Cusp Forms Research Articles

The sup-norm problem for automorphic cusp forms of 𝑃𝐺𝐿(𝑛,ℤ[𝕚

Let ϕ \phi be an L 2 L^2 -normalized Hecke–Maaß cusp form on the locally symmetric space X ≔ P G L n ( Z [ i ] ) ∖ P G L n ( C ) / P U n X≔\mathrm {PGL}_n(\mathbb {Z}[i])\backslash \mathrm {PGL}_n(\mathbb {C}) / \mathrm {PU}_n . If Ω \Omega is a compact subset of X X , then we prove the bound ‖ ϕ | Ω ‖ ∞ ≪ Ω λ ϕ n ( n − 1 ) / 4 − δ \|\phi |_{\Omega }\|_{\infty }\ll _{\Omega } \lambda _{\phi }^{n(n-1)/4-\delta } for some δ > 0 \delta >0 depending only on n n , where λ ϕ \lambda _{\phi } is the Laplace eigenvalue of ϕ \phi .

Ergodic averages with the Hecke eigenvalue square weights and the Piltz divisor function weights

In this paper, we prove that the Hecke eigenvalue square for a holomorphic cusp form and the Piltz divisor functions are good weighting functions for the pointwise ergodic theorem. Additionally, we prove similar results for various other arithmetical functions in the last section.

GL(2) Weyl bound via a multiplicative character delta method

We use a trivial delta method with multiplicative characters for congruence detection to prove the Weyl bound for GL(2) in the [Formula: see text]-aspect for a holomorphic or Hecke–Maass cusp form of arbitrary level and nebentypus. This parallels the work of Aggarwal [2] in 2018, with the difference being the multiplicative character has a more natural connection to the twisted [Formula: see text]-function. This provides another viewpoint to understand and explore the trivial and other delta methods.

Hodge-Elliptic Genera, K3 Surfaces and Enumerative Geometry

K3 surfaces play a prominent role in string theory and algebraic geometry. The properties of their enumerative invariants have important consequences in black hole physics and in number theory. To a K3 surface, string theory associates an Elliptic genus, a certain partition function directly related to the theory of Jacobi modular forms. A multiplicative lift of the Elliptic genus produces another modular object, an Igusa cusp form, which is the generating function of BPS invariants of K3×E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{K3} \ imes E$$\\end{document}. In this note, we will discuss a refinement of this chain of ideas. The Elliptic genus can be generalized to the so-called Hodge-Elliptic genus which is then related to the counting of refined BPS states of K3×E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{K3} \ imes E$$\\end{document}. We show how such BPS invariants can be computed explicitly in terms of different versions of the Hodge-Elliptic genus, sometimes in closed form, and discuss some generalizations.

BCOV cusp forms of lattice polarized K3 surfaces

We introduce the BCOV formula for the lattice polarized K3 surfaces. We find that it yields cusp forms expressed by certain eta products for many families of rank 19 lattice polarized K3 surfaces over P1. Moreover, for Clingher-Doran's family of U⊕E8(−1)⊕E7(−1)-polarized K3 surfaces, we obtain the Igusa cusp forms χ10 and χ12 from the formula. Inspired by the arithmetic properties of mirror maps studied by Lian-Yau, we also derive the K3 differential operators for all the genus zero groups of type Γ0(n)+.

Scalar‐valued depth two Eichler–Shimura integrals of cusp forms

AbstractGiven cusp forms and of integral weight , the depth two holomorphic iterated Eichler–Shimura integral is defined by , where is the Eichler integral of and are formal variables. We provide an explicit vector‐valued modular form whose top components are given by . We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by , that can be seen as a higher depth generalization of the scalar‐valued Eichler integral 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 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 . This allows for effective computation of .

The Mahler measure of x + 1/x + y + 1/y + 4 ± 42 and Beilinson’s conjecture

In this note we study the Mahler measures of reciprocal polynomials [Formula: see text] for [Formula: see text]. We prove identities relating the Mahler measure of [Formula: see text] to special values of the [Formula: see text]-functions of weight 2 cusp forms of level 64. We also express the Beilinson regulator which is a [Formula: see text] determinant of Mahler measures as special value of [Formula: see text]-function of the elliptic curve defined by [Formula: see text].

On the Rankin–Selberg problem, II

Abstract In this paper, we improve our bounds on the Rankin–Selberg problem. That is, we obtain a smaller error term of the second moment of Fourier coefficients of a GL(2) cusp form (both holomorphic and Maass).

