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We prove a variety of quantum unique ergodicity (QUE) results for Eisenstein series in the level aspect. A new feature of this variant of QUE is that the main term involves the logarithmic derivative of a Dirichlet L-function on the 1-line. A zero of this L-function near the 1-line can thus have a distorting effect on the main term. We obtain quantitative control on the test function and thereby prove an asymptotic formula in the level aspect version of the problem with test functions of shrinking support. Surprisingly, this asymptotic formula shows some obstruction to equidistribution that may retrospectively be interpreted as being caused by the growth of Eisenstein series in the cusps. We also make some coarse descriptions on the unevenness of the mass distribution of level N Eisenstein series on the fibers of the canonical projection map from Y 0 (N) to Y 0 (1) . [ABSTRACT FROM AUTHOR]

Let r Q (n) be the representation number of a nonnegative integer n by an integral positive definite quadratic form Q in 2 k variables. Let N be the level of Q. For a fixed prime p ∤ N , we assume that r Q (n) satisfies the relation r Q (1) r Q (p 2 n) = r Q (p 2) r Q (n) for all positive integers n with p ∤ n. We actually show that this relation holds for any prime p ∤ N when (− 1) k N is a fundamental discriminant. [ABSTRACT FROM AUTHOR]

The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra T consisting of symmetric polynomials in the part sizes and multiplicities. [ABSTRACT FROM AUTHOR]

We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a 'hidden' invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier–Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory. [ABSTRACT FROM AUTHOR]

In this work, we use the Rankin–Selberg method to obtain results on the analytic properties of the standard L -function attached to vector-valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted L -function beyond the usual range of absolute convergence. Our results include also the case of metaplectic Siegel modular forms. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series. [ABSTRACT FROM AUTHOR]

We give a lower bound of the number of zeros of the modular forms E k + E ℓ + − E k + ℓ + for large enough k and ℓ on each of the arc boundary and on the vertical boundary of the fundamental domain for the Fricke group Γ 0 + (2) , where E k + is the Eisenstein series of weight k for Γ 0 + (2). [ABSTRACT FROM AUTHOR]

Newform Dedekind sums are a class of crossed homomorphisms that arise from newform Eisenstein series. We initiate a study of the kernel of these newform Dedekind sums. Our results can be loosely described as showing that these kernels are neither "too big" nor "too small." We conclude with an observation about the Galois action on Dedekind sums that allows for significant computational efficiency in the numerical calculation of Dedekind sums. [ABSTRACT FROM AUTHOR]

The article discusses the p-adic Waldspurger formula involving a p-stabilization equation and the newform of weight w for the anticyclotomic p-adic L-function. Also cited are the key elements of the formula including the Coleman's theory of p-adic integration of 1-forms on p-adic analytic spaces and the p-adic variation of p-adic Maass-Shimura derivatives, as well as the p-adic Abel-Jacobi maps.

The article discusses the supersingular Rankin-Selberg p-adic L-functions by constructing a 1-variable anticyclotomic p-adic L-function linked to a given normalized eigenform like a newform or normalized Eisenstein series. Also cited are the use of an imaginary quadratic field and the fundamental discriminant of K to create a Heegner hypothesis for N, and the equation for a complex-valued Hecke character.

The article discusses the construction of p-adic L-functions over imaginary quadratic fields K based on the works of N. Katz, M. Bertolini, H. Darmon and K. Prasanna. Topics include Katz's theory of p-adic modular forms, the application of p-adic differential operators in modular forms, and the differential operator Atkin-Serre operator.

The Diophantine exponent of an action of a group on a homogeneous space, as defined by Ghosh, Gorodnik, and Nevo, quantifies the complexity of approximating the points of the homogeneous space by the points on an orbit of the group. We show that the Diophantine exponent of the SL n (Z [ 1 / p ]) -action on the generalized upper half-space SL n (R) / SO n (R) , lies in [ 1 , 1 + O (1 / n) ] , substantially improving upon Ghosh–Gorodnik–Nevo's method which gives the above range to be [ 1 , n − 1 ]. We also show that the exponent is optimal , i.e. equals one, under the assumption of Sarnak's density hypothesis. The result, in particular, shows that the optimality of Diophantine exponents can be obtained even when the temperedness of the underlying representations, the crucial assumption in Ghosh–Gorodnik–Nevo's work, is not satisfied. The proof uses the spectral decomposition of the homogeneous space and bounds on the local L 2 -norms of the Eisenstein series. [ABSTRACT FROM AUTHOR]

