Holomorphic Modular Forms Research Articles

Non-holomorphic modular forms from zeta generators

We study non-holomorphic modular forms built from iterated integrals of holomorphic modular forms for SL(2, ℤ) known as equivariant iterated Eisenstein integrals. A special subclass of them furnishes an equivalent description of the modular graph forms appearing in the low-energy expansion of string amplitudes at genus one. Notably the Fourier expansion of modular graph forms contains single-valued multiple zeta values. We deduce the appearance of products and higher-depth instances of multiple zeta values in equivariant iterated Eisenstein integrals, and ultimately modular graph forms, from the appearance of simpler odd Riemann zeta values. This analysis relies on so-called zeta generators which act on certain non-commutative variables in the generating series of the iterated integrals. From an extension of these non-commutative variables we incorporate iterated integrals involving holomorphic cusp forms into our setup and use them to construct the modular completion of triple Eisenstein integrals. Our work represents a fully explicit realisation of the modular graph forms within Brown’s framework of equivariant iterated Eisenstein integrals and reveals structural analogies between single-valued period functions appearing in genus zero and one string amplitudes.

Special values of shifted Dirichlet series and quasi-modular forms

Multiplication by a given modular form can be viewed as a linear map on the space of modular forms. By computing its adjoint operator, one can obtain certain cusp forms whose Fourier coefficients are special values of Dirichlet series of Rankin-Selberg type associated to modular forms. We generalize this idea to the space of almost holomorphic modular forms with some cuspidal conditions. We prove that the generating function of special values of the Dirichlet series at certain points is a quasi-modular form.

Modular linear differential operators and generalized Rankin-Cohen brackets

The aim of this paper is to give expressions for modular linear differential operators (MLDOs) of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of S L 2 ( R ) SL_2(\mathbb {R}) , as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms.

Modular forms and hierarchical Yukawa couplings in heterotic Calabi-Yau compactifications

We study the modular symmetry in heterotic string theory on Calabi-Yau threefolds. In particular, we examine whether moduli-dependent holomorphic Yukawa couplings are described by modular forms in the context of heterotic string theory with standard embedding. We find that SL(2, ℤ) modular symmetry emerges in asymptotic regions of the Calabi-Yau moduli space. The instanton-corrected holomorphic Yukawa couplings are then given by modular forms under SL(2, ℤ) or its congruence subgroups such as Γ0(3) and Γ0(4). In addition to the modular symmetry, it turns out that another coupling selection rule controls the structure of holomorphic Yukawa couplings. Furthermore, the coexistence of both the positive and negative modular weights for matter fields leads to a hierarchical structure of matter field Kähler metric. Thus, these holomorphic modular forms and the matter field Kähler metric play an important role in realizing a hierarchical structure of physical Yukawa couplings.

Algebraic aspects of holomorphic quantum modular forms

Matrix-valued holomorphic quantum modular forms are intricate objects associated to 3-manifolds (in particular to knot complements) that arise in successive refinements of the volume conjecture of knots and involve three holomorphic, asymptotic and arithmetic realizations. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the (-2,3,7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-2,3,7)$$\\end{document}-pretzel knot.

Structure of the space of polyharmonic Maass forms with an application to L-values

A polyharmonic Maass form introduced by Lagarias and Rhoades is a generalization of a harmonic Maass form in a sense. In this paper, we construct some bases for the spaces of polyharmonic Maass forms from the structure of the spaces of weakly holomorphic modular forms for prime levels or square-free levels. Furthermore, we relate the polyharmonic Maass forms of depth 1 or depth 3/2 to the L-series of holomorphic cusp forms for square-free levels.

The modularity of Siegel’s zeta functions

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of representations, and play an important role in the arithmetic theory of quadratic forms. In a 1938 paper, Siegel made a comment to the effect that the modularity of his zeta functions would be proved with the help of a suitable converse theorem. In the present paper, we accomplish Siegel’s original plan by using a Weil-type converse theorem for Maass forms, which has appeared recently. It is also shown that “half” of Siegel’s zeta functions correspond to holomorphic modular forms.

A basis for the space of weakly holomorphic Drinfeld modular forms of level T

In this article, we explicitly construct a canonical basis for the space of certain weakly holomorphic Drinfeld modular forms for Γ0(T) (resp., for Γ0+(T)) and compute the generating function satisfied by the basis elements. We also give an explicit expression for the action of the Θ-operator, which depends on the divisor of meromorphic Drinfeld modular forms.

Log concavity for unimodal sequences

In this paper, we prove that the number of unimodal sequences of size n is log-concave. These are coefficients of a mixed false modular form and have a Rademacher-type exact formula due to recent work of the second author and Nazaroglu on false theta functions. Log-concavity and higher Turán inequalities have been well-studied for (restricted) partitions and coefficients of weakly holomorphic modular forms, and analytic proofs generally require precise asymptotic series with error term. In this paper, we proceed from the exact formula for unimodal sequences to carry out this calculation. We expect our method applies to other exact formulas for coefficients of mixed mock/false modular objects.

