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We study the values of | L ( 1 , F ) | for Hecke–Maass cusp forms F on S L ( n , Z ) ( n ≥ 3 ) of large Langlands parameters. New unconditional results on the extreme values and conditional results on the size range are derived, which determine precisely the order of magnitude of L ( 1 , F ) . In addition, we enhance the new average estimate toward the Ramanujan Conjecture due to Matz and Templier. An application of the Hecke multiplicativity to the Littlewood–Richardson rule for a product of two Schur polynomials is cultivated. [ABSTRACT FROM AUTHOR]

In previous works [18] and [19] , we described algorithms to compute the number field cut out by the mod ℓ representation attached to a modular form of level N = 1 . In this article, we explain how these algorithms can be generalised to forms of higher level N . As an application, we compute the Galois representations attached to a few forms which are supersingular or admit a companion mod ℓ with ℓ = 13 and ℓ = 41 , and we obtain previously unknown number fields of degree ℓ + 1 whose Galois closure has Galois group PGL 2 ( F ℓ ) and a root discriminant that is so small that it beats records for such number fields. Finally, we give a formula to predict the discriminant of the fields obtained by this method, and we use it to find other interesting examples, which are unfortunately out of our computational reach. [ABSTRACT FROM AUTHOR]

Let f be a nonzero cusp form of even integral weight k ⩾ 2 on the Hecke congruence subgroup Γ 0 ( N ) with N squarefree. Suppose that the normalized Fourier coefficients λ f ( n ) of f are real. We prove that the first sign change of λ f ( n ) occurs in the range n ≪ ( k N ) 2 + ε . This improves upon the earlier result of Choie and Kohnen [2] . [ABSTRACT FROM AUTHOR]

With the method of moments and the mollification method, we study the central L -values of GL(2) Maass forms of weight 0 and level 1 and establish a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law. As an application of this result and a formula of Katok–Sarnak, we give a nonvanishing result on the first Fourier coefficients of Maass forms of weight 1 2 and level 4 in the Kohnen plus space. [ABSTRACT FROM AUTHOR]

The denominator formula for the Monster Lie algebra is the product expansion for the modular function J ( z ) − J ( τ ) given in terms of the Hecke system of SL 2 ( Z ) -modular functions j n ( τ ) . It is prominent in Zagier's seminal paper on traces of singular moduli, and in the Duncan–Frenkel work on Moonshine. The formula is equivalent to the description of the generating function for the j n ( z ) as a weight 2 modular form with a pole at z . Although these results rely on the fact that X 0 ( 1 ) has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the X 0 ( N ) modular curves. We use these functions to study divisors of modular forms. [ABSTRACT FROM AUTHOR]

We use the theory of modular forms to give formulas for the number of representations of n by all sextenary quadratic forms with coefficients 1 , 2 , 3 , 6 . We also apply our results to write newforms in S 3 ( Γ 0 ( 24 ) , χ ) in terms of eta quotients. [ABSTRACT FROM AUTHOR]

Assume that α , β are arbitrary arithmetic functions, and that γ is an arithmetic function, which does not correlate with additive characters. By introducing a simple argument, we are able to give a general upper bound for the triple correlation ∑ n ∈ ( X , 2 X ] α ( n ) β ( n − h ) γ ( n + h ) averaging over a short variable h . More precise estimates can be obtained by taking γ to be Fourier coefficients of cusp forms, which improve and streamline the results of Lin and Singh. [ABSTRACT FROM AUTHOR]

Let A f ( 1 , n ) be the normalized Fourier coefficients of a G L ( 3 ) Hecke–Maass cusp form f and let a g ( n ) be the normalized Fourier coefficients of a G L ( 2 ) cusp form g . Let λ ( n ) be either A f ( 1 , n ) or the triple divisor function d 3 ( n ) . It is proved that for any ϵ > 0 , any integer r ≥ 1 and r 5 / 2 X 1 / 4 + 7 δ / 2 ≤ H ≤ X with δ > 0 , 1 H ∑ h ≥ 1 W ( h H ) ∑ n ≥ 1 λ ( n ) a g ( r n + h ) V ( n X ) ≪ X 1 − δ + ϵ , where V and W are smooth compactly supported functions, and the implied constants depend only on the associated forms and ϵ . [ABSTRACT FROM AUTHOR]

We construct certain Casimir operators and study the spectral properties of their resolvents on L 2 ( Γ \ Sp 2 ( R ) ) . We define non-holomorphic multi-variable Poincaré series of exponential type for symplectic groups and continue them analytically in case of genus two for the small weight four using the above resolvents. We apply our results to describe the holomorphic projection to the weight four holomorphic discrete series in terms of Fourier coefficients by using Sturm's operator. This paper is a study of a prototype for symplectic groups and special orthogonal groups. [ABSTRACT FROM AUTHOR]

