In this paper, we evaluate Fine type integrals of level 30. For an example, [Formula: see text] In the process, we discover certain new Eisenstein series identities of level 30.

We introduce the algebra of formal multiple Eisenstein series and study its derivations. This algebra is motivated by the classical multiple Eisenstein series, introduced by Gangl-Kaneko-Zagier as a hybrid of classical Eisenstein series and multiple zeta values. In depth one, we obtain formal versions of the Eisenstein series satisfying the same algebraic relations as the classical Eisenstein series. In particular, they generate an algebra whose elements we call formal quasimodular forms. We show that the algebra of formal multiple Eisenstein series is an sl 2 -algebra by formalizing the usual derivations for quasimodular forms and extending them naturally to the whole algebra. Additionally, we introduce some families of derivations for general quasi-shuffle algebras, providing a broader context for these derivations. Further, we prove that a quotient of this algebra is isomorphic to the algebra of formal multiple zeta values. This gives a novel and purely formal approach to classical (quasi)modular forms and builds a new link between (formal) multiple zeta values and modular forms.

The integral model of a G U ( n − 1 , 1 ) \mathrm {GU}(n-1,1) Shimura variety carries a universal abelian scheme over it, and the dual top exterior power of its Lie algebra carries a natural hermitian metric. We express the arithmetic volume of this metrized line bundle, defined as an iterated self-intersection in the arithmetic Chow ring, in terms of logarithmic derivatives of Dirichlet L L -functions. We also determine the arithmetic volumes of Kudla-Rapoport divisors and relate them to coefficients of Eisenstein series.

Rationality of the periods of Eisenstein series

Using the relations between rational functions and Eisenstein series, as well as the inferences for cotangent sums and period polynomials, we work out a precise description for Eisenstein series whose L L -series vanish at certain critical values. This is possible for small weights compared to the level of the Eisenstein series. For large weights we give a partial result and determine subspaces with simultaneous vanishing properties.

Abstract Let G be an affine or hyperbolic rank 2 Kac–Moody group over a finite field 𝔽 q {\mathbb{F}_{q}} . Let X = X q + 1 {X=X_{q+1}} be the Tits building of G , the ( q + 1 ) {(q+1)} -homogeneous tree, and let Γ be a non-uniform lattice in G . When Γ is a standard parabolic subgroup for the negative BN -pair, we define Eisenstein series on Γ \ X {\Gamma\backslash X} and prove its convergence in a half space using Iwasawa decomposition of the Haar measure on G . A crucial tool is a description of the vertices of X in terms of Iwasawa cells. We also prove meromorphic continuation of the Eisenstein series. This requires us to construct an integral operator on the Tits building X and a truncation operator for the Eisenstein series. We also develop the functional analytic framework necessary for proving meromorphic continuation in our setting, by refining and extending Bernstein’s Continuation Principle.

Motivated by the fact that the classical Jacobi theta function is the exponential generating function of the Eisenstein series, we study the exponential Taylor coefficients (in the elliptic variable) of a related natural partial theta function, as well as a false theta function corresponding to the Dedekind eta function. We prove that the space spanned by these objects is closed under differentiation, analogous to the space of quasimodular forms, and that it contains the quasimodular forms themselves. We further provide their Fourier expansions, establish quasimodular completions, and derive a recursive formula for the Taylor coefficients of the logarithm of the unimodal rank generating function, expressed as partition traces of the false and partial objects.

In this paper, we introduce the zeta function of the prehomogeneous vector space of binary cubic forms, twisted by the real analytic Eisenstein series. We prove the meromorphic continuation of this zeta function and identify its poles and their residues. We also identify the poles and residues of the zeta function when restricted to irreducible binary cubic forms. This zeta function can be used to prove the equidistribution of the lattice shape of cubic rings.

In this paper, we give a list of 113 holomorphic eta-quotients of integral weight (66 of which are primitive) and provide a uniform closed formula for their Fourier coefficients [Formula: see text] where [Formula: see text] with some fixed [Formula: see text]. The proof involves Wohlfahrt’s extension of Hecke operators and a dimension formula for spaces of modular forms of general multiplier system. We further provide the expansions of these eta-quotients as linear combinations of standard Eisenstein series.

In a recent preprint entitled “Holomorphy of Eisenstein series – a new method and applications in the case of the general linear group”, the author has developed a new method for proving holomorphy of degenerate Eisenstein series, based on the Franke filtration of spaces of automorphic forms. In this paper, the method is applied in the case of degenerate Eisenstein series on the symplectic group of rank two. Although the analytic properties of Eisenstein series in that case are already known, the goal is to exhibit the method in a simple setting, in which all additional technical details are peeled off.