Simultaneous non-vanishing of central values of GL(2)×GL(3) and GL(3) L-functions

We study simultaneous non-vanishing of L(12,Ϝ) and L(12,g⊗Ϝ), when Ϝ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for SL(3,Z) and g is a fixed SL(2,Z) holomorphic Hecke cusp form of weight k≡0(mod4).

On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift

On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift

Fourier coefficients of automorphic L-functions

We prove two unconditional results on Fourier coefficients of modular forms: Let [Formula: see text] be a Hecke eigen cusp form of weight [Formula: see text] and level [Formula: see text] ([Formula: see text] square free), and [Formula: see text] be the normalized Hecke eigenvalues. Let [Formula: see text] be the least prime [Formula: see text] such that [Formula: see text]. Then we show that in a family of modular forms of weight [Formula: see text] and level [Formula: see text], except for a density zero set, [Formula: see text] for some [Formula: see text]. Second, we show that outside a density zero set, a modular form is determined by Fourier coefficients up to [Formula: see text] for some [Formula: see text].

An approximation formula for the shifted cubic moment of automorphic L-functions in the weight aspect

Abstract Consider the family of automorphic L-functions associated with primitive cusp forms of level one, ordered by weight k. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of shifted L-values over this family which relates it to the fourth moment of the Riemann zeta function. More precisely, the formula includes a conjectural main term, the fourth moment of the Riemann zeta function and error terms of size smaller than that predicted by the recipe conjectures.

Paramodular groups and theta series

For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling or pullback of Eisenstein series method. For this we give a new geometric proof of Garrett’s double coset decomposition which works in our more general situation.

Averages of long Dirichlet polynomials with modular coefficients

AbstractWe study the moments ofL‐functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of alongDirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindelöf Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.

Sign changes of coefficients of half-integral weight Hecke eigenforms

Let f \mathfrak {f} be a cusp form of half-weight k + 1 / 2 k+1/2 and at most quadratic nebentype character whose Fourier coefficients are denoted as a f ( n ) \mathfrak {a}_\mathfrak {f}(n) . We study sign changes of the family { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} where t t is a square-free number and p p runs through the prime numbers. By taking use of the relationship between the cusp form f \mathfrak {f} of half-integral weight and the Shimura lift f t f_t , we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence { a f ( t p 2 ) } p ∈ P \{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}} .

Higher level cusp forms on exceptional group of type E7

By using new techniques with the degenerate Whittaker functions developed by Ikeda and Yamana, we construct a higher-level cusp form on GE7,3 (exceptional similitude group of type E7), called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercuspidal local components. This generalizes our earlier results on level-1 forms. But there is a new phenomenon in higher levels; we can handle cusp forms with nontrivial central characters. We also compute the degree-133 adjoint L-function of the Ikeda type lift.

On a family of Siegel Poincaré series

Let [Formula: see text] be a congruence subgroup of [Formula: see text]. Using Poincaré series of [Formula: see text]-finite matrix coefficients of integrable discrete series representations of [Formula: see text], we construct a spanning set for the space [Formula: see text] of Siegel cusp forms of weight [Formula: see text]. We prove the non-vanishing of certain elements of this spanning set using Muić’s integral non-vanishing criterion for Poincaré series on locally compact Hausdorff groups. Moreover, using the representation theory of [Formula: see text], we study the Petersson inner products of corresponding cuspidal automorphic forms, thereby recovering a representation-theoretic proof of some well-known results on the reproducing kernel function of [Formula: see text]. Our results are obtained by generalizing representation-theoretic methods developed by Muić in his work on holomorphic cusp forms on the upper half-plane to the setting of Siegel cusp forms of a higher degree.

Equivalent criterion for the grand Riemann hypothesis associated with Maass cusp forms

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated with the primitive Maass cusp form over $\Gamma _0(N)$.

Quantum ergodicity for Eisenstein series on hyperbolic surfaces of large genus

We give a quantitative estimate for the quantum mean absolute deviation on hyperbolic surfaces of finite area in terms of geometric parameters such as the genus, number of cusps and injectivity radius. It implies a delocalisation result of quantum ergodicity type for eigenfunctions of the Laplacian on hyperbolic surfaces of finite area that Benjamini-Schramm converge to the hyperbolic plane. We show that this is generic for Mirzakhani’s model of random surfaces chosen uniformly with respect to the Weil-Petersson volume. Depending on the particular sequence of surfaces considered this gives a result of delocalisation of most cusp forms or of Eisenstein series.

The cubic moment of Hecke–Maass cusp forms and moments of L-functions

The cubic moment of Hecke–Maass cusp forms and moments of L-functions