Two-dimensional theta functions were found by the Borwein brothers to work on Gauss and Legendre's arithmetic-geometric mean iteration. In this paper, some new Eisenstein series identities are obtained by using (p , k)-parametrization in terms of Borweins' theta functions. [ABSTRACT FROM AUTHOR]

In this paper we study the analytic properties of the standard L -function attached to vector valued quaternionic modular forms using the Rankin-Selberg method. This involves the construction of vector valued theta series, which we obtain by applying some differential operators on Jacobi-theta series studied by Krieg. Such differential operators are obtained from the Howe-Weyl duality for the pair S p n (C) × G L 2 n (C). [ABSTRACT FROM AUTHOR]

We apply the methods of Fukaya, Kato and Sharifi to refine Mazur's study of the Eisenstein ideal. Given prime numbers N and p ≥ 5 such that p | ϕ (N) , we study the quotient of the cohomology group of modular curve X 0 (N) by the square of the Eisenstein ideal. We study two invariants b , c attached to this quotient and compute c. We propose a conjecture about the invariant b which relates the structure of the ray class group of conductor N to the modular symbols of X 0 (N). Assuming this conjecture, we compute the invariant b. [ABSTRACT FROM AUTHOR]

This paper continues work of the earlier articles with the same title. For two classes of modular forms f : • para-Eisenstein series α k and • coefficient forms ℓ k a , where k ∈ N and a is a non-constant element of F q [ T ] , the growth behavior on the fundamental domain and the zero loci Ω (f) as well as their images BT (f) in the Bruhat-Tits building BT are studied. We obtain a complete description for f = α k and for those of the forms ℓ k a where k ≤ deg ⁡ a. It turns out that in these cases, α k and ℓ k a are strongly related, e.g., BT (ℓ k a ) = BT (α k) , and that BT (α k) is the set of Q -points of a full subcomplex of BT with nice properties. As a case study, we present in detail the outcome for the forms α 2 in rank 3. [ABSTRACT FROM AUTHOR]

We present a novel expression for an integrated correlation function of four superconformal primaries in SU(N) N = 4 supersymmetric Yang-Mills (N = 4 SYM) theory. This integrated correlator, which is based on supersymmetric localisation, has been the subject of several recent developments. In this paper the correlator is re-expressed as a sum over a two dimensional lattice that is valid for all N and all values of the complex Yang-Mills coupling τ = θ / 2 π + 4 πi / g YM 2 . In this form it is manifestly invariant under SL(2, ℤ) Montonen-Olive duality. Furthermore, it satisfies a remarkable Laplace-difference equation that relates the SU(N) correlator to the SU(N + 1) and SU(N − 1) correlators. For any fixed value of N the correlator can be expressed as an infinite series of non-holomorphic Eisenstein series, E s τ τ ¯ with s ∈ ℤ, and rational coefficients that depend on the values of N and s. The perturbative expansion of the integrated correlator is an asymptotic but Borel summable series, in which the n-loop coefficient of order (gYM/π)2n is a rational multiple of ζ(2n + 1). The n = 1 and n = 2 terms agree precisely with results determined directly by integrating the expressions in one-loop and two-loop perturbative N = 4 SYM field theory. Likewise, the charge-k instanton contributions (|k| = 1, 2,...) have an asymptotic, but Borel summable, series of perturbative corrections. The large-N expansion of the correlator with fixed τ is a series in powers of N 1 2 − ℓ (ℓ ∈ ℤ) with coefficients that are rational sums of E s τ τ ¯ with s ∈ ℤ + 1/2. This gives an all orders derivation of the form of the recently conjectured expansion. We further consider the 't Hooft topological expansion of large-N Yang-Mills theory in which λ = g YM 2 N is fixed. The coefficient of each order in the 1/N expansion can be expanded as a series of powers of λ that converges for |λ| < π2. For large λ this becomes an asymptotic series when expanded in powers of 1 / λ with coefficients that are again rational multiples of odd zeta values, in agreement with earlier results and providing new ones. We demonstrate that the large-λ series is not Borel summable, and determine its resurgent non-perturbative completion, which is O exp − 2 λ . [ABSTRACT FROM AUTHOR]

