In this article, we derive a sub convexity estimate of Hecke eigen cusp forms associated to certain cocompact arithmetic subgroups of SL(2,R). The main result can be considered as the holomorphic version of the estimate of Hecke eigen Maass forms, derived in a famous paper of Iwaniec and Sarnak. A stronger estimate was derived by Khayutin and Steiner in arXiv:2009.07194. However, techniques used in both the papers are very different.
Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.
We correct a mistake in the paper "On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields " by A. Borel, J.-P. Labesse, J. Schwermer that appeared in Compositio Mathematica, 102 (1996), 1-40.
In this paper, we describe the asymptotic distribution of Hecke eigenvalues in the Laplace eigenvalue aspect for certain families of Hecke-Maass forms on compact arithmetic quotients. Instead of relying on the trace formula, which was the primary tool in preceding studies on the subject, we use Fourier integral operator methods. This allows us to treat not only spherical, but also non-spherical Hecke-Maass forms with corresponding remainder estimates. Our asymptotic formulas are available for arbitrary simple and connected algebraic groups over number fields with cocompact arithmetic subgroups…
In this article, using the heat kernel approach from \cite{bouche}, we derive sup-norm bounds for cusp forms of integral and half integral weight. Let $Γ\subset \mathrm{PSL}_{2}(\mathbb{R})$ be a cocompact Fuchsian subgroup of first kind. For $k\in\frac{1}{2}\mathbb{Z}$ (or $k\in 2\mathbb{Z}$), let $S^{k}(Γ)$ denote the complex vector space of weight-$k$ cusp forms. Let $\lbrace f_{1},\ldots,f_{j_{k}} \rbrace$ denote an orthonormal basis of $S^{k}(Γ)$. In this article, we show that as $k\rightarrow \infty,$ the sup-norm for $\sum_{i=1}^{j_{k}}y^{k}|f_{i}(z)|^{2}$ is bounded by $O(k)$, where th…
We generalize a result of Sury and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.
Let $Γ\subseteq\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting on the upper half-plane $\mathbb{H}$. Consider the $d$-dimensional space of cusp forms $\mathcal{S}_{k}^Γ$ of weight $2k$ for $Γ$, and let $\{f_{1},\ldots,f_{d}\}$ be an orthonormal basis of $\mathcal{S}_{k}^Γ$ with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity $S_{k}^Γ(z):=\sum_{j=1}^{d}| f_{j}(z)|^{2}\,\mathrm{Im}(z)^{2k}$ is bounded as $O_Γ(k)$ in the cocompact setting, and as $O_Γ(k^{3/2})$ in the cofinite case, where the implied constants depend so…
Let $Γ$ be an arithmetic subgroup of $SU(d,1)$ with cusps, and let $X_Γ$ be the associated locally symmetric space. We prove that if the first inner cohomology group $H^1_!(X_Γ,\mathbb{C})$ is non-zero then the pre-image of $Γ$ in each connected cover of $SU(d,1)$ is residually finite. We also give an example of a such a group $Γ$ for which $H^1_!(X_Γ,\mathbb{C})$ is non-zero.
We prove an arithmetic Hilbert-Samuel type theorem for semi-positive singular hermitian line bundles of finite height. In particular, the theorem applies to the log-singular metrics of Burgos-Kramer-Kühn. Our theorem is thus suitable for application to some non-compact Shimura varieties with their bundles of cusp forms. As an application, we treat the case of Hilbert modular surfaces, establishing an arithmetic analogue of the classical result expressing the dimensions of spaces of cusp forms in terms of special values of Dedekind zeta functions.
We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also discuss a higher-rank analogue.
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 show that if n>5, PU(n-1,1) does not contain a cocompact arithmetic subgroup with the same Euler-Poincare characteristic (in the sense of C.T.C. Wall) as the complex projective space of dimension n-1, and show that if n=5, there are at least four such subgroups, which are in fact torsion-free. This, in particular, leads to examples of a fake projective space of dimension 4. Analogous results for arithmetic fake grassmannians Gr(m,n) with n>3 odd are also obtained.