Suppose k is a positive integer. In this work, we establish formulas for the number of representations of integers by the quadratic forms x 1 2 + ⋯ + x k 2 + m ( x k + 1 2 + ⋯ + x 2 k 2 ) for m ∈ { 2 , 4 } . [ABSTRACT FROM AUTHOR]

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty [24] to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an L -function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL ( 2 ) holomorphic cusp forms, GL ( 2 ) Maass forms, and GL ( 3 ) Maass forms. Building on [13,14] , we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL ( 2 ) holomorphic cusp forms and Maass forms. [ABSTRACT FROM AUTHOR]

Let f and g be weight k holomorphic cusp forms and let S f ( n ) and S g ( n ) denote the sums of their first n Fourier coefficients. Hafner and Ivić [9] proved asymptotics for ∑ n ≤ X | S f ( n ) | 2 and proved that the Classical Conjecture, that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ , holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series D ( s , S f × S g ) = ∑ S f ( n ) S g ( n ) ‾ × n − ( s + k − 1 ) and D ( s , S f × S g ‾ ) = ∑ n S f ( n ) S g ( n ) n − ( s + k − 1 ) . We then prove asymptotics for the smoothed second moment sums ∑ S f ( n ) S g ( n ) ‾ e − n / X , giving a smoothed generalization of [9] . We also attain asymptotics for analogous sums of normalized Fourier coefficients. Our methodology extends to a wide variety of weights and levels, and comparison with [4] indicates very general cancellation between the Rankin–Selberg L -function L ( s , f × g ) and convolution sums of the coefficients of f and g . In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on | S f ( n ) | 2 is true on short intervals, and to prove sign change results on { S f ( n ) } n ∈ N . [ABSTRACT FROM AUTHOR]

Let f be a weight k holomorphic cusp form of level one, and let S f ( n ) denote the sum of the first n Fourier coefficients of f . In analogy with Dirichlet's divisor problem, it is conjectured that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ . Understanding and bounding S f ( X ) has been a very active area of research. The current best bound for individual S f ( X ) is S f ( X ) ≪ X k − 1 2 + 1 3 ( log ⁡ X ) − 0.1185 from Wu [17] . Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for S f ( X ) holds on average over intervals of length X . Jutila [10] improved this result to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 3 4 + ϵ . Building on the results and analytic information about ∑ | S f ( n ) | 2 n − ( s + k − 1 ) from our recent work [9] , we further improve these results to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 2 3 ( log ⁡ X ) 1 6 . [ABSTRACT FROM AUTHOR]

Under a slightly better zero-free region of the corresponding modular L -function, we get a very small bound for the size of first sign change of Hecke eigenvalues for the classical modular forms. When it comes to the prime argument, we derive, for almost all primitive forms, a small bound for the first sign change on prime numbers. [ABSTRACT FROM AUTHOR]

In this paper, we establish a general summation formula for the coefficients of a class of L -functions, without assuming the generalized Ramanujan conjecture. As an application, we consider integral power moments of Fourier coefficients of Hecke–Maass cusp forms. [ABSTRACT FROM AUTHOR]

Zagier defined lifts of weakly holomorphic modular functions to weakly holomorphic modular forms of weight 3/2. Duke and Jenkins extended Zagier-lifts for weakly holomorphic modular forms of negative-integral weights. In this paper, we show that half-integral weight harmonic weak Maass forms can be interpreted as Zagier-lifts of negative-integral weight harmonic weak Maass forms. [ABSTRACT FROM AUTHOR]

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group S L 2 ( Z ) . Denote by λ sym m f ( n ) the n th normalized coefficient of the Dirichlet expansion of the m th symmetric power L -function associated to f . In this paper, we establish some bounds for ∑ n ⩽ x λ sym m f ( n ) , ∑ n ⩽ x λ f ( n m ) , which improve the corresponding results of Lau & Lü [8] . [ABSTRACT FROM AUTHOR]

In recent work of Bringmann, Guerzhoy, and the first author, p -adic modular forms were constructed from mock modular forms. We look at a specific case, starting with the weight −10 mock modular form, and prove explicit congruences. [ABSTRACT FROM AUTHOR]

In this paper, we study the exponential sums involving Fourier coefficients of cusp forms for SL ( 2 , Z ) over primes. We are able to establish some new results. In addition, similar results have been obtained for the coefficients of symmetric-square lifts. [ABSTRACT FROM AUTHOR]