We construct (n + 1)-variable Hida families of Klingen Eisenstein series on Sp(2n + 2) for n-variable Hida families on Sp(2n), and relate their images under the Siegel operator to p-adic L-functions. We also carry out some preliminary calculations of the non-degenerate Fourier coefficients of the constructed Klingen Eisenstein families. [ABSTRACT FROM AUTHOR]

We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function ℘ (z | τ) as a blueprint, we will derive their representations in terms of q-series and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups Γ 0 (N) and Γ 1 (N) . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider the computation of the Fourier coefficients of real-analytic Eisenstein series associated to general cusps of Hecke congruence subgroups of arbitrary level. Explicit formulas for some coefficients in the Fourier expansion of the Eisenstein series of weight zero for general cusps are given. [ABSTRACT FROM AUTHOR]

Using results from spectral theory of Eisenstein series, we prove a formula for the second moment of the Siegel transform when averaged over the subspace of symplectic lattices. This generalizes the classical formula of Rogers for the second moment in the full space of unimodular lattices. Using this new formula we give very strong bounds for the discrepancy of the number of lattice points in a Borel set, which hold for generic symplectic lattices. [ABSTRACT FROM AUTHOR]

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions σ ψ sm (n). More precisely, in terms of the weight 2 quasimodular Eisenstein series E 2 (τ) and a generic Shimura theta function 𝜃 ψ (τ) , we show that there is a constant α ψ for which ε ψ + (τ) : = α ψ ⋅ E 2 (τ) 𝜃 ψ (τ) + 1 𝜃 ψ (τ) ∑ n = 1 ∞ σ ψ sm (n) q n is a half integral weight (polar) mock modular form. These include generating functions for combinatorial objects such as the Andrews s p t -function and the "consecutive parts" partition function. Finally, in analogy with Serre's result that the weight 2 Eisenstein series is a p -adic modular form, we show that these forms possess canonical congruences with modular forms. [ABSTRACT FROM AUTHOR]

In this paper we introduce two general identities relating Eisenstein series on split classical groups, as well as double covers of symplectic groups. The first identity can be viewed as an extension of the doubling construction introduced in [CFGK19]. The second identity is a generalization of the descent construction studied in [GRS11]. [ABSTRACT FROM AUTHOR]

In [1] we have studied the single-particle free energy of a class of Little String Theories of A-type, which are engineered by N parallel M5-branes on a circle. To leading instanton order (from the perspective of the low energy U(N) gauge theory) and partially also to higher order, a decomposition was observed, which resembles a Feynman diagrammatic expansion: external states are given by expansion coefficients of the N = 1 BPS free energy and a quasi-Jacobi form that governs the BPS-counting of an M5-brane coupling to two M2-branes. The effective coupling functions were written as infinite series and similarities to modular graph functions were remarked. In the current work we continue and extend this study: working with the full non-perturbative BPS free energy, we analyse in detail the cases N = 2, 3 and 4. We argue that in these cases to leading instanton order all coupling functions can be written as a simple combination of two-point functions of a single free scalar field on the torus. We provide closed form expressions, which we conjecture to hold for generic N. To higher instanton order, we observe that a decomposition of the free energy in terms of higher point functions with the same external states is still possible but a priori not unique. We nevertheless provide evidence that tentative coupling functions are still combinations of scalar Greens functions, which are decorated with derivatives or multiplied with holomorphic Eisenstein series. We interpret these decorations as corrections of the leading order effective couplings and in particular link the latter to dihedral graph functions with bivalent vertices, which suggests an interpretation in terms of disconnected graphs. [ABSTRACT FROM AUTHOR]

We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the N = 4 SU(N) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the N = 2∗ theory with respect to the squashing parameter b and mass parameter m, evaluated at the values b = 1 and m = 0 that correspond to the N = 4 theory on a round sphere. At each order in the 1/N expansion, these fourth derivatives are modular invariant functions of (τ, τ ¯ ). We present evidence that at half-integer orders in 1/N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1/N, they are certain "generalized Eisenstein series" which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS5× S5. [ABSTRACT FROM AUTHOR]

Properties of divisor functions σ k (n) , defined as sums of k-th powers of all divisors of n, are studied through the analysis of Ramanujan's differential equations. This system of three differential equations is singular at x = 0 . Solution techniques suitable to tackle this singularity are developed and the problem is transformed into an analysis of a dynamical system. Number theoretical consequences of the presented dynamical system analysis are then discussed, including recursive formulas for divisor functions. [ABSTRACT FROM AUTHOR]

