Our principal aim in the present article is to establish a uniform hybrid bound for individual values on the critical line of Hecke $L$-functions associated with cusp forms over the full modular group. This is rendered in the statement that for $t\ge0$ $$ \eqalignno{H_j(\txt{1\over2}+it)&\ll(κ_j+t)^{1/3+ε},& (1.1)\cr H_{j,k}(\txt{1\over2}+it)&\ll (k+t)^{1/3+ε},&(1.2)\cr} $$ with the common notation to be made precise in the course of discussion. Talks on this and relevant results were delivered by the authors at MF Oberwolfach on September 20, 2004, and at General Assembry of Math. Soc. Japan on March 30, 2005.
We compute the first moment of twisted Hecke $L$-functions of prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.
Shifted convolution sums play a prominent rôle in analytic number theory. We investigate pointwise bounds, mean-square bounds, and average bounds for shifted convolution sums for Hecke eigenforms.
Fix an integer $N$ and a prime $p\nmid N$ where $p\geq 5$. We show that the number of newforms $f$ (up to a scalar multiple) of level $N$ and even weight $k$ such that $\mathcal{T}_p(f)=0$ is bounded independently of $k$, where $\mathcal{T}_p(f)$ is the Hecke operator.
We prove that the fourth moment of holomorphic Hecke cusp forms is bounded provided that the Riemann Hypothesis holds for an appropriate degree 8 L-function. We accomplish this using Watson's formula, which translates the question in hand into a moment problem for L-functions which is amenable to the techniques of Soundararajan and Harper on obtaining sharp bounds for moments of the Riemann zeta function.
We prove an upper bound for the fifth moment of Hecke L-functions associated to holomorphic Hecke cusp forms of full level and weight k in a dyadic interval K < k < 2K, as K tends to infinity. The bound is sharp on Selberg's eigenvalue conjecture.
In this paper, we prove uniform bounds for $\rm GL (3)\times GL(2)$ $L$-functions in the $\rm GL(2)$ spectral aspect and the $t$ aspect by a delta method. More precisely, let $ϕ$ be a Hecke--Maass cusp form for $\rm SL(3,\mathbb{Z})$ and $f$ a Hecke--Maass cusp form for $\rm SL(2,\mathbb{Z})$ with the spectral parameter $t_f$. Then for $t\in\mathbb{R}$ and any $\varepsilon>0$, we have \[ L(1/2+it,ϕ\times f) \ll_{ϕ,\varepsilon} (t_f+|t|)^{27/20+\varepsilon}. \] Moreover, we get subconvexity bounds for $L(1/2+it,ϕ\times f)$ whenever $|t|-t_f \gg (|t|+t_f)^{3/5+\varepsilon}$.
We prove sub-convex bounds on the fourth moment of Hecke-Laplace eigenforms on $S^3$. As a corollary, we get a bound on the sup-norm on an individual eigenform, which constitutes an improvement over what is achievable through employing the Iwaniec-Sarnak amplifier.
In Monstrous moonshine, genus 0 property and the notion of replicability are strongly connected. With regards to recent developments of moonshine, we investigate a higher genus generalization of replicability for a general automorphic form. Specifically, we extend the definitions of replicates and a Hecke operator to harmonic Maass functions on modular curves of higher genera to obtain number theoretic generalizations of important results in Monstrous moonshine. Furthermore, we show the utility of the extended notions in yielding uniform proofs for numerous arithmetic properties of Fourier coe…
Let πbe a unitary cuspidal automorphic representation for GL(n) over a number field. We establish upper bounds on the number of Hecke eigenvalues of πequal to a fixed complex number. For GL(2), we also determine upper bounds on the number of Hecke eigenvalues with absolute value equal to a fixed number γ; in the case γ=0, this answers a question of Serre. These bounds are then improved upon by restricting to non-dihedral representations. Finally, we obtain analogous bounds for a family of cuspidal automorphic representations for GL(3).
We derive an explicit upper bound for the number of systems of Hecke eigenvalues coming from Siegel modular forms (mod p) of dimension g and level N relatively prime to p. In the special case of elliptic modular forms (g=1), our result agrees with recent work of G. Herrick.
We obtain an asymptotic formula for the fourth moment of central values of a family of quadratic Hecke $L$-functions in the Gaussian field under the generalized Riemann hypothesis (GRH). We also establish lower bounds unconditionally and upper bounds under GRH for higher moments of the same family.
We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_2(Γ_0(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_0(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. For prime levels $N\equiv 7\mod 8$ our method yields an unconditional bound of $d\ge\log…
Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of $G:=H(\mathbb{R})$, and $Γ\subset H(\mathbb{Q})$ a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke-Maass forms on the locally symmetric space $Γ\backslash G/K$ to corresponding bounds on the arithmetic quotient $Γ\backslash G$ for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial $K$-types, yielding subconvex bounds for new classes of automorphic representations, a…
We give improved uniform estimates for the rate of convergence to Plancherel measure of Hecke eigenvalues of holomorphic forms of weight 2 and level N. These are applied to determine the sharp cutoff for the non-backtracking random walk on arithmetic Ramanujan graphs and to Serre's problem of bounding the multiplicities of modular forms whose coefficients lie in number fields of degree d.