Let π be a unitary automorphic cuspidal representation of GL 2 ( Q A ) with Fourier coefficients λ π ( n ) . Asymptotic expansions of certain sums of λ π ( n ) are proved using known functorial liftings from GL 2 , including symmetric powers, isobaric products, exterior squares, and base change. These asymptotic expansions are manifestation of the underlying functoriality and reflect value distribution of λ π ( n ) on integers, squares, cubes and fourth powers. [ABSTRACT FROM AUTHOR]

Let g ( z ) be a Maass cusp form for SL ( 2 , Z ) , and let λ g ( n ) be the n -th Fourier coefficient of g ( z ) . In this paper we investigate the nonlinear exponential sum ∑ n ∼ X n ≡ l mod q λ g ( n ) e ( α n β ) twisted by Fourier coefficient over the arithmetic progression. We prove an asymptotic formula when β = 1 / 2 and α is close to ± 2 k q , k ∈ Z + . [ABSTRACT FROM AUTHOR]

Let f be a primitive holomorphic or Maass cusp form for the group SL ( 2 , Z ) , and a f ( n ) its n th normalized Fourier coefficient. It is proved that, for any α , β ∈ R , there exists an effective positive constant c such that ∑ n ≤ N Λ ( n ) a f ( n ) e ( α n 2 + β n ) ≪ f N exp ⁡ ( − c log ⁡ N ) . [ABSTRACT FROM AUTHOR]

In this note, we first see that the isomorphism in [26] between vector-valued modular forms and scalar-valued modular forms holds for more general discriminant forms. With this established, we shall prove the Zagier duality for canonical bases for spaces of weakly holomorphic modular forms satisfying some ϵ -condition, thus obtain the complete grids for the duality. Finally we raise a question on certain integrality of the Fourier coefficients of reduced modular forms and provide a partial solution. [ABSTRACT FROM AUTHOR]

Let F be a totally real field and χ a primitive narrow ray class character of F . We prove a formula for the Fourier coefficients of the Siegel Eisenstein series of degree n , weight k and character χ under certain conditions on χ . We also prove the existence of Λ -adic Siegel Eisenstein series of degree n . [ABSTRACT FROM AUTHOR]

In this article, we prove that the resultant of the standard multiplication polynomials A n , B n of an elliptic curve in the form y 2 = x 3 + a x + b is ( 16 Δ ) n 2 ( n 2 − 1 ) 6 , where Δ = − ( 4 a 3 + 27 b 2 ) is the discriminant of the curve. We give an application to good reduction of an associated Lattès map. We also prove a similar result for the discriminant of the largest squarefree factor of B n . [ABSTRACT FROM AUTHOR]

Let ϕ ( z ) denote a holomorphic or Maass cusp form for the full modular group Γ = SL ( 2 , Z ) . And let λ Sym 2 ϕ ( n ) be the n -th coefficient of symmetric square L -function associated with ϕ ( z ) . We establish the uniform upper bound for the summatory function ∑ n ≤ x λ Sym 2 ϕ ( n ) , which improves the results of Ichihara [4] , Lü [10] , Sankaranarayanan [13] and Tang [14] . [ABSTRACT FROM AUTHOR]

Let π be an irreducible unitary cuspidal representation for GL m ( A Q ) , and let L ( s , π ) be the automorphic L -function attached to π , which has a Dirichlet series expression in the half-plane ℜ e s > 1 . When π is self-contragredient, all the coefficients in the Dirichlet series expression are real. In this paper we give non-trivial lower bounds for the number of positive and negative coefficients, respectively. [ABSTRACT FROM AUTHOR]

The Kuznetsov formula provides a deep connection between the spectral theory in hyperbolic Riemann surfaces and some exponential sums of arithmetic nature that has been extremely fruitful in modern number theory. Unfortunately the application of the Kuznetsov formula is by no means easy in practice because it involves oscillatory integral transforms with kernels given by special functions in non-standard ranges. In this paper we introduce a new formulation of the Kuznetsov formula that rules out these complications reducing the integral transforms to something almost as simple as a composition of two Fourier transforms. This formulation admits a surprisingly short and clean proof that does not require any knowledge about special functions, solving in this way another of the disadvantages of the classical approach. Moreover the reversed formula becomes more natural and in the negative case it reduces to a direct application of Fourier inversion. We also show that our approach is more convenient in applications and gives some freedom to play with explicit test functions. [ABSTRACT FROM AUTHOR]

We formulate an analogue of the Ichino–Ikeda conjectures for the Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups. This sharpens the conjectures of Sakellaridis–Venkatesh in the case at hand. [ABSTRACT FROM AUTHOR]

An error in Proposition 5.2 of “Weak Maass–Poincaré series and weight 3/2 mock modular forms” [J. Number Theory 133 (8) (2013) 2567–2587] is corrected. [ABSTRACT FROM AUTHOR]