COUNTING DISCRETE, LEVEL- $1$ , QUATERNIONIC AUTOMORPHIC REPRESENTATIONS ON $G_2$

AbstractQuaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $\mathrm {GL}_2$ . Here, we use ‘hyperendoscopy’ techniques to develop a general trace formula and understand them on an arbitrary group. Then we specialize this general formula to study quaternionic automorphic representations on the exceptional group $G_2$ , eventually getting an analog of the Eichler–Selberg trace formula for classical modular forms. We finally use this together with some techniques of Chenevier, Renard and Taïbi to compute dimensions of spaces of level- $1$ quaternionic representations. On the way, we prove a Jacquet–Langlands-style result describing them in terms of classical modular forms and automorphic representations on the compact-at-infinity form $G_2^c$ .The main technical difficulty is that the quaternionic discrete series that quaternionic automorphic representations are defined in terms of do not satisfy a condition of being ‘regular’. A real representation theory argument shows that regularity miraculously does not matter for specifically the case of quaternionic discrete series.We hope that the techniques and shortcuts highlighted in this project are of interest in other computations about discrete-at-infinity automorphic representations on arbitrary reductive groups instead of just classical ones.

Knots and Their Related $q$-Series

We discuss a matrix of periodic holomorphic functions in the upper and lower half-plane which can be obtained from a factorization of an Andersen-Kashaev state integral of a knot complement with remarkable analytic and asymptotic properties that defines a ${\rm PSL}_2({\mathbb Z})$-cocycle on the space of matrix-valued piecewise analytic functions on the real numbers. We identify the corresponding cocycle with the one coming from the Kashaev invariant of a knot (and its matrix-valued extension) via the refined quantum modularity conjecture of [arXiv:2111.06645] and also relate the matrix-valued invariant with the 3D-index of Dimofte-Gaiotto-Gukov. The cocycle also has an analytic extendability property that leads to the notion of a matrix-valued holomorphic quantum modular form. This is a tale of several independent discoveries, both empirical and theoretical, all illustrated by the three simplest hyperbolic knots.

Analogues of the Bol operator for half-integral weight weakly holomorphic modular forms

We define an analogue of the Bol operator on spaces of weakly holomorphic modular forms of half-integral weight. We establish its main properties and relation with other objects.

GENERALISED AUTOMORPHIC SHEAVES AND THE PROPORTIONALITY PRINCIPLE OF HIRZEBRUCH-MUMFORD

AbstractWe axiomatise the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalised automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery, we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.

Algebraic de Rham theory for the relative completion of 𝑆𝐿₂(ℤ)

We develop an explicit algebraic de Rham theory for the relative completion of S L 2 ( Z ) {\mathrm {SL}}_2(\mathbb {Z}) . This allows the construction of iterated integrals involving modular forms of the second kind, generalizing iterated integrals of holomorphic modular forms that were previously studied by Brown and Manin. These newly constructed iterated integrals provide all ‘multiple modular values’ defined by Brown.

On the Rankin-Selberg problem in families

In this paper, we investigate the Rankin-Selberg problem over short intervals in families of holomorphic modular forms and Hecke-Maass cusp forms. We assume a Lindelöf-on-average bound for holomorphic modular forms. On the other hand, we make no assumptions for Hecke-Maass cusp forms.

Modular forms on {{,textrm{SU},}}(2,1) with weight frac{1}{3}

In this note, we describe several new examples of holomorphic modular forms on the group {{,textrm{SU},}}(2,1). These forms are distinguished by having weight frac{1}{3}. We also describe a method for determining the levels at which one should expect to find such fractional weight forms.

Generalized matter parities from finite modular symmetries

AbstractWe classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries ΓN. Since all the couplings in supersymmetric theories of finite modular symmetries ΓN are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight 1/M, we find $\mathbb {Z}_{2M}$ symmetries, including the generalized baryon and lepton parities, R-parity, $\mathbb {Z}_3$ baryon triality and $\mathbb {Z}_6$ proton hexality. Such $\mathbb {Z}_{2M}$ symmetries are enlarged to $\mathbb {Z}_{2M} \rtimes \mathbb {Z}_2^{\text{CP}}$ symmetries together with the CP transformation.

A note on Fourier eigenfunctions in four dimensions

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem.

Siegel Eisenstein series of level two and its applications

In this paper, we construct a holomorphic Siegel modular form of weight 2 and level 2, and compute its Fourier coefficients explicitly. Moreover, we prove that this modular form equals the generating function of the representative number [Formula: see text] associated with the maximal order in the quaternion algebra [Formula: see text]. As a corollary, we can give a new proof of the famous formula for the sums of three squares. As applications, we give an explicit formula for the numbers of solutions of two systems of Diophantine equations related with Sun’s “1-3-5 conjecture”. Furthermore, we show that “a perfect square” in the integral condition version of Sun’s conjecture can be replaced by “a power of 4”.

An extension of Venkatesh’s converse theorem to the Selberg class

Abstract We extend Venkatesh’s proof of the converse theorem for classical holomorphic modular forms to arbitrary level and character. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree $2$ gamma factors of Selberg class type.