The cuspidal cohomology groups of arithmetic groups in certain infinite dimensional Modules are computed. As a result we get a simultaneous generalization of the Patterson-Conjecture and the Lewis-Correspondence.
Let $Γ$ be a cocompact, discrete, and irreducible subgroup of $\mathrm{PSL}_{2}(\mathbb{R})^{n}$. Let $ν$ be a unitary character of $Γ$. For $k\in1\slash 2\,\mathbb{Z}$, let $\sknu$ denote the complex vector space of cusp forms of weight-$\tk=\k$ and nebentypus $ν^{2k}$ with respect to $Γ$. We assume that $ω_{X,ν}$, the line bundle of cusp forms of weight-$\tilde{1\slash 2}:=(1\slash 2,\ldots,1\slash2)$ with nebentypus $ν$ over $X$ exists. Let $\lbrace f_{1},\ldots,f_{j_{\tk}} \rbrace$ denote an orthonormal basis of $\sknu$. In this article, we show that as $k\rightarrow \infty$, the sum $\sum…
We prove an equidistribution result for the Satake parameters of Maass cusp forms on $GL_3$ with respect to the $p$-adic Plancherel measure by using an application of the Kuznetsov trace formula. The techniques developed in this paper deal with the removal of arithmetic weight $L(1,F,Ad)^{-1}$ in the Kuznetsov trace formula on $GL_3$.
In this article, for $n\geq 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}\big((n,1),\mathbb{C}\big)$. The main result of the article is the following result. Let $Γ\subset \mathrm{SU}\big((2,1),\mathcal{O}_{K}\big)$ be a torsion-free subgroup of finite index, where $K$ is a totally imaginary field. Let $\mathcal{B}_Γ^{k}$ denote the Bergman kernel associated to the $\mathcal{S}_{k}(Γ)$, complex vector space of weight-$k$ cusp forms with respect to $Γ$. Let $\m…
Mazur, Rubin, and Stein have recently formulated a series of conjectures about statistical properties of modular symbols in order to understand central values of twists of elliptic curve $L$-functions. Two of these conjectures relate to the asymptotic growth of the first and second moments of the modular symbols. We prove these on average by using analytic properties of Eisenstein series twisted by modular symbols. Another of their conjectures predicts the Gaussian distribution of normalized modular symbols ordered according to the size of the denominator of the cusps. We prove this conjecture…
Based on the values of the Weierstrass elliptic function $\wp(z|τ)$ at $z=πτ/2$, $(π+πτ)/{2}, (π+πτ)/{4},(π+2πτ)/{4}$ and the theory of modular forms on the arithmetic group $Γ_0(2)$, we decompose $θ_2^{2n}(τ)$ as sum of Eisenstein series and a cusp forms. Using the recurrence relation of $\wp^{(2n)}(z|τ)$, we provide an algorithm to determine the exact form of these cusp forms.
We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce the following theorem: Given an arithmetically defined cocompact subgroup of SL(2,C), provided the underlying quaternion algebra meets some conditions, there is a decreasing sequence of finite index subgroups of such that the first Betti number grows at least as fast as the squa…
We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from $1$. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold $Γ\backslash X$ by showing the existence of a positive constant $δ(X)>0$ such that squares of lengths of closed geodesics shorter than $δ$ must be pairwise linearly dependent over $\mathbb Q$.
We extend some results of Gun, Murty, and Rath on elliptic modular forms. We take ANY Fuchsian triangle group with a cusp and look at power series expansions in a natural parameter around that cusp. Consider the automorphic forms for such a triangle group whose power series expansions in the natural parameter have algebraic coefficients. We show that the zeros of such forms are either transcendental or are "CM." By "CM," we mean they correspond to abelian varieties with complex multiplication. This result is the first of its kind in the case of non-arithmetic groups.
