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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 ⁡ (M k) 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. [ABSTRACT FROM AUTHOR]

We describe the foundations of a Hecke theory for the orthogonal group S O + (2 , n + 2). In particular we consider the Hermitian modular group of degree 2 as a special example of S O + (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. [ABSTRACT FROM AUTHOR]

We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod p singular forms of nonquadratic nebentypus. Here we consider the case where the Fourier coefficients of the modular forms are algebraic integers, and we emphasize that p is a rational prime. Moreover, we construct some examples of mod p singular forms of nonquadratic nebentypus using the Eisenstein series studied by Takemori. [ABSTRACT FROM AUTHOR]

Very recently, Pushpa and Vasuki (Arab. J. Math. 11, 355–378, 2022) have proved Eisenstein series identities of level 5 of weight 2 due to Ramanujan and some new Eisenstein identities for level 7 by the elementary way. In their paper, they introduced seven restricted color partition functions, namely P ∗ (n) , M (n) , T ∗ (n) , L (n) , K (n) , A (n) , and B(n), and proved a few congruence properties of these functions. The main aim of this paper is to obtain several new infinite families of congruences modulo 2 a · 5 ℓ for P ∗ (n) , modulo 2 3 for M(n) and T ∗ (n) , where a = 3 , 4 and ℓ ≥ 1 . For instance, we prove that for n ≥ 0 , P ∗ (5 ℓ (4 n + 3) + 5 ℓ - 1) ≡ 0 (mod 2 3 · 5 ℓ). In addition, we prove witness identities for the following congruences due to Pushpa and Vasuki: M (5 n + 4) ≡ 0 (mod 5) , T ∗ (5 n + 3) ≡ 0 (mod 5). [ABSTRACT FROM AUTHOR]

This study seeks to establish connections between Ramanujan-type Eisenstein series and Borweins’ cubic theta functions, employing the (p, k)-parametrization method introduced by Alaca. Furthermore, the derived identities are applied to provide a novel representation for the evaluation of the discrete convolution sum P 2l+3m=n δ(i)δ(j). [ABSTRACT FROM AUTHOR]

We give a complete description of the distribution of the critical points of the classical Eisenstein series E4 (τ). An application to the spectrum of the Lamé operator is also given. [ABSTRACT FROM AUTHOR]

Viazovska proved that the [Formula: see text] lattice sphere packing is the densest sphere packing in [Formula: see text] dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.

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

Let p ∈ { 3 , 5 , 7 } and let Δ denote the weight twelve modular form arising from Ramanujan's tau function. We show that Δ is congruent to an Eisenstein series E k , χ , ψ modulo p for explicit choices of k and Dirichlet characters χ and ψ . We then prove formulae describing the Iwasawa invariants of the Mazur–Tate elements attached to Δ , confirming numerical data gathered by the authors in a previous work. [ABSTRACT FROM AUTHOR]

Let Γ be a Fuchsian group acting on the hyperbolic upper half-plane H , such that Γ \ H is a geometrically finite Riemann surface with respect to the natural hyperbolic metric induced from H. If γ is hyperbolic then following [J. Kudla and S. Millson, Harmonic differentials and closed geodesics on a Riemann surface, Invent. Math. 54 (1979) 193–211; J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293 (1977) 143–203], there is a corresponding hyperbolic Eisenstein series. In this paper, we study the limiting behavior of hyperbolic Eisenstein series on a degenerating family of geometrically finite hyperbolic surfaces. In particular, we give a partial lightening to a question of Ji, concerning the approximation of Eisenstein series during degeneration (see Proposition 5.2 and Theorem 5.2). [ABSTRACT FROM AUTHOR]

We provide for primes p ≥ 5 a method to compute valuations appearing in the "formal" Katz expansion of the family E κ ⁎ V (E κ ⁎) derived from the family of Eisenstein series E κ ⁎. We will describe two algorithms: the first one to compute the Katz expansion of an overconvergent modular form and the second one, which uses the first algorithm, to compute valuations appearing in the "formal" Katz expansion. Based on data obtained using these algorithms we make a precise conjecture about a constant appearing in the overconvergence rates related to the classical Eisenstein series at level p. The study of these overconvergence rates of the members of this family goes back to a conjecture of Coleman. [ABSTRACT FROM AUTHOR]

The objective of this paper is to establish six new identities which depict interrelationships between Eisenstein series identities, Göllnitz-Gordon identities and combinatorial partition identities. [ABSTRACT FROM AUTHOR]

