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Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups O (b + , b -) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule Π , which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to Π as theta liftings of holomorphic Mp 2 (R) cusp forms by using the Borcherds' method (Invent Math 132:491–562, 1998). We propose a special choice of Schwartz functions to define the liftings, which yields precise descriptions of their Fourier expansions. [ABSTRACT FROM AUTHOR]

Let f be any normalized Hecke eigen forms with even integral weight k ≥ 2 for the full modular group S L (2 , Z) , and χ be a primitive Dirichlet character modulo q. Let L f (s , χ) be the automorphic L-function attached to f and χ . In this paper, we study the mean-square estimate of L f (s , χ) weighted by incomplete character sums, via using properties of Fourier coefficients and analytic methods, and establish an asymptotic formula which refines the previous results. [ABSTRACT FROM AUTHOR]

Let f, g be two distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 for the full modular group Γ = S L (2 , Z) , respectively. In this paper, we are interested in the upper bounds for the integral mean square of error terms involving λ f 2 i 1 (n) , λ f 2 (n j 1) and λ f 2 (n i 2) λ g 2 (n j 2) on average in terms of f, g respectively. Here i 1 ≥ 2 , j 1 ≥ 3 and i 2 , j 2 ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

In this paper, we study shells of the D 4 lattice with a slight generalization of spherical t-designs due to Delsarte–Goethals–Seidel, namely, the spherical design of harmonic index T (spherical T-design for short) introduced by Delsarte-Seidel. We first observe that, for any positive integer m, the 2m-shell of D 4 is an antipodal spherical { 10 , 4 , 2 } -design on the three dimensional sphere. We then prove that the 2-shell, which is the D 4 root system, is a tight { 10 , 4 , 2 } -design, using the linear programming method. The uniqueness of the D 4 root system as an antipodal spherical { 10 , 4 , 2 } -design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the D 4 lattice in terms of orthogonal transformations of the D 4 root system, and the uniqueness of the D 4 lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the D 4 lattice and non-vanishing of the Fourier coefficients of a certain newform of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed. [ABSTRACT FROM AUTHOR]

We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups Γ 0 (N) with N odd square-free. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem [Thm. 6]. These results are the generalization of the paper [1] results to odd square-free level. [ABSTRACT FROM AUTHOR]

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that lim x → ∞ # { 1 ≤ n ≤ x ∣ τ (n) ≠ 0 } x > 1 - 1.15 × 10 - 12 , where τ (n) is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that τ (n) ≠ 0 . [ABSTRACT FROM AUTHOR]

Suppose that H ∗ is the set of all primitive cusp forms f of even integral weight k ≥ 2 for the full modular group S L 2 (Z) . In this paper, we establish asymptotic formulas for the second moment of Fourier coefficients of the triple product L-function L (s , f ⊗ f ⊗ f) and the related L-function L (s , sym 2 f ⊗ f) attached to f on average, which improves previous results. [ABSTRACT FROM AUTHOR]

Let f, g be two distinct normalized primitive holomorphic cusp forms of weights k 1 , k 2 and levels N 1 , N 2 , respectively. Denote by λ f (n) and λ g (n) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the first change of sequences { λ f (n i) } n ∈ N and { λ f (n j) λ g (n l) } n ∈ N with positive integers i ≥ 2 and j , l ≥ 1 . And we also derive the lower bounds for the sequences { λ f (n i) } n ∈ N , { λ f (n j) λ f (n l) } n ∈ N of the same signs in the interval [1, x], which remains the best possible results in terms of order of magnitude, where i ≥ 2 , j , l ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

Let f and g be distinct primitive holomorphic cusp forms of even integral weights k 1 and k 2 for the full modular group Γ = S L (2 , Z) , respectively. Denote by λ f (n) and λ g (n) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we consider short sums of isotypic trace functions associated to some sheaves modulo primes q of bounded conductor, twisted by the multiplicative function λ f (n i) λ f (n j) and λ f (n i) λ g (n j) for any integers i , j ≥ 1 . We are able to establish non-trivial bounds for these algebraic twisted sums with intervals of length of at least q 1 2 + ε for arbitrarily small ε > 0 . In the similar manner, We also establish the nontrivial bounds for short sums of isotypic trace functions twisted by the coefficients λ f × f × f (n) and λ f × f × g (n) of triple product L-functions L (f × f × f , s) and L (f × f × g , s) , respectively. [ABSTRACT FROM AUTHOR]

We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer n by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form. [ABSTRACT FROM AUTHOR]

We investigate the linear dependence among odd periods of modular forms. We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three odd periods are linearly independent. We also prove an asymptotic result for an arbitrary number of odd periods. The results are achieved by studying cuspidal projections of products of two Eisenstein series. [ABSTRACT FROM AUTHOR]

We formulate a precise conjecture about the size of the L ∞ -mass of the space of Jacobi forms on H n × C g × n of matrix index S of size g. This L ∞ -mass is measured by the size of the Bergman kernel of the space. We prove the conjectured lower bound for all such n, g, S and prove the upper bound in the k aspect when n = 1 , g ≥ 1 . When n = 1 and g = 1 , we make a more refined study of the sizes of the index-(old and) new spaces, the latter via the Waldspurger's formula. Towards this and with independent interest, we prove a power saving asymptotic formula for the averages of the twisted central L-values L (1 / 2 , f ⊗ χ D) with f varying over newforms of level a prime p and even weight k as k , p → ∞ and D being (explicitly) polynomially bounded by k, p. Here χ D is a real quadratic Dirichlet character. We also prove that the size of the space of Saito-Kurokawa lifts (of even weight k) is k 5 / 2 by three different methods (with or without the use of central L-values), and show that the size of their pullbacks to the diagonally embedded H × H is k 2 . In an appendix, the same question is answered for the pullbacks of the whole space S k 2 , the size here being k 3 . [ABSTRACT FROM AUTHOR]

We extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h (log X) c , with h = h (X) → ∞ and where c = c f ≥ 0 is determined by the distribution of { | f (p) | } p in an explicit way. We give three applications. First, we show that the classical Rankin–Selberg-type asymptotic formula for partial sums of | λ f (n) | 2 , where { λ f (n) } n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length h log X , if h = h (X) → ∞ . We also generalize this result to sequences { | λ π (n) | 2 } n , where λ π (n) is the nth coefficient of the standard L-function of an automorphic representation π with unitary central character for G L m , m ≥ 2 , provided π satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments { | λ f (n) | α } n over intervals of length h (log X) c α , with c α > 0 explicit, for any α > 0 , as h = h (X) → ∞ . Finally, we show that the (non-multiplicative) Hooley Δ -function has average value ≫ log log X in typical short intervals of length (log X) 1 / 2 + η , where η > 0 is fixed. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the masses over CM points on Drinfeld–Stuhler modular curves and definite Shimura curves over global function fields. After connecting the CM masses with the modified Hurwitz class numbers, we show that the generating function of these masses is a particular metaplectic theta series. Applying the Poisson summation formula, we are able to describe explicitly the meromorphic continuation of the Mellin transform of the generating theta series. This enables us to derive not only a Gauss-type mean value formula of the average masses in question, but also provides “new” class number relations in the positive characteristic world. [ABSTRACT FROM AUTHOR]

Our goal in this paper is to prove an asymptotic estimate for sums of Fourier coefficients of holomorphic cusp forms over integers without large prime factors by using a Chebyshev-type functional equation. [ABSTRACT FROM AUTHOR]

Let j ≥ 2 be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = S L (2 , Z) . Denote by λ sym j f (n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L (sym j f , s) attached to f. In this paper, we are interested in the average behaviour of the following sum ∑ a 2 + b 2 + c 2 + d 2 ≤ x (a , b , c , d) ∈ Z 4 λ sym j f 2 (a 2 + b 2 + c 2 + d 2) , where x is sufficiently large, which improves and generalizes the recent work of Sharma and Sankaranarayanan [52]. By analogy, we also consider the analogous results for higher moments of normalized Fourier coefficients and the second moment of normalized coefficients of two symmetric power L-functions attached to two distinct cusp forms of the same sequence. [ABSTRACT FROM AUTHOR]

In this article, we obtain finite field analogues of classical summation identities connecting F 3 -Appell series and 4 F 3 -classical hypergeometric series. As an application, we establish a new summation formula satisfied by the 4 F 3 -Gaussian hypergeometric series. We further express certain special values of 4 F 3 -Gaussian hypergeometric series in terms of traces of the Frobenius endomorphisms of certain families of elliptic curves. We also explicitly find some special values of 4 F 3 -Gaussian hypergeometric series. [ABSTRACT FROM AUTHOR]

In this paper we estimate sums of the form ∑ n ≤ X | a Sym m π (| f (n) |) | , for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f (x) ∈ Z [ x ] of arbitrary degree. We give new upper bounds for these sums under certain natural assumptions on f. Our results are unconditional when deg (f) ≤ 4 . Moreover, we study the analogous sum over polynomials in several variables. We obtain an estimate for all cubic polynomials in two variables that define elliptic curves. [ABSTRACT FROM AUTHOR]

The goal of this paper is to provide a complete and refined study of the standard L-functions L (π , Std , s) for certain non-generic cuspidal automorphic representations π of G 2 (A) . For a cuspidal automorphic representation π of G 2 (A) that corresponds to a modular form φ of level one and of even weight on G 2 , we explicitly define the completed standard L-function, Λ (π , Std , s) . Assuming that a certain Fourier coefficient of φ is nonzero, we prove the functional equation Λ (π , Std , s) = Λ (π , Std , 1 - s) . Our proof proceeds via a careful analysis of a Rankin-Selberg integral that is due to an earlier work of Gurevich and Segal. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of orthogonal non-holomorphic Eisenstein series at their harmonic points by using theta lifts. In the case of singular weight, we show that the orthogonal non-holomorphic Eisenstein series that can be written as a theta lift have a simple pole at s = 1 whose residues yield holomorphic orthogonal modular forms that are Eisenstein series on the boundary. Moreover, we will investigate the image of this construction and give sufficient conditions for the surjectivity. [ABSTRACT FROM AUTHOR]

Suppose ∑ n = 1 ∞ a (n) n - s be a Dirichlet series in the Selberg class of degree d and let E(x) be the arithmetical error term of ∑ n ⩽ x a (n) . By the truncated Tong-type formula of E(x), we can get two kinds of the mean square estimates of E(x) in short intervals of Jutila's type. Using the estimates, we are able to improve some previous results established by Lü and Wang [9]. [ABSTRACT FROM AUTHOR]

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group G (A K) , which we refer to as Fourier coefficients associated to the data of a 'Whittaker pair'. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are 'Levi-distinguished' Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients. [ABSTRACT FROM AUTHOR]

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ0(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum and show that for every k ⩾ 1 and ε > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ⩾ 5. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form , where the sum involves restricted coefficients of some suitable half integral weight modular forms. [ABSTRACT FROM AUTHOR]