The purpose of this paper is to give an explicit dimension formula for the spaces of vector valued Siegel cusp forms of degree two with respect to a certain kind of arithmetic subgroups of the non-split Q-forms of Sp(2,R). We obtain our result by using Hashimoto and Ibukiyama's results in [HI80],[HI83] and Wakatsuki's formula in [Wak]. Our result is a generalization of formulae in [Has84,Theorem 4.1] and [Wak,Theorem 6.1].
We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution which comes from the asymptotic mixed distribution of Salié sums.
Let $\bf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let $Γ=\bf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\ge 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to $Γ$ from a Hecke cusp form in $S_{2k-8}(SL_2(\mathbb{Z}))$. This lift is an analogue of Ikeda's construction.
We discuss certain Eisenstein series on arithmetic quotients of loop groups, G^, which are associated to cusp forms on finite-dimensional groups associated with maximal parabolics of G^.
We show that, in a restricted range, the divisor function of integers in residue classes modulo a prime follows a Gaussian distribution, and a similar result for Hecke eigenvalues of classical holomorphic cusp forms. Furthermore, we obtain the joint distribution of these arithmetic functions in two related residue classes. These results follow from asymptotic evaluations of the relevant moments, and depend crucially on results on the independence of monodromy groups related to products of Kloosterman sums.
Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. In this note we compare the signs of the…
We classify Siegel modular cusp forms of weight two for the paramodular group K(p) for primes p< 600. We find that weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian varieties defined over the rationals of conductor p. The arithmetic classification is in a companion article by A. Brumer and K. Kramer. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, is a partial extension to degree 2 of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the…
In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form $\mathfrak{f}$ with half-integral weight:\begin{itemize}\item[--]The first negative coefficient of the sequence $\{\mathfrak{a}\_{\mathfrak{f}}(tn^2)\}\_{n\in \N}$,\item[--]The number of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ of same signs,\item[--]Non-vanishing of coefficients $\mathfrak{a}\_{\mathfrak{f}}(tn^2)$ in short intervals and in arithmetic progressions,\end{itemize}where $\mathfrak{a}\_{\mathfrak{f}}(n)$ is the $n$-th Fourier coefficient of $\mathfrak{f}$ and $t$ is a square-f…
Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ normalized Fourier coefficient. Assuming that all the coefficients $a(n)$ are real, we study the sign of $a(n)$ when $n$ runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers $n\le x$ such that $a(n)>n^{-α}$ where $x$ and $α$ are positive and $f$ is not necessarily a Hecke eigenform.
Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler--Shimura isomorphism and contain information about automorphic $L$-functions. In this paper we prove that central values of additive twists of the $L$-function associated to a holomorphic cusp form $f$ of even weight $k$ are asymptotically normally distributed. This generalizes (to $k\geq 4$) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore we give as an application an asymptotic formula for the averages of certain 'wide' fam…
This is the second of two coupled papers estimating the mean values of multiplicative functions, of unknown support, on arithmetic progressions with large differences. Applications are made to the study of primes in arithmetic progression and to the Fourier coefficients of automorphic cusp forms.
