We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k\to\infty$, that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family.

Abstract We propose a novel framework that simultaneously addresses three critical issues: tiny neutrino masses and their mixing patterns, dark matter, and the strong CP problem. Our model extends the Peccei–Quinn (PQ) symmetry by incorporating modular $$S_3$$ S 3 symmetry, which plays a central role in explaining the observed neutrino mixing structure. The field content includes two vector-like colored fermions and three colored scalars as $$S_3$$ S 3 singlets, an isospin doublet inert scalar and a singlet PQ scalar, each assigned appropriate modular weights. We show that such an extension, together with a suitable assignment of modular weights to the fields, can lead to holomorphic modular forms of Yukawa interactions, which can be derived from a superpotential. Furthermore, we explore an extension of the model to include non-holomorphic Yukawa interactions in the non-supersymmetric framework and show that the results are distinct from the holomorphic case. Tiny neutrino masses are generated radiatively through colored mediators, while the KSVZ-type axion appears to dynamically resolve the strong CP problem. We investigate the phenomenology of lepton flavor violation and the muon $$g-2$$ g - 2 anomaly within this framework. Additionally, we explore the axion’s properties and its role as dark matter.

Abstract We establish the first case of the surprising correlation phenomenon observed in the recent works of He, Lee, Oliver, Pozdnyakov, and Sutherland between Fourier coefficients in families of modular forms and their root numbers. We give a complete description of the resulting correlation functions for holomorphic modular forms of any fixed weight $k$ k and examine the asymptotic properties of these functions.

In this paper, we study sign changes of weakly holomorphic modular forms which are given as [Formula: see text]-quotients. We give representative examples for forms of negative weight, weight zero, and positive weight.

In this article, we show that Fourier eigenmeasures supported on spheres with radii given by a locally finite sequence, which we call k-spherical measures, correspond to Fourier series exhibiting a modular-type transformation behaviour with respect to the metaplectic group. A familiar subset of such Fourier series comprises holomorphic modular forms. This allows us to construct k-spherical eigenmeasures and derive Poisson-type summation formulas, thereby recovering formulas of a similar nature established by Cohn–Gonçalves, Lev–Reti, and Meyer, among others. Additionally, we extend our results to higher dimensions, where Hilbert modular forms yield higher-dimensional k-spherical measures.

Recently, Amdeberhan, Griffin, Ono, and Singh started the study of “traces of partition Eisenstein series” and used it to give explicit formulas for many interesting functions. In this note we determine the precise spaces in which they lie, find modular completions, and show how they are related via operators.

Let Γ denote the subgroup Γ_0^± (N) of GL_2(Z), N prime. Let V be the space of holomorphic modular forms for Γ. Let V_α ⊂ V denote the various Hecke eigenspaces, with the last V_α denoting the Eisenstein subspace. If M ∈ V is a modular symbol, define the type of M to be (t_1,...,t_k,t_E) where t_α = 1 if the projection of M to V_α is nonzero, and t_α = 0 otherwise. For each N ≤ 100, we compute the types of the modular symbols in an increasing series of concentric boxes. We prove an obstruction for a given type to occur, related to the existence of “Eisenstein primes.” For any given type that survives this obstruction, we give computational evidence that the proportion of its occurrence in a box stabilizes as the boxes grow larger. We interpret the limit of this ratio (assuming it exists) as the box size goes to infinity as the probability that a random modular symbol will have this type. Contrary to our original expectation, it does not appear to be the case that with probability 1 a random symbol will project nontrivially to each Vi. Whether the limit referred to in the previous paragraph actually exists, and why the limits have the various values that appear in our computations, are open questions.

Generalized Hecke operators, originating from the replication formula in Monstrous Moonshine, were extended in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] to apply to harmonic Maass functions on modular curves of higher genera, building on works in [M. Koike, On replication formula and Hecke operators, Nagoya University, preprint]. Their action was further applied to weakly holomorphic modular functions, deriving numerous arithmetic properties of Fourier coefficients. In this paper, we extend these operators to weakly holomorphic modular forms of arbitrary even non-positive weights. In the process, we show [Proposition 3.1 of L. Beneish and M. H. Mertens, On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras, Math. Z. 297(1–2) (2021) 59–80], which is used to obtain dimension formulas for certain vertex operator algebras, can be derived from our results. Additionally, we identify the conditions under which the action of the generalized Hecke operator preserves holomorphicity. Moreover, we show that this action can be expressed as a linear combination of same-level or lower-level forms, refining the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717]. Finally, we establish more general congruence relations on Fourier coefficients, with the results in [D. Jeon, S.-Y. Kang and C. H. Kim, The Hecke system of harmonic Maass functions and applications to modular curves of higher genera Ramanujan J. 62(3) (2023) 675–717] emerging as a special case.

