Eisenstein Series Research Articles

Double Eisenstein series and modular forms of level 4

Double Eisenstein series and modular forms of level 4

On the common zeros of quasi-modular forms for Γ+ 0(N) of level N = 1, 2, 3

Abstract In this article, we study common zeros of the iterated derivatives of the Eisenstein series for Γ 0 + ( N ) {\Gamma }_{0}^{+}\left(N) of level N = 1 , 2 N=1,2 , and 3, which are quasi-modular forms. More precisely, we investigate the common zeros of quasi-modular forms and prove that all the zeros of the iterated derivatives of the Eisenstein series θ m E k ( N ) {\theta }^{m}{E}_{k}^{\left(N)} of weight k = 2 , 4 , 6 k=2,4,6 for Γ 0 + ( N ) {\Gamma }_{0}^{+}\left(N) of level N = 2 , 3 N=2,3 are simple by generalizing the results of Meher and Gun-Oesterlé for SL 2 ( Z ) {{\rm{SL}}}_{2}\left({\mathbb{Z}}) .

On the convergence of Kac–Moody Eisenstein series

On the convergence of Kac–Moody Eisenstein series

Reciprocal Hyperbolic Series of Ramanujan Type

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of z=F12(1/2,1/2;1;x) and z′=dz/dx. When a certain parameter in these series is equal to π, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented.

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

Zero-Dimensional Shimura Varieties and Central Derivatives of Eisenstein Series

Abstract We formulate and prove a version of the arithmetic Siegel–Weil formula for (zero dimensional) Shimura varieties attached to tori, equipped with some additional data. More precisely, we define a family of “special” divisors in terms of Green functions at archimedean and non-archimedean places and prove that their degrees coincide with the Fourier coefficients of the central derivative of an Eisenstein series. The proof relies on the usual Siegel–Weil formula to provide a direct link between both sides of the identity, and in some sense, offers a more conceptual point of view on prior results in the literature.

Eisenstein degeneration of Euler systems

Abstract We discuss the theory of Coleman families interpolating critical-slope Eisenstein series. We apply it to study degeneration phenomena at the level of Euler systems. In particular, this allows us to prove relations between Kato elements, Beilinson–Flach classes and diagonal cycles, and also between Heegner cycles and elliptic units. We expect that this method could be extended to construct new instances of Euler systems.

𝑝-adic limit of the Eisenstein series on the exceptional group of type 𝐸_{7,3}

In this paper, we show that the p p -adic limit of a family of Eisenstein series on the exceptional domain where the exceptional group of type E 7 , 3 E_{7,3} acts is an ordinary modular form for a congruence subgroup.

Concyclicity of the zeros of polynomials associated to derivatives of the L-functions of Eisenstein series

Concyclicity of the zeros of polynomials associated to derivatives of the L-functions of Eisenstein series

Exact results for giant graviton four-point correlators

We study the four-point correlator O2O2DD\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left\\langle {\\mathcal{O}}_2{\\mathcal{O}}_2\\mathcal{DD}\\right\\rangle $$\\end{document} in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory (SYM) with SU(N) gauge group, where O2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{O}}_2 $$\\end{document} represents the superconformal primary operator with dimension two, while D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{D} $$\\end{document} denotes a determinant operator of dimension N, which is holographically dual to a giant graviton D3-brane extending along S5. We analyse the integrated correlator associated with this observable, obtained after integrating out the spacetime dependence over a supersymmetric invariant measure. Similarly to other classes of integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM, this integrated correlator can be computed through supersymmetric localisation on the four-sphere. Employing matrix-model recursive techniques, we demonstrate that the integrated correlator can be reformulated as an infinite sum of protected three-point functions with known coefficients. This insight allows us to circumvent the complexity associated with the dimension-N determinant operator, significantly streamlining the large-N expansion of the integrated correlator. In the planar limit and beyond, we derive exact results for the integrated correlator valid for all values of the ’t Hooft coupling, and investigate the resurgent properties of their strong coupling expansion. Additionally, in the large-N expansion with fixed (complexified) Yang-Mills coupling, we deduce the SL(2, ℤ) completion of these results in terms of the non-holomorphic Eisenstein series. The proposed modular functions are confirmed by explicit instanton calculations from the matrix model, and agree with expectations from the holographic dual picture of known results in type IIB string theory.