In 1997 the author found a criterion for the Riemann hypothesis for the Riemann zeta function, involving the nonnegativity of certain coefficients associated with the Riemann zeta function. In 1999 Bombieri and Lagarias obtained an arithmetic formula for these coefficients using the ``explicit formula'' of prime number theory. In this paper, the author obtains an arithmetic formula for corresponding coefficients associated with the Euler product of Hecke polynomials, which is essentially a product of L-functions attached to weight 2 cusp forms (both newforms and oldforms) over Hecke congruence…
Let $f$ be a primitive Maass cusp form for a congruence subgroup $Γ_0(D) \subset $ SL($2,\mathbb{Z}$) and $λ_f(n)$ its $n$-th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many $λ_f(n)$ one can often solve for the level $D$, and in some cases, estimate the Laplace eigenvalue to arbitrarily high precision. This is done by analyzing the resonance and rapid decay of smoothly weighted sums of $λ_f(n)e(αn^β)$ for $X \leq n \leq 2X$ and any choice of $α\in \mathbb{R}$, and $β>0$. The methods include the Voronoi summation formula, asymptotic expansions of Bessel …
Let $π$ traverse a sequence of cuspidal automorphic representations of GL(2) with large prime level, unramified central character and bounded infinity type. For G either of the groups GL(1) or PGL(2), let H(G) denote the assertion that subconvexity holds for G-twists of the adjoint $L$-function of $π$, with polynomial dependence upon the conductor of the twist. We show that H(GL(1)) implies H(PGL(2)). In geometric terms, H(PGL(2)) corresponds roughly to an instance of arithmetic quantum unique ergodicity with a power savings in the error term, H(GL(1)) to the special case in which the relevant…
Let $K$ be a global field of characteristic $p>0$. We study the cohomology of arithmetic subgroups $Γ$ of $SL_{n+1}(K)$ (with respect to a fixed place of $K$), under the hypothesis that these groups have no $p'$-torsion (any arithmetic group possesses a normal subgroup of finite index without $p'$-torsion). We define the cohomology of $Γ$ with compact supports and values in ${\Bbb Z}[1/p]$, and we relate it to spaces of harmonic cocycles, also with compact supports (§3). We give a description of the locus of these supports, in particular by introducing a notion of cusp in dimension $n\geq 1$ (…
We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coeff…
Let $M = \mathbb{B}^2 / Γ$ be a smooth ball quotient of finite volume with first betti number $b_1(M)$ and let $\mathcal{E}(M) \ge 0$ be the number of cusps (i.e., topological ends) of $M$. We study the growth rates that are possible in towers of finite-sheeted coverings of $M$. In particular, $b_1$ and $\mathcal{E}$ have little to do with one another, in contrast with the well-understood cases of hyperbolic $2$- and $3$-manifolds. We also discuss growth of $b_1$ for congruence arithmetic lattices acting on $\mathbb{B}^2$ and $\mathbb{B}^3$. Along the way, we provide an explicit example of a l…
We prove an equidistribution statement for the Satake parameters of the local representations attached to Siegel cusp forms of degree $2$ of increasing level and weight, counted with a certain arithmetic weight. We then apply this to compute the symmetry type of a similarly weighted distribution of the low-lying zeros of $L$-functions attached to these cusp forms.
Let $\G\subset \mathrm{SL}_{2}(\R)$ be a cofinite Fuchsian subgroup, and let $i\infty$ be a cusp of $\G$. For $k\in\Z_{\geq 0}$, let $\Sk$ denote the complex vector space of cusp forms of weight-$k$, with respect to the Fuchsian subgroup $\G$. Let $f\in\Sk$ be a cusp form of weight-$k$, which is normalized, with respect to the Petersson inner-product on $\Sk$. For any $n\in\Z_{\geq 1}$, let $a_{n}$ denote the $n$-th Fourier coefficient of $f$ at $i\infty$. Then, for any $k\in\Z_{\geq 5}$, we show that \begin{align*} \big|a_{n}\big|=O_{f,\G}\big(n^{\frac{k-1}{2}+\frac{2}{k}}\big), \end{align*} …
There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group PSL(2,Z) including the following statements: The ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weight four and six; the smallest weight cusp form Delta has weight twelve and can be written as a polynomial in E4 and E6; and the Hauptmodul j can be written as a multiple of E4 cubed divided by Delta. The goal of the present article is to seek generalizations of these results to some other genus zero arithmetic groups, namely those generate…