We revisit a theorem of Ram Murty about the number of initial Fourier coefficients that two cuspidal eigenforms of different weights can have in common. We prove an explicit upper bound on this number, and give better conditional and unconditional asymptotic upper bounds. Finally, we describe a numerical experiment testing the sharpness of the upper bound in the case of forms of level one.
This work contains a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a SL(3,Z) Hecke-Maass cusp form restricted to a compact set.
We give the best possible lower bounds in order of magnitude for the number of positive and negative Hecke eigenvalues. This improves upon a recent work of Kohnen, Lau & Shparlinski. Also, we study an analogous problem for short intervals.
We prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon of Deuring and Heilbronn for Hecke $L$-functions. In forthcoming work of the second author, these estimates will be used to establish explicit bounds on the least norm of a prime ideal in a congruence class group and improve upon existing explicit bounds for the least norm of a prime ideal in the Chebotarev density theorem.
In this article, the authors give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. They also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus were dealt in an earlier work of Choie, the first author and Kohnen.
We prove a new upper bound for the $L^4$-norm of a holomorphic Hecke newform of large fixed weight and prime level $q\to \infty$. This is achieved by proving a sharp mean value estimate for a related $L$-function on GL(6)
A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the l-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level l and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.
We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $Γ\backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $Γ<G$ is a lattice associated to a division algebra over $\mathbb{Q}$ of prime degree $d$. More generally, we introduce a new method of proving positive entropy of quantum limits, which applies to higher-rank groups. The result on AQUE is obtained by combining this with a measure-rigidity theorem due to Einsiedler-Katok, following a strategy first pioneer…
We construct an Euler system of generalized Heegner cycles to bound the Selmer group associated to a modular form and an algebraic Hecke character. The main argument is based on Kolyvagin's machinery explained by Gross while the key object of the Euler system, the generalized Heegner cycles, were first considered by Bertolini, Darmon and Prasanna.
Let F in S_k(Sp(2g, Z)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues mu_F(n). Suppose that the associated automorphic representation pi_F is locally tempered everywhere. For each c>0 we consider the set of primes p for which |mu_F(p)| >= c and we provide an explicit upper bound on the density of this set. In the case g=2, we also provide an explicit upper bound on the density of the set of primes p for which mu_F(p) >= c.
We bound Selmer groups attached to Grossencharacters of CM elliptic curves by the appropriate $L$-value. Our method is to use Kato's explicit reciprocity law and the Main Conjecture as proved by Rubin. These results are then used together with work of Han and fundamental results of Nekovar to bound the torsion in the image of the $p$-adic cycle class map for self products of CM elliptic curves. Our results are valid for Chow groups of arbitrary codimension for almost all $p$.
Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary number field, $(p)=\mathfrak{p}\mathfrak{p}'$ be a split odd prime and $ψ$ be a Hecke character of conductor $\mathfrak{p}$. Let $L(s,ψ)$ be the associated $L$-function. We prove the Burgess bound in $t$-aspect and a hybrid bound in conductor aspect, \begin{equation*} L(1/2+it,ψ)\ll_{D,\varepsilon} (1+|t|)^{3/8+\varepsilon}p^{1/8} \end{equation*} for $p\ll t$. In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice struct…
We prove a non-vanishing result for families of $\GL_n\times\GL_n$ Rankin-Selberg $L$-functions in the critical strip, as one factor runs over twists by Hecke characters. As an application, we simplify the proof, due to Luo, Rudnick, and Sarnak, of the best known bounds towards the Generalized Ramanujan Conjecture at the infinite places for cusp forms on $\GL_n$. A key ingredient is the regularization of the units in residue classes by the use of an Arakelov ray class group.
In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $Γ$ be a congruence subgroup of $\SL_2(\Om_K)$ and $M_{(k_1,k_2)}(Γ)$ the space of Hilbert modular forms of weight $(k_1,k_2)$ for $Γ$. The first main result is an algorithm to construct a finite set $S$, depending on $K$, $Γ$ and $(k_1,k_2)$, such that if the Fourier expansion coefficients of a form $G \in M_{(k_1,k_2)}(Γ)$ vanish on the set $S$, then $G$ is the zero form. The second result corresponds to th…
We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck constant N=1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are << N^epsilon for all…
We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.
We use a `trivial' delta method to prove the Weyl bound in $t$-aspect for the $\rm L$-function of a holomorphic or a Hecke-Maass cusp form of arbitrary level and nebentypus. In particular, this extends the results of Meurman and Jutila for the $t$-aspect Weyl bound, and the recent result of Booker, Milinovich and Ng to Hecke cusp forms of arbitrary level and nebentypus.