With the help of so called pre-weak functions, we formulate a very general transformation law for some holomorphic functions on the upper half plane and motivate the term of a generalized Eisenstein series with real-exponent Fourier expansions. Using the transformation law in the case of negative integers k, we verify a close connection between finite cotangent sums of a specific type and generalized L-functions at integer arguments. Finally, we expand this idea to Eichler integrals and period polynomials for some types of modular forms. [ABSTRACT FROM AUTHOR]

Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series E k , especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized Petersson's approach via Poincaré series, to handle the general case. We introduce a third approach. We attach recursively defined polynomials to reciprocals of Eisenstein series. This provides easy access to the signs of the Fourier coefficients of reciprocals of Eisenstein series, sheds some light on reciprocals of E k of general weight, and provides some upper and lower bounds for their growth. [ABSTRACT FROM AUTHOR]

We define the space of nearly holomorphic automorphic forms on a connected reductive group G over Q such that the homogeneous space G (R) 1 / K ∞ ∘ is a Hermitian symmetric space. By Pitale, Saha and Schmidt's study, there are the classification of indecomposable (g , K ∞) -modules which occur in the space of nearly holomorphic elliptic modular forms and Siegel modular forms of degree 2. This paper studies global representations of the adele group G (A Q) which occur in the space of nearly holomorphic Hilbert modular forms. In the case of elliptic modular forms, the result of this paper is an adelization of Pitale, Saha and Schmidt's result. [ABSTRACT FROM AUTHOR]

We give a precise determination of the algebraicity of the critical values of L -functions associated to Siegel modular forms of half-integral weight and arbitrary degree. We generalise and improve on similar results for the integral-weight case by adapting the Rankin-Selberg method to this setting, with the aid of Shimura's theory of half-integral weight modular forms and recent work on precise algebraicity results by Bouganis. An essential ingredient of this work is a proof of a new analogue of Garrett's conjecture on the algebraicity of Klingen Eisenstein series, a result which is also of independent interest. [ABSTRACT FROM AUTHOR]

Let E k (z) be the normalized Eisenstein series of weight k for the modular group SL (2 , Z). We study the zeros of E k to prove that the equation ∏ i = 1 n E k i = ∏ j = 1 m E ℓ j has no solutions, except for those given by known relationships between E 4 , E 6 , E 8 , E 10 , and E 14. We go on to discuss some implications of this result. [ABSTRACT FROM AUTHOR]

In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on |$\operatorname{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$|⁠. The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain L-functions. [ABSTRACT FROM AUTHOR]

In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative "period polynomials" in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our "period polynomials". This is suggested by numerical experiments on families of perturbed "period polynomials" (Section 5.3) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series. Although not much is currently known about derivatives higher than first order ones for general modular forms, celebrated recent work of Yun and Zhang established the analogues of the Gross–Zagier formula for higher L-derivatives in the function field case. A critical role in their work was played by a notion of "super-positivity", which, as recently shown by Goldfeld and Huang, holds in infinitely many cases for classical modular forms. As will be discussed, this is similar to properties which were required by Jin, Ma, Ono, and Soundararajan in their proof of the Riemann Hypothesis for Period Polynomials, thus suggesting a connection between the analytic nature of our conjectures here and the framework of Yun and Zhang. [ABSTRACT FROM AUTHOR]

In this note we shall show that a lattice Zω1 + Zω2 in C has Q-linearly dependent quasi-periods if and only if ω2/ω1 is equivalent to a zero of the Eisenstein series E2 under the action of SL2(Z) on the upper half plane of C. [ABSTRACT FROM AUTHOR]

In this paper, we study hybrid subconvexity bounds for class group L-functions associated to quadratic extensions K/ℚ (real or imaginary). Our proof relies on relating the class group L-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the uniform sup norm bound for Eisenstein series E(z, 1/2 + i t) ≪ εy1/2(|t| + 1)1/3+ε, y ≫ 1, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series. [ABSTRACT FROM AUTHOR]