The equation |$x^2 + 1 = 0\mod p$| has solutions whenever p  = 2 or |$4n + 1$|⁠. A famous theorem of Fermat says that these primes are exactly the ones that can be described as a sum of two squares. The roots of the former equation are equidistributed is a beautiful theorem of Duke, Friedlander and Iwaniec. The angles associated to the representation of such prime as a sum of squares are equidistributed is a famous theorem of Hecke. We give a natural way to associate between roots and angles and prove that the joint equidistribution of the sequence of pairs of roots and angles is equidistributed as well. Our approach involves an automorphic interpretation, which reduces the problem to the study of certain Poincare series on an arithmetic quotient of |$SL_2(\mathbb{R})$|⁠. Since our Poincare series have a nontrivial dependence on their Iwasawa θ -coordinate, they do not factor into functions on the upper half plane, as in the case studied by Duke et al. Spectral analysis on these higher dimensional varieties involves the nonspherical spectrum, making this paper the first complete study of a nonspherical equidistribution problem, with an arithmetic application. A couple of notable challenges we had to overcome were that of obtaining pointwise bounds for nonspherical Eisenstein series and utilizing a non-spherical analogue of the Selberg inversion formula, which we believe may have further implications beyond this work. [ABSTRACT FROM AUTHOR]

Ramanujan, in his lost notebook on page 188, delineated unique categories of remarkable infinite series, expressing them through the framework of Eisenstein series. The aim of this paper is to explore diverse differential identities concerning classical η-functions and h-functions. Furthermore, we introduce specific identities involving Eisenstein series of various levels and h-functions, which serve to establish connections between class one infinite series and h-functions. [ABSTRACT FROM AUTHOR]

We give a general conjecture concerning the existence of Eisenstein congruences between weight k ≥ 3 newforms of square-free level NM and weight k new Eisenstein series of square-free level N. Our conjecture allows the forms to have arbitrary character χ of conductor N. The special cases M = 1 and M = p prime are fully proved, with partial results given in general. We also consider the relation with the Bloch–Kato conjecture, and finish with computational examples demonstrating cases of our conjecture that have resisted proof. [ABSTRACT FROM AUTHOR]

For even integers k ≥ ℓ ≥ 4 , we consider the modular forms E k + E ℓ + + E k + ℓ + for the Fricke group Γ 0 + (2) , where E k + is the Eisenstein series of weight k for Γ 0 + (2) , and we prove that if 26630 ≤ ℓ < k ≤ 77 ℓ or k = ℓ ≥ 10 , then all of their zeros in the fundamental domain F + for Γ 0 + (2) lie on the arc boundary of F + . [ABSTRACT FROM AUTHOR]

We give explicit expressions for MacMahon's generalized sums-of-divisors q-series A r and C r by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular forms, vertex algebras, N = 4 SU(N) Super–Yang–Mills theory, and the study of congruences of partitions. We relate them to a broader mathematical framework and give explicit expressions for both q-series in terms of Eisenstein series and their odd variants. [ABSTRACT FROM AUTHOR]

On page 188 of his lost notebook, Ramanujan introduced distinct classes of exquisite infinite series, representing them in terms of Eisenstein series. The objective of this paper is to develop various differential identities related to Borwein's cubic theta functions and h-functions. Also, we present certain identities containing Eisenstein series of level 6 and h-functions that are used to establish relations involving class one infinite series and h-functions. Additionally, we present a straightforward approach to evaluate a discrete convolution sum by employing relationships involving Eisenstein series and h-functions. [ABSTRACT FROM AUTHOR]

The (p, k)-parametrization method, as introduced by Alaca, offers an innovative approach to derive a variety of new Eisenstein series identities that incorporate Borweins' theta functions. This paper presents new Eisenstein series identities that involve Borweins' cubic theta functions, achieved through the (p, k)-parameterization technique. [ABSTRACT FROM AUTHOR]

In this work, Ramanujan type congruences modulo powers of primes p ≥ 5 are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and subsume generating functions for t -core partitions known to satisfy Ramanujan type congruences for p = 5 , 7 , 11. The vectors of exponents corresponding to products that are modular forms for Γ 1 (p) are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for Γ 1 (p) of weights 1 ≤ k ≤ 5 for primes 5 ≤ p ≤ 19 and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. For each product satisfying a congruence, cyclic permutations of the exponents determine additional products satisfying congruences. Common forms among the exponent sets lead to products satisfying Ramanujan type congruences for a broad class of primes, including p > 19. Canonical bases for modular forms of level 5 ≤ p ≤ 19 are constructed by summing weight one Hecke Eisenstein series of levels 5 ≤ p ≤ 19 and expressing the result as a quotient of Klein forms. Generating sets for the graded algebras of modular forms for Γ 1 (p) and Γ (p) are formulated in terms of permutations of the exponent sets. A sieving process is described by decomposing the space of modular forms of weight 1 for Γ 1 (p) as a direct sum of subspaces of modular forms for Γ (p) of the form q r / p Z [ [ q ] ]. Since the relevant bases generate the graded algebra of modular forms for these groups, the weight one decompositions determine series dissections for modular forms of higher weight that lead to additional classes of congruences. [ABSTRACT FROM AUTHOR]