Abstract We introduce the L-series of weakly holomorphic modular forms using Laplace transforms and give their functional equations. We then determine converse theorems for vector-valued harmonic weak Maass forms, Jacobi forms, and elliptic modular forms of half-integral weight in Kohnen plus space.

Griffin, the second author, and Molnar studied coefficient duality for canonical bases for a broad range of spaces of weakly holomorphic modular forms, showing that the Fourier coefficients of canonical basis elements appear as negatives of Fourier coefficients for elements of a canonical basis of a related space of forms. We investigate the effect of the trace operator on this duality for modular forms for [Formula: see text] of genus zero and show exactly when duality still holds after applying the trace operator.

Abstract For p ∈ { 2 , 3 } p\in \left\{2,3\right\} and an even integer k k , let W k − 2 − ( p ) {W}_{k-2}^{-}\left(p) be the space of period polynomials of weight k − 2 k-2 on Γ 0 + ( p ) {\Gamma }_{0}^{+}\left(p) with eigenvalue − 1 -1 under the Fricke involution. We determine the dimension formula for W k − 2 − ( p ) {W}_{k-2}^{-}\left(p) and construct an explicit basis for it using period functions for weakly holomorphic modular forms. Furthermore, for a quadratic form Q Q , we define the function F − ( z , Q ) {F}^{-}\left(z,Q) on the complex upper half-plane as a generating function of the cycle integrals of the canonical basis elements for the space of weakly holomorphic modular forms of weight k k and eigenvalue − 1 -1 under the Fricke involution on Γ 0 ( p ) {\Gamma }_{0}\left(p) . We also show that F − ( z , Q ) {F}^{-}\left(z,Q) is a modular integral on Γ 0 + ( p ) {\Gamma }_{0}^{+}\left(p) . Our approach focuses on calculating cycle integrals within Γ 0 ( p ) {\Gamma }_{0}\left(p) rather than Γ 0 + ( p ) {\Gamma }_{0}^{+}\left(p) , which allows us to overcome certain technical challenges. This study extends earlier work by Choi and Kim (Rational period functions and cycle integrals in higher level cases, J. Math. Anal. Appl. 427 (2015), no. 2, 741–758) which focused on eigenvalue +1, providing new insights by examining eigenvalue − 1 -1 cases in the theory of rational period functions and cycle integrals in this setting.

Some years ago, it was conjectured by the first author that the Chern–Simons perturbation theory of a 3-manifold at the trivial flat connection is a resurgent power series. We describe completely the resurgent structure of the above series (including the location of the singularities and their Stokes constants) in the case of a hyperbolic knot complement in terms of an extended square matrix (x, q)-series whose rows are indexed by the boundary parabolic SL2(C)-flat connections, including the trivial one. We use our extended matrix to describe the Stokes constants of the above series, to define explicitly their Borel transform and to identify it with state–integrals. Along the way, we use our matrix to give an analytic extension of the Kashaev invariant and of the colored Jones polynomial and to complete the matrix valued holomorphic quantum modular forms as well as to give an exact version of the refined quantum modularity conjecture of Zagier and the first author. Finally, our matrix provides an extension of the 3D-index in a sector of the trivial flat connection. We illustrate our definitions, theorems, numerical calculations and conjectures with the two simplest hyperbolic knots.

The notion of formal Siegel modular forms for an arithmetic subgroup Γ \Gamma of the symplectic group of genus n n is a generalization of symmetric formal Fourier-Jacobi series. Assuming an upper bound on the affine covering number of the Siegel modular variety associated with Γ \Gamma , we prove that all formal Siegel modular forms are given by Fourier-Jacobi expansions of classical holomorphic Siegel modular forms. We also show that the required upper bound is always met if 2 ≤ n ≤ 4 2\leq n \leq 4 . As an application we consider the case of the paramodular group of squarefree level and genus 2 2 .

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.

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.

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.

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.

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.

There is a sophisticated theory of nearly holomorphic Siegel modular forms by Shimura. Using previous results by Nagaoka and myself on Rankin-Cohen operators and theta-operators we will present a proof that quasimodular forms (defined as constant terms or as holomorphic part of a nearly holomorphic Siegel modular form) are always p-adic. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.

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