Abstract In this paper, we consider the decomposition of theta series for lattice cosets of ternary lattices. We show that the natural decomposition into an Eisenstein series, a unary theta function, and a cuspidal form which is orthogonal to unary theta functions correspond to the theta series for the genus, the deficiency of the theta series for the spinor genus from that of the genus, and the deficiency of the theta series for the class from that of the spinor genus, respectively. These three pieces are hence invariants of the genus, spinor genus, and class, respectively, extending known results for lattices and verifying a conjecture of the first author and Haensch. We furthermore extend the definition of p -neighbors to include lattice cosets and construct an algorithm to compute representatives for the classes in the genus or spinor genus via the p -neighborhoods.

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest H1,11,1↔H0,22,0, with H denoting the Fox H-function.

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.

Zeta generators are derivations associated with odd Riemann zeta values that act freely on the Lie algebra of the fundamental group of Riemann surfaces with marked points. The genus-zero incarnation of zeta generators are Ihara derivations of certain Lie polynomials in two generators that can be obtained from the Drinfeld associator. We characterize a canonical choice of these polynomials, together with their non-Lie counterparts at even degrees wge 2, through the action of the dual space of formal and motivic multizeta values. Based on these canonical polynomials, we propose a canonical isomorphism that maps motivic multizeta values into the f-alphabet. The canonical Lie polynomials from the genus-zero setup determine canonical zeta generators in genus one that act on the two generators of Enriquez’ elliptic associators. Up to a single contribution at fixed degree, the zeta generators in genus one are systematically expanded in terms of Tsunogai’s geometric derivations dual to holomorphic Eisenstein series, leading to a wealth of explicit high-order computations. Earlier ambiguities in defining the non-geometric part of genus-one zeta generators are resolved by imposing a new representation-theoretic condition. The tight interplay between zeta generators in genus zero and genus one unravelled in this work connects the construction of single-valued multiple polylogarithms on the sphere with iterated-Eisenstein-integral representations of modular graph forms.

A bstract We extend the framework of non-holomorphic modular flavor symmetry to include the odd weight polyharmonic Maaß forms. The integer weight polyharmonic Maaß forms of level N can be arranged into multipltets of the homogeneous finite modular group $$ {\Gamma}_N^{\prime } $$ Γ N ′ . We propose to construct the integer weight, including weight one, non-holomorphic polyharmonic Maaß forms from the non-holomorphic Eisenstein series. The previous results of even weight polyharmonic Maaß forms are reproduced. We apply this formalism to address the flavor structure of the standard model. An example lepton model based on the modular group $$ {\Gamma}_3^{\prime}\cong {T}^{\prime } $$ Γ 3 ′ ≅ T ′ is constructed, where neutrino masses are generated via type-I seesaw mechanism with two right-handed neutrinos. This model can accommodate the experimental data for both normal and inverted neutrino mass orderings. We further extend this model to include quarks, so that the masses and mixing parameters of both quark and lepton sectors can be successfully described in terms of only thirteen real free parameters. It is the modular invariant model with the smallest number of free parameters so far, only normal ordering neutrino mass is viable after including quarks, and the correlations among the input parameters and flavor observables are analyzed.

A bstract It is known that the holographic thermal propagator in 4 spacetime dimensions can be related to the Nekrasov-Shatashvili limit of the Ω-deformed $$\mathcal{N}=2$$ supersymmetric SU(2) Yang-Mills theory with N f = 4 hypermultiplets. There are two expansions involved: one is the expansion in small temperature which in the Seiberg-Witten language is equivalent to the semiclassical expansion in inverse powers of the large adjoint vev and the second is the expansion in instanton numbers. Working in the simplified case of zero energy, we find that the latter expansion gives rise to quasi-modular forms which can be resummed as functions of Eisenstein series. The so obtained series in positive powers of small temperature shows clear signs of being asymptotic.

Zeros of cuspidal projections of products of Eisenstein series

On the Borisov–Gunnells relations for products of Eisenstein series

Abstract We prove the second adjointness in the setting of the categorical local Langlands correspondence. Moreover, we study the relation between Eisenstein series and cuspidal supports and present a conjectural characterization of irreducible smooth representations with supercuspidal 𝐿-parameters regarding geometric constant terms. The main technical ingredient is an induction principle for geometric Eisenstein series which allows us to reduce to the situations already treated in the literature.

Joint cubic moment of Eisenstein series and Hecke-Maass cusp forms