In this paper, we construct higher-order modular differential equations for elliptic modular forms, holomorphic Jacobi forms, and skew-holomorphic Jacobi forms by using the Rankin-Cohen brackets extended to the Eisenstein series of weight two. Rankin-Cohen brackets are typical tools in the construction of modular forms by differential operators. We introduce the extended Rankin-Cohen brackets for holomorphic or skew-holomorphic Jacobi forms paired with elliptic modular forms. They provide a unified method of construction of higher-order modular differential equations for the modular forms above. In particular, with these extended Rankin-Cohen brackets, we can reproduce the modular differential equations of previous studies. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Ramanujan (1916) and Shen (1999) discovered differential equations for classical Eisenstein series. Motivated by them, we derive new differential equations for Eisenstein series of level 2 from the second kind of Jacobi theta function. This gives a new characterization of a system of differential equations by Ablowitz–Chakravarty–Hahn (2006), Hahn (2008), Kaneko–Koike (2003), Maier (2011), Nidelan (2022) and Toh (2011). As application, we show some arithmetic results on Ramanujan's tau function. [ABSTRACT FROM AUTHOR]

Let Γ = PSL 2 (O) be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight k period polynomials for Γ is "dual" to the space of weight k modular symbols for Γ , reflecting the duality between the first and second cohomology groups of the arithmetic group Γ . Using this result, we describe the action of Hecke operators on the space of period polynomials for Γ via the Heilbronn matrices. As in the classical case, spaces of Bianchi period polynomials are related to parabolic cohomology of Bianchi groups, and in turn, via the Eichler–Shimura–Harder Isomorphism, to spaces of Bianchi modular forms. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the above mentioned Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature. [ABSTRACT FROM AUTHOR]

We use mass formulas to construct minimal parabolic Eisenstein congruences for algebraic modular forms on reductive groups compact at infinity. For unitary groups of prime degree, this construction yields Eisenstein congruences for non-endoscopic cuspidal automorphic forms on quasi-split unitary groups. In supplementary sections, we also generalize previous weight 2 Eisenstein congruences for Hilbert modular forms, and prove some special congruence mod p results between cusp forms on U(p). [ABSTRACT FROM AUTHOR]

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 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. [ABSTRACT FROM AUTHOR]

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 ρ (n , m , r) associated with the maximal order in the quaternion algebra (− 1 , − 1) ℚ . 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". [ABSTRACT FROM AUTHOR]

Let Γ be a subgroup of finite index of SL 2 ⁡ (퐙) . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ of the corresponding modular curve X Γ . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ (2) , where we can consider convenient generalized Jacobians instead of J Γ . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves. [ABSTRACT FROM AUTHOR]

The (p, fc)-parametrization technique introduced by Alaca provides an alternative approach for formulating several new Eisenstein series identities involving Borwein's theta functions. As a practical example, we have computed the convolution sum of divisor functions, which exhibit a connection with Eisenstein series. [ABSTRACT FROM AUTHOR]

A composition law, inspired by the Farey addition, is introduced on the set of Pythagorean triples. We study some of its properties as well as two symmetric matrices naturally associated to a given Pythagorean triple. Several examples are discussed, some of them involving the degenerated Pythagorean triple (1, 0, 1). The case of Eisenstein triples is also presented. [ABSTRACT FROM AUTHOR]

We prove under certain conditions (local-global compatibility and vanishing of modulo |$p$| cohomology), a generalization of a theorem of Galatius and Venkatesh. We consider the case of |$\operatorname{\textsf{GL}}(N)$| over a CM field; we construct a Hecke-equivariant injection from the divisible group associated to the first fundamental group of a derived deformation ring to the Selmer group of the twisted dual adjoint motive with divisible coefficients and we identify its cokernel as the first Tate-Shafarevich group of this motive. Actually, we also construct similar maps for higher homotopy groups with values in exterior powers of Selmer groups, although with less precise control on their kernel and cokernel. By a result of Y. Cai generalizing previous results by Galatius-Venkatesh on the graded cohomology group of a locally symmetric space, our maps relate the (non-Eisenstein) localization of the graded cohomology group for a locally symmetric space to the exterior algebra of the Selmer group of the Tate dual of the adjoint representation. We generalize this to Hida families as well. [ABSTRACT FROM AUTHOR]

