an application, we consider sums of coefficients of the Rankin–Selberg L-function and the triple product L-function, and prove that∑n⩽xλf(n)λg(n)≪x35(logx)−2θ3,∑n⩽xλf(n)λg(n)λh(n)≪x79(logx)−θ3, and other similar results.

On Subsets of Q(&amp;amp;radic;&amp;lt;span style=&amp;quot; text-decoration:overline;&amp;quot;&amp;gt;m&amp;lt;/span&amp;gt;) Q under the Action of Hecke Groups H(&amp;amp;lambda;&amp;lt;sub&amp;gt;q&amp;lt;/sub&amp;gt;)

is the disjoint union of for all , where is the set of all roots of primitive second degree equations , with reduced discriminant equal to k2m. We study the action of two Hecke groups—the full modular group and the group of linear-fractional transformations on . In particular, we investigate the action of on for finding different orbits.

EICHLER COHOMOLOGY THEOREM FOR VECTOR-VALUED MODULAR FORMS

We prove the Eichler cohomology theorem for vector-valued modular forms of large integer weights on the full modular group.

Eichler integrals and harmonic weak Maass forms

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more general groups, namely, H-groups by employing the theory of supplementary functions introduced and developed by M.I. Knopp and S.Y. Husseini. In particular, we show that the set of Eichler integrals, which have polynomial period functions, is the same as the set of holomorphic parts of harmonic weak Maass forms of which the non-holomorphic parts are certain period integrals of cusp forms. From this we deduce relations among period functions for harmonic weak Maass forms.

A short note on sign changes

In this paper, we present a quantitative result for the number of sign changes for the sequences $\{a(n^j)\}_{n\ge 1}, j=2,3,4$ of the Fourier coefficients of normalized Hecke eigencusp forms for the full modular group SL 2(ℤ). We also prove a similar kind of quantitative result for the number of sign changes of the q-exponents $c(p)~(p \mbox{~vary ~over ~primes})$ of certain generalized modular functions for the congruence subgroup Γ0(N), where N is square-free.

Average behavior of Fourier coefficients of cusp forms over sum of two squares

Let λ f ( n ) denote the n th normalized Fourier coefficient of the classical holomorphic cusp form of even integral weight k ≥ 2 for the full modular group SL ( 2 , Z ) . In this paper, we investigate the average behavior of the power sum ∑ a 2 + b 2 ≤ x λ f ( a 2 + b 2 ) ℓ for x ≥ 1 , 2 ≤ ℓ ≤ 8 and a , b , ℓ ∈ Z .

Some remarks on zeros of quasimodular forms

We study the zeros of quasimodular forms on the full modular group \({SL_2(\mathbb{Z})}\). In particular, we prove the existence of infinitely many \({SL_2(\mathbb{Z})}\)-inequivalent zeros for any quasimodular form of depth 1. We also prove the non-existence of common zeros between some well known quasimodular forms.

Lower bound for the higher moment of symmetric square L -functions

Let H k ( Γ ) be the space of all normalized holomorphic Hecke-eigen cusp forms of even integral weight k for the full modular group Γ = SL ( 2 , Z ) , and denote by L ( s , sym 2 f ) the corresponding symmetric square L -function associated to f ∈ H k ( Γ ) . In this paper, the lower bound of the higher moment of L ( 1 / 2 , sym 2 f ) is established, i.e., for any even number r ∈ Z + , ∑ f ∈ H k ( Γ ) ω f − 1 L ( 1 / 2 , sym 2 f ) r ≫ r ( log k ) r ( r + 1 ) 2 holds for k → ∞ , where ω f = k − 1 2 π 2 L ( 1 , sym 2 f ) .