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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]

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 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]

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]

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]

We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing Fourier coefficients of basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Strömbergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Strömberg. [ABSTRACT FROM AUTHOR]

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change. [ABSTRACT FROM AUTHOR]

Let f, g be distinct normalized Hecke eigenforms of weights k 1 , k 2 lying in the subspace of newforms with Fourier coefficients { n (k 1 - 1) / 2 λ f (n) } n ∈ N and { n (k 2 - 1) / 2 λ g (n) } n ∈ N respectively. For such newforms f, g of CM type and primes p, we study the natural density of the set S = { p | λ f (p) = λ g (p) }. We show that the upper natural density of S is ≤ 3 / 4 if f ≠ g and it is equal to 1/2 when f and g have different weights and have the same associated CM (quadratic) field. Further, f and g have different associated CM (quadratic) fields if and only if the natural density of S is 1/4. When at least one of f, g is a non-CM form, we study the natural density of the sets S + (x , α) = { p ≤ x | θ f (p) + θ g (p) = α } and S - (x , β) = { p ≤ x | θ f (p) - θ g (p) = β } where θ f (p) , θ g (p) ∈ [ 0 , π ] are the angles associated to the p-th Hecke eigen values of f, g respectively and α ∈ [ 0 , 2 π ] , β ∈ [ - π , π ] . In this case, we show that S + (x , α) and S - (x , β) have natural density zero when f and g are distinct and not twists of each other. Finally, we establish an explicit link between the elements of these sets and sign changes of Fourier coefficients at prime powers which allows us to improve a number of existing results in this set-up. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class t Z 2 , we investigate the sign changes in the t p 2 -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level. [ABSTRACT FROM AUTHOR]

We study canonical bases for spaces of weakly holomorphic modular forms of level 4 and weights in $$\mathbb {Z}+\frac{1}{2}$$ and show that almost all modular forms in these bases have the property that many of their zeros in a fundamental domain for $$\Gamma _0(4)$$ lie on a lower boundary arc of the fundamental domain. Additionally, we show that at many places on this arc, the generating function for Hurwitz class numbers is equal to a particular mock modular Poincaré series, and show that for positive weights, a particular set of Fourier coefficients of cusp forms in this canonical basis cannot simultaneously vanish. [ABSTRACT FROM AUTHOR]