Emergence of R4-terms in M-theory

It has been recently suggested that the strong Emergence Proposal is realized in M-theory limits by integrating out all light towers of states with a typical mass scale not larger than the species scale, i.e. the eleventh dimensional Planck mass. Within the BPS sector, these are transverse M2- and M5-branes, that can be wrapped and particle-like, carrying Kaluza-Klein momentum along the compact directions. We provide additional evidence for this picture by revisiting and investigating further the computation of R4-interactions in M-theory à la Green-Gutperle-Vanhove. A central aspect is a novel UV-regularization of Schwinger-like integrals, whose actual meaning and power we clarify by first applying it to string perturbation theory. We consider then toroidal compactifications of M-theory and provide evidence that integrating out all light towers of states via Schwinger-like integrals thus regularized yields the complete result for R4-interactions. In particular, this includes terms that are tree-level, one-loop and space-time instanton corrections from the weakly coupled point of view. Finally, we comment on the conceptual difference of our approach to earlier closely related work by Kiritsis-Pioline and Obers-Pioline, leading to a correspondence between two types of constrained Eisenstein series.

Norm bounds on Eisenstein series

We study the sup-norm bound (both individually and on average) for Eisenstein series on certain arithmetic hyperbolic orbifolds producing sharp exponents for the modular surface and Picard 3-fold. The methods involve bounds for Epstein zeta functions, and counting restricted values of indefinite quadratic forms at integer points.

On p-adic Siegel–Eisenstein series from a point of view of the theory of mod $$p^m$$ singular forms

On p-adic Siegel–Eisenstein series from a point of view of the theory of mod $$p^m$$ singular forms

Lambda-invariants of Mazur–Tate elements attached to Ramanujan’s tau function and congruences with Eisenstein series

Lambda-invariants of Mazur–Tate elements attached to Ramanujan’s tau function and congruences with Eisenstein series

Critical points of the Eisenstein series $E_{4}$ and application to the spectrum of the Lamé operator

We give a complete description of the distribution of the critical points of the classical Eisenstein series E_{4}(\tau) . An application to the spectrum of the Lamé operator is also given.

Ensemble averages of ℤ2 orbifold classes of Narain CFTs

In this work we study families of ℤ2 orbifolds of toroidal conformal field theories based on both factorizable and non-factorizable target space tori. For these classes of theories, we analyze their moduli spaces, and compute their partition functions. Building on previous work, we express the calculated partition functions in terms of suitable Siegel-Narain theta functions that allow us to determine their ensemble averages. We express the derived averaged partition functions of the studied families of conformal field theories in a manifest modular invariant finite sum of products of real analytic Eisenstein series. We speculate on a tentative holographic three-dimensional dual bulk interpretations for the considered ℤ2 orbifold classes of ensembles of conformal field theories.

Macmahon’s sums-of-divisors and their connection to multiple Eisenstein series

Macmahon’s sums-of-divisors and their connection to multiple Eisenstein series

Higher d Eisenstein series and a duality-invariant distance measure

The Petersson inner product is a natural inner product on the space of modular invariant functions. We derive a formula, written as a convergent sum over elementary functions, for the inner product Es(G, B) of the real analytic Eisenstein series \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${E}_{s}\\left(\ au ,\\overline{\ au }\\right)$$\\end{document} and a general point in Narain moduli space. We also discuss the utility of the Petersson inner product as a distance measure on the space of 2d CFTs, and apply our procedure to evaluate this distance in various examples.

Hecke theory for SO+(2,n + 2)

We describe the foundations of a Hecke theory for the orthogonal group SO+(2,n+2). In particular we consider the Hermitian modular group of degree 2 as a special example of SO+(2,4). As an application we show that the attached Maaß space is invariant under Hecke operators. This implies that the Eisenstein series belongs to the Maaß space. If the underlying lattice is even and unimodular, our approach allows us to reprove the explicit formula of its Fourier coefficients.

On the modulo p zeros of modular forms congruent to theta series

For a prime p larger than 7, the Eisenstein series of weight p−1 has some remarkable congruence properties modulo p. Those imply, for example, that the j-invariants of its zeros (which are known to be real algebraic numbers in the interval [0,1728]), are at most quadratic over the field with p elements and are congruent modulo p to the zeros of a certain truncated hypergeometric series. In this paper we introduce “theta modular forms” of weight k≥4 for the full modular group as the modular forms for which the first dim⁡(Mk) Fourier coefficients are identical to certain theta series. We consider these theta modular forms for both the Jacobi theta series and the theta series of the hexagonal lattice. We show that the j-invariant of the zeros of the theta modular forms for the Jacobi theta series are modulo p all in the ground field with p elements. For the theta modular form of the hexagonal lattice we show that its zeros are at most quadratic over the ground field with p elements. Furthermore, we show that these zeros in both cases are congruent to the zeros of certain truncated hypergeometric functions.