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We denote by S l ( N , ψ ) (resp. S k ( N , χ ) ) the space of cusp forms of weight l (resp. k) for Γ 0 ( N ) with nebentypus ψ (resp. χ). We give an arithmetic expression for the critical values of the Rankin–Selberg L-series D ( s , f ⊗ g ) associated with a pair ( f , g ) of modular forms for S L 2 ( Z ) . It is also shown that a cusp form g ∈ S l ( N , ψ ) is uniquely determined by certain critical values of the family of Rankin–Selberg L-series D ( s , f ⊗ g ) , where f runs over a fixed orthogonal basis of S k ( N , χ ) .

Iwaniec, Luo, and Sarnak proposed Hypothesis S and its generalization which predicts non-trivial bounds for a smooth sum of the product of an arithmetic sequence { a n } and a fractional exponential function. When a n is the Fourier coefficient λ f ( n ) of a fixed holomorphic cusp form f, however, a resonance phenomenon prohibits any improvement of the bound beyond a barrier. It is believed that this resonance barrier could be overcome when the weight k of f tends to infinity. The present paper is a first step toward this goal by proving non-trivial bounds for this sum when k and the summation length X both tend to infinity. No such non-trivial bounds are previously known if the form f is allowed to move. Similar bounds are also proved for linear phases and for Maass forms. The main technology is improved large sieve inequalities over a short interval.

We prove that for a Hecke cuspform f ∈ S k ( Γ 0 ( N ) , χ ) and a prime l > max ⁡ { k , 6 } such that l ∤ N , there exists an infinite family { k r } r ≥ 1 ⊆ Z such that for each k r , there is a cusp form f k r ∈ S k r ( Γ 0 ( N ) , χ ) such that the Deligne representation ρ f k r , l is a crystalline and potentially diagonalizable lift of ρ ‾ f , l . When f is l-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to Bockle and a local dimension result by Kisin. We discuss the motivation and tentative future applications of our result in ongoing research on the automorphy of GL 2 n -type representations in the higher level case.

Let $$k\ge 2$$ be an even integer, and let M be a positive integer. We prove a hybrid subconvexity bound for $$L(1/2,\text {sym}^2 f\otimes \chi )$$ with the associated parameters k, M explicitly determined, where f is a primitive holomorphic cusp form of weight k and level 1, and $$\chi $$ is a primitive Dirichlet character modulo M satisfying $$1< k\le M^{1/26}$$ .

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 $${{\cal S}_k}(X): = \sum\limits_{n < X} {\sum\limits_{n = {n_1},{n_2}, \ldots ,{n_k}} {{\lambda _f}({n_1}){\lambda _f}({n_2}) \ldots {\lambda _f}({n_k})}}$$ and show that $${{\cal S}_k}(X){\ll _{f,\varepsilon }}{X^{1 - 3/(2(k + 3)) + \varepsilon}}$$ for every k ⩾ 1 and e > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lu (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 $${{\cal S}_k}(X)$$ , where the sum involves restricted coefficients of some suitable half integral weight modular forms.

We examine an arithmetical function defined by recursion relations on the sequence {f(pk)}k∈ℕ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form.

Recently Luo and Zhou showed that for a fixed GL(2) Hecke-Maass cusp form, the natural density of primes at which the Satake parameters fail the Ramanujan Conjecture does not exceed 1/35. In this short note, we investigate the GL(3) case and obtain two similar (conditional) results.

We prove that given any $$\epsilon >0$$ , a non-zero adelic Hilbert cusp form $${\mathbf {f}}$$ of weight $$k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n$$ and square-free level $$\mathfrak {n}$$ with Fourier coefficients $$C_{{\mathbf {f}}}(\mathfrak {m})$$ , there exists a square-free integral ideal $$\mathfrak {m}$$ with $$N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }$$ such that $$C_{{\mathbf {f}}}(\mathfrak {m})\ne 0$$ . The implied constant depends on $$\epsilon , F$$ .

We consider the Fourier expansion of a Hecke (resp. Hecke–Maaß) cusp form of general level N at the various cusps of $$\Gamma _{0}(N)\backslash \mathbb {H}$$ Γ 0 ( N ) \ H . We explain how to compute these coefficients via the local theory of p-adic Whittaker functions and establish a classical Voronoï summation formula allowing an arbitrary additive twist. Our discussion has applications to bounding sums of Fourier coefficients and understanding the (generalised) Atkin–Lehner relations.

In this paper we prove that the first Fourier-Jacobi coefficient of a non-zero Siegel cusp form and a Hecke eigenform of degree 2, weight k, for S p 4 ( Z ) is not identically zero.

We obtain certain estimates for averages of shifted convolution sums involving the Fourier coefficients of a normalized Hecke–Maass eigenform and holomorphic cusp form.

We study the argument of L(s, f) associated with holomorphic cusp form in both weight aspect and t-aspect. We prove that for −1 ≤ σ ≤ 2 and t ≥ 4, argL(s, f) ≪ log kt, where s = σ + it. Assuming the generalized Riemann hypothesis (GRH), we have arg L(s, f) ≪ log kt/log log kt for σ ≥ 1/2 and t ≥ 4.

Let $\mathfrak{q}>2$ be a prime number, $\chi$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$ L(1/2,f\otimes \chi)\ll \mathfrak{q}^{1/2-1/12+\varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifying trivial delta method developed in [1]., Comment: 12 pages

For a positive integer [Formula: see text], we say that [Formula: see text] is a Weierstrass point on the modular curve [Formula: see text] if there is a non-zero cusp form of weight [Formula: see text] on [Formula: see text] which vanishes at [Formula: see text] to order greater than the genus of [Formula: see text]. If [Formula: see text] is a prime with [Formula: see text], Ogg proved that [Formula: see text] is not a Weierstrass point on [Formula: see text] if the genus of [Formula: see text] is [Formula: see text]. We prove a similar result for even weights [Formula: see text]. We also study the space of weight [Formula: see text] cusp forms on [Formula: see text] vanishing to order greater than the dimension.

The degeneracies of single-centered dyonic $\frac14$-BPS black holes (BH) in Type II string theory on K3$\times T^2$ are known to be coefficients of certain mock Jacobi forms arising from the Igusa cusp form $\Phi_{10}$. In this paper we present an exact analytic formula for these BH degeneracies purely in terms of the degeneracies of the perturbative $\frac12$-BPS states of the theory. We use the fact that the degeneracies are completely controlled by the polar coefficients of the mock Jacobi forms, using the Hardy-Ramanujan-Rademacher circle method. Here we present a simple formula for these polar coefficients as a quadratic function of the $\frac12$-BPS degeneracies. We arrive at the formula by using the physical interpretation of polar coefficients as negative discriminant states, and then making use of previous results in the literature to track the decay of such states into pairs of $\frac12$-BPS states in the moduli space. Although there are an infinite number of such decays, we show that only a finite number of them contribute to the formula. The phenomenon of BH bound state metamorphosis (BSM) plays a crucial role in our analysis. We show that the dyonic BSM orbits with $U$-duality invariant $\Delta, Comment: 58 pages, 9 figures, 3 tables