We prove new bounds for weighted mean values of sums involving Fourier coefficients of cusp forms that are automorphic with respect to a Hecke congruence subgroup Γ=Γ_0(q) of the group SL(2,Z[i]), and correspond to exceptional eigenvalues of the Laplace operator on the space L^2(Γ\SL(2,C)/SU(2)). These results are, for certain applications, an effective substitute for the generalised Selberg eigenvalue conjecture. We give a proof of one such application, which is an upper bound for a sum of generalised Kloosterman sums (of significance in the study of certain mean values of Hecke zeta-function…
Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it \right) \ll_{f, ε} \left( 1 + |t|\right)^{1/3 + ε}, \] for any $ε> 0.$
We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.
Let $f $ be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it, f \right) \ll_{f, ε} \left( 2 + |t|\right)^{1/3 + ε}, \] for any $ε> 0.$
Let $F$ be a $G L(3)$ Hecke-Maass cusp form of prime level $P_1$ and let $f$ be a $G L(2)$ Hecke-Maass cuspform of prime level $P_2$. In this article, we will prove a subconvex bound for the $G L(3) \times G L(2)$ Rankin-Selberg $L$-function $L(s,F\times f)$ in the level aspect for certain ranges of the parameters $P_1$ and $P_2$.
Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The main new ingredient, possibly of independent interest, is a large sieve for n-th order characters. As further applications of this tool, we derive several results concerning L(s,χ) for n-th order Hecke characters: an estimate of the second moment on the critical line, a non-vanishing result at the central point, and a zero-density theorem.
For a cuspidal Hecke eigenform $F$ for $Sp_n(Z)$ and a Dirichlet character $χ$ let $L(s,F,χ,St)$ be the standard $L$-function of $F$ twisted by $χ$. Boecherer showed the boundedness of denominators of the algebraic part of $L(m,F,χ,St)$ at a critical point $m$ when $χ$ varies. In this paper, we give a refined version of his result We also prove a similar result for the products of Hecke $L$ functions of primitive forms for $SL_2(Z)$.
We use recently obtained bounds for sums of Kloosterman sums to bound the sum $\sum_{-D\leq d\leq D} \int_{-D}^D |ζ(1/2+it,λ^d)|^4| \sum_{0<|μ|^2\leq M} A(μ)λ^d((μ)) |μ|^{-2it}|^2 {\rm d}t$, where $λ^d$ is the groessencharacter satisfying $λ^d((α)) = λ^d(α{\Bbb Z}[i]) = (α/|α|)^{4d}$, for $0\neqα\in{\Bbb Z}[i]$, and $ζ(s,λ^d)$ is the Hecke zeta function that satisfies $ζ(s,λ^d) =(1/4)\sum_{0\neqα\in{\Bbb Z}[i]} λ^d((α)) |α|^{-2s}$ for $\Re(s)>1$, while the numbers $D,M\in(0,\infty)$ and function $A:{\Bbb Z}[i]-\{0\}\rightarrow{\Bbb C}$ are arbitrary (though it is only in respect of cases in wh…
Let $π$ be a Hecke-Maass cusp form for $\mathrm{SL(3, \mathbb{Z})}$ and $f$ be a holomorphic cusp form for $\mathrm{SL(2,\mathbb{Z})}$ of weight $k$ or a Hecke-Maass cusp form corresponding to the Laplacian eigenvalue $1/4+k^2$, $k\geq 1$, for $\mathrm{SL(2,\mathbb{Z})}$. In this paper, we prove the following subconvexity bound \begin{align*} L\left({1}/{2},\ π\times f\right) \ll_{π,\,ε} k^{\frac{3}{2}-\frac{1}{51}+ε}. \end{align*}
In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K/\mathbb{Q}$ (real or imaginary). Our proof relies on relating the class group $L$-functions to Eisenstein series evaluated at Heegner points using formulas due to Hecke. The main technical contribution is the following uniform sup norm bound for Eisenstein series $E(z,1/2+it)\ll_\varepsilon y^{1/2} (|t|+1)^{1/3+\varepsilon}, y\gg 1$, extending work of Blomer and Titchmarsh. Finally, we propose a uniform version of the sup norm conjecture for Eisenstein series.
We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight $k$ with respect to a finite index subgroup $Γ$ of $\mathrm{SL}_2(\mathbb{Z})$. We also prove corresponding bounds for the supremum over a compact set. We achieve this by extending to a sum over an orthonormal basis $\sum_j y^k|f_j(z)|^2$ and analysing this sum by means of a Bergman kernel and the Fourier coefficients of Poincaré series. As such our results are valid without any assumption that the forms are Hecke eigenfunctions. Under some weak assumptions we further prove the right order of magnitude of $…
We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(\sqrt{-q}), conditionally subject to the existence of many small (subnormal) gaps between zeros of the L-function associated with a character of the class group associated with this field. In particular, we prove that if the gap between consecutive zeros of the L-function is somewhat smaller than the average for sufficiently many pairs of zeros on the critical line, then h >> \sqrt q (log q)^{-A} for some constant A > 0. For the trivial character, the L-function is the Dedekind zeta-functio…