We extend the work of Ash and Stevens [Ash-Stevens 97] on p-adic analytic families of p-ordinary arithmetic cohomology classes for GL(N,Q) by introducing and investigating the concept of p-adic rigidity of arithmetic Hecke eigenclasses. An arithmetic eigenclass is said to be "rigid" if (modulo twisting) it does not admit a nontrivial p-adic deformation containing a Zariski dense set of arithmetic specializations. This paper develops tools for explicit investigation into the structure of eigenvarieties for GL(N). We use these tools to prove that known examples of non-sefldual cohomological cusp…
The purpose of this semi-expository article is to give another proof of a classical theorem of Shimura on the critical values of the standard L-function attached to a Hilbert modular form. Our proof is along the lines of previous work of Harder and Hida (independently). What is different is an organizational principle based on the period relations proved by Raghuram and Shahidi for periods attached to regular algebraic cuspidal automorphic representations. The point of view taken in this article is that one need only prove an algebraicity theorem for the most interesting L-value, namely, the c…
In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers. The dimensions are expressed by special values of Shintani zeta functions for spaces of symmetric matrices at non-positive integers. This formula was given by Shintani for only a small part of the geometric side of the trace formula. To be precise, it is the contribution of unipotent elements corresponding to the partitions $(2^j,1^{2n-2j})$, where $n$ denotes the degree and $0\leq j \leq n$. Hence, ou…
We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a suitable range, generalizing the case N=2 treated by E. Fouvry, S. Ganguly, E. Kowalski and P. Michel. Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.
We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-compact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp forms. From geometrical considerations, we find evidence for the existence of small eigenvalues. We search for these small eigenvalues and compute the corresponding Maass cusp forms numerically.
We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of Saito-Kurokawa type, we prove an improved bound in the $k$-aspect for the smallest prime at which its Hecke eigenvalue is negative. Secondly, we show that there are infinitely many sign changes among the Hecke eigenvalues of $F$ at primes lying in an arithmetic progression. Third, we show that there are infinitely many positive as well as infinitely many negat…
We prove non-vanishing modulo p, for a prime $\ell$ different from p, of central critical Rankin-Selberg L-values with anticyclotomic twists of $\ell$-power conductor. The L-function is Rankin product of a cusp form and a theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with the case when the weight of Hecke character is greater than that of cusp form, so the L-value is essentially different in nature from the one in the landmark work of Vatsal and Cornut-Vatsal on the same theme.
We construct local charts for the ramified cusps of the Hilbert modular variety of level $Γ_1(\mathfrak{c},\mathfrak{n})$. This allows us, following closely M. Rapoport and C.-L. Chai, to construct arithmetic toroidal and minimal compactifications of these modular varieties. Some applications to modular forms are given. See our joint work with J. Tilouine for other applications.
Michael HARRIS defined the arithmetic automorphic periods for certain cuspidal representations of $GL_{n}$ over quadratic imaginary fields in his Crelle paper 1997. He also showed that critical values of automorphic L-functions for $GL_{n}\times GL_{1}$ can be interpreted in terms of these arithmetic automorphic periods. In the thesis, we generalize his results in two ways. Firstly, the arithmetic automorphic periods have been defined over general CM fields. We also prove that these periods factorize as products of local periods over infinity places. Secondly, we show that critical values of a…
For a prime number $p\ge 5$, we consider three classical cusp eigenforms $f_j(z)$ of weights $k_1, k_2, k_3$, of conductors $N_1, N_2, N_3$, and of nebentypus characters $ψ_j \bmod N_j$. According to H.Hida and R.Coleman, one can include each $f_j$ into a {$p$-adic analytic family} $k_j \mapsto \{f_{j,k_j}\}$ of cusp eigenforms $f_{j,k_j}$ of weights $k_j$ in such a way that $f_{j,k_j}=f_j$, and that all their Fourier coefficients $a_n(f_{j, k_j})$ are given by certain $p$-adic analytic functions $k_j{}\mapsto a_{n, j}(k_j{})$. The purpose of this paper is to describe a four variable $p$-adic …