For m = 3, 4, ..., let λm = 2 cos π{m and let Jmpm = 3, 4, ...) be triangle functions for the Hecke groups Gpλmq with Fourier expansions Jm(τ) = °8 n=-1 an(m)qnm, where)mpτ) = exp 2πiτ {λm. (When normalized appropriately, J3 becomes Klein's j-invariant jpτ) = 1{e2πiτ - 744 - ... .) For n = -1, 0, 1, 2 and 3, Raleigh gave polynomials Pnpxq such that a-1(m)n2n-2 m anpmq = Pnpmq for m = 3, 4, ..., and conjectured that similar relations hold for all positive integers n. This was proved by Akiyama. We apply work of Hecke to study experimentally similar polynomial interpolations of the Jm Fourier coefficents and the Fourier coefficients of other, positive weight, modular forms for G(λm). We connect these polynomials (again, only empirically) with variants of Dedekind's eta function, with the Fourier expansions of some standard Hauptmoduln, and, in the case of analogues of Eisenstein series for SLp2,Zq, with certain divisor sums. [ABSTRACT FROM AUTHOR]

In this note, we use a basic identity, derived from the generalized doubling integrals of [ 2 ], in order to explain the existence of various global Rankin–Selberg integrals for certain |$L$| -functions. To derive these global integrals, we use the identities relating Eisenstein series in [ 11 ], together with the process of exchanging roots. We concentrate on several well-known examples, and explain how to obtain them from the basic identity. Using these ideas, we also show how to derive a new global integral. [ABSTRACT FROM AUTHOR]

We propose a characterisation of the field into which quotients of non-critical values of L-functions associated to a cusp form belong. The construction involves shifted convolution series of divisor sums and a certain double Eisenstein series that induces a kernel of L-functions. [ABSTRACT FROM AUTHOR]

We compute an asymptotic expression for the Arakelov self-intersection number of the relative dualizing sheaf of Edixhoven's minimal regular model for the modular curve X 0 (p 2) over Q . The computation of the self-intersection numbers are used to prove an effective version of the Bogolomov conjecture for the semi-stable models of modular curves X 0 (p 2) and obtain a bound on the stable Faltings height for those curves in a companion article (Banerjee and Chaudhuri in Isr J Math, 2020). [ABSTRACT FROM AUTHOR]

We obtain an asymptotic for the regularized 4th moment of the Eisenstein series for the full modular group, in agreement with the Random Wave Conjecture. [ABSTRACT FROM AUTHOR]

The second author formulated quantum unique ergodicity for Eisenstein series in the prime level aspect in "Equidistribution of Eisenstein series in the level aspect", Commun. Math. Phys. 289(3) 1150 (2009). We point out errors and correct the proofs with partially weakened claims. [ABSTRACT FROM AUTHOR]

For primitive nontrivial Dirichlet characters χ 1 and χ 2 , we study the weight zero newform Eisenstein series E χ 1 , χ 2 (z , s) at s = 1. The holomorphic part of this function has a transformation rule that we express in finite terms as a generalized Dedekind sum. This gives rise to the explicit construction (in finite terms) of elements of H 1 (Γ 0 (N) , ℂ). We also give a short proof of the reciprocity formula for this Dedekind sum. [ABSTRACT FROM AUTHOR]

We study the four-point function of the lowest-lying half-BPS operators in the N = 4 SU(N) super-Yang-Mills theory and its relation to the flat-space four-graviton amplitude in type IIB superstring theory. We work in a large-N expansion in which the complexified Yang-Mills coupling τ is fixed. In this expansion, non-perturbative instanton contributions are present, and the SL(2, ℤ) duality invariance of correlation functions is manifest. Our results are based on a detailed analysis of the sphere partition function of the mass-deformed SYM theory, which was previously computed using supersymmetric localization. This partition function determines a certain integrated correlator in the undeformed N = 4 SYM theory, which in turn constrains the four-point correlator at separated points. In a normalization where the two-point functions are proportional to N2− 1 and are independent of τ and τ ¯ , we find that the terms of order N and 1 / N in the large N expansion of the four-point correlator are proportional to the non-holomorphic Eisenstein series E 3 2 τ τ ¯ and E 5 2 τ τ ¯ , respectively. In the flat space limit, these terms match the corresponding terms in the type IIB S-matrix arising from R4 and D4R4 contact inter-actions, which, for the R4 case, represents a check of AdS/CFT at finite string coupling. Furthermore, we present striking evidence that these results generalize so that, at order N 1 2 − m with integer m ≥ 0, the expansion of the integrated correlator we study is a linear sum of non-holomorphic Eisenstein series with half-integer index, which are manifestly SL(2, ℤ) invariant. [ABSTRACT FROM AUTHOR]