In this paper, we prove a formula, realizing certain residual Eisenstein series on symplectic groups as regularized kernel integrals. These Eisenstein series, as well as the kernel integrals, are attached to Speh representations. This forms an initial step to a new type of a regularized Siegel-Weil formula that we propose. This new formula bears the same relation to the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan, as does the regularized Siegel-Weil formula to the doubling integrals of Piatetski-Shapiro and Rallis. [ABSTRACT FROM AUTHOR]

We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the r th central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of "virtual dimension 0" on the moduli stack of hermitian shtukas with r legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series. [ABSTRACT FROM AUTHOR]

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg's later proofs. [ABSTRACT FROM AUTHOR]

Srinivasa Ramanujan developed seventeen fast convergent series for 1/π. Motivated by Ramanujan's series for 1/π many mathematicians have developed many theories for deriving new series for 1/π. In this article we obtain new series for 1/π using Eisenstein series of level three. [ABSTRACT FROM AUTHOR]

The main theorem of the paper [2] requires a restriction on the prime p. This statement is being modified based on a certain conjecture on the p -adic zeta functions. [ABSTRACT FROM AUTHOR]

Let p be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form E k ∗ V (E k ∗) where E k ∗ is a classical, normalized Eisenstein series on Γ 0 (p) and V the p-adic Frobenius operator. In particular, we extend our previous paper to the primes 2 and 3. For these primes our main theorem improves somewhat upon earlier results by Emerton, Buzzard and Kilford, and Roe. We include a detailed discussion of those earlier results as seen from our perspective. We also give some improvements to our earlier paper for primes p ≥ 5 . Apart from establishing these improvements, our main purpose here is also to show that all of these results can be obtained by a uniform method, i.e., a method where the main points in the argumentation is the same for all primes. We illustrate the results by some numerical examples. [ABSTRACT FROM AUTHOR]

Given cusp forms f and g of integral weight k ≽ 2, the depth two holomorphic iterated Eichler-Shimura integral If,g is defined by ∫i∞τ ƒ(z)(X - z)k-2 Ig (z; Y) dz, where Ig is the Eichler integral of g and X,Y are formal variables. We provide an explicit vector-valued modular form whose top components are given by If,g. We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by εf,g, that can be seen as a higher depth generalization of the scalar-valued Eichler integral εf of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol-Popa. We show that εf,g can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form Δ. This allows for effective computation of εf,g. [ABSTRACT FROM AUTHOR]

Few new Eisenstein series identities involving Borwein’s theta functions are formulated using (p, k)- parametrization technique which is introduced by Alaca. As an application, we have evaluated the convolution sum of divisor functions which are related to Eisenstein series. [ABSTRACT FROM AUTHOR]

We study the number of two‐dimensional sublattices of Z4$\mathbb {Z}^4$ of a fixed covolume and construct the associated Dirichlet series. The latter is shown to be related to Eisenstein series on both GL4$\operatorname{GL}_4$ and its metaplectic double cover. [ABSTRACT FROM AUTHOR]

In this work, we give an explicit formula for the Fourier coefficients of Eisenstein series corresponding to certain arithmetic lattices acting on hyperbolic n+1$n+1$‐space. As a consequence, we obtain results on location of all poles of these Eisenstein series as well as their supremum norms. We use this information to get new results on counting rational points on spheres. [ABSTRACT FROM AUTHOR]

This article focuses on the application of the Ramanujan-type Eisenstein series to the formation of numerous differential identities. In this paper, utilizing Alaca’s (p,k) parametrization, we explore a few additional interesting links between Eisenstein Series and Borewein’s cubic theta functions. Further, we build a set of higher-order nonlinear differential equations that include Ramanujan’s function k(q). In addition, we formulate identities relating the Class one infinite series and Ramanujan’s k-function, using the relationship derived between Eisenstein and the Class one infinite series. [ABSTRACT FROM AUTHOR]

We describe the p-divisibility transposition for the Fourier coefficients of Siegel modular forms. This provides a supplement to the result by Wilton for p-divisibility satisfied by the Ramanujan τ -function. [ABSTRACT FROM AUTHOR]