This note contains an attempt to relate Hecke's presentation of an ideal class zeta function in a real quadratic field as an integral of the nonholomorphic Eisenstein series along the loop on modular curve and Zagier's decomposition of this zeta function into the sum of zeta functions of indefinite quadratic forms. (See D.Zagier, A Kronecker limit formula for real quadratic field.) It is a preliminary version.
If E(z,s) is the nonholomorphic Eisenstein series on the upper half plane, then for all y sufficiently large, E(z,s) has a "Siegel zero." That is E(z,β)=0 for a real number βjust to the left of one. We give a generalization of this result to Eisenstein series formed with real valued automorphic forms on a finite central covering of the adele points of a connected reductive algebraic group over a global field.
We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E_2*(z), we obtain CRT-based algorithms that compute the class polynomials H_D(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form F_p(z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under t…
We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral transforms which begins with the heat kernel, obtains a Poisson and wave kernel, and then yields the hyperbolic and ellipt…
For any number $m \equiv 0,1 \, (4)$ we correct the generating function of Hurwitz class number sums $\sum_r H(4n - mr^2)$ to a modular form (or quasimodular form if $m$ is a square) of weight two for the Weil representation attached to a binary quadratic form of discriminant $m$ and determine its behavior in the Petersson scalar product. This modular form arises through holomorphic projection of the zero-value of a nonholomorphic Jacobi Eisenstein series of index $1/m$. When $m$ is prime, we recover the classical Hirzebruch-Zagier series whose coefficients are intersection numbers of curves o…
We obtain some basic partial differential operators connected with nonholomorphic automorphic forms on $Γ\backslash U(2, 1)/K$. We give the corresponding Eisenstein series of weight $k$ and automorphic Green functions of weight $k$. Moreover, we get the product formulas connected with them. As a consequence of the product formulas, we give the functional equations of Eisenstein series associated to a convex cocompact subgroup of U(2,1).
Let $Γ$ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane $\mathbb H$, and let $M = Γ\backslash \mathbb H$ be the associated finite volume hyperbolic Riemann surface. If $γ$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $γ$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbol…
We consider certain double series of Eisenstein type involving hyperbolic-sine functions. We define certain generalized Hurwitz numbers, in terms of which we evaluate those double series. Our main results can be regarded as a certain generalization of well-known results of Hurwitz, Herglotz, Katayama and so on. Our results also include recent formulas of the third-named author which are double analogues of the formulas of Cauchy, Mellin, Ramanujan, Berndt and so on, about certain Dirichlet series involving hyperbolic functions. As an application, we give some evaluation formulas for $q$-zeta f…
The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for $Γ_0(N)$ as regularized theta …
We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of $\mathrm{SL}(2,\mathbb{R})$ due to Bernstein and Reznikov to study triple products of Eisenstein series of arbitrary (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.
In this paper we consider certain classes of generalized double Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formula for some double Eisenstein series. We can find that these double Eisenstein series are reducible to infinite series involving hyperbolic functions. Moreover, some interesting new examples are given.
We define Eisenstein series on rank 2 hyperbolic Kac--Moody groups over R, induced from quasi--characters. We prove convergence of the constant term and hence the almost everywhere convergence of the Eisenstein series. We define and calculate the degenerate Fourier coefficients. We also consider Eisenstein series induced from cusp forms and show that these are entire functions.
Let $Γ\subset\mathrm{PSL}_{2}(\mathbb{R})$ be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane $\mathbb{H}$, and let $M=Γ\backslash\mathbb{H}$ be the associated finite volume hyperbolic Riemann surface. Associated to any cusp of $M$, there is the classically studied non-holomorphic (parabolic) Eisenstein series. In 1979, Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series associated to any closed geodesic on $Γ\backslash\mathbb{H}$. In 2004, Jorgenson and Kramer introduced so-called elliptic Eisenstein series associa…
The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of p…
We extend results of Bhagwat and Rajan on a strong multiplicity one property for length spectrum to hyperbolic manifolds with cusps, showing that for two even dimensional hyperbolic manifolds of finite volume, if all but finitely many closed geodesics have the same length, then all closed geodesics have the same length. When the set of exceptional lengths is infinite, but sufficiently sparse, we can show that the two manifolds must have the same volume, and in low dimensions also the same number of cusps. A main ingredient in our proof is a generalization of a result of Selberg on the distribu…
We derive the Fourier expansion of scalar-valued Eisenstein series for O(2, n+2) using classical methods of Siegel, Braun, Zagier, Bruinier and others. We assume that the underlying lattice splits two hyperbolic planes. Finally we prove for the Eisenstein series at the standard cusp that they belong to the Maass space for O(2, n + 2), an analogue of the "Spezialschar" for Siegel modular forms introduced by Maass, at least at all local places p, where the localization of the underlying lattice is maximal.
Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce vectors whose integral transforms achieve arbitrarily weighted moments. Applications include general Motohashi-type identities and Weyl-type subconvex bounds for some families of $L$-functions, extending some results of Conrey--Iwaniec and Petrow--Young to the number field setting. We deduce improved estimates for representation numbers of ternary quadratic for…
In the former part of this paper, we summarize our previous results on infinite series involving the hyperbolic sine function, especially, with a focus on the hyperbolic sine analogue of Eisenstein series. Those are based on the classical results given by Cauchy, Mellin and Kronecker. In the latter part, we give new formulas for some infinite series involving the hyperbolic cosine function.
We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with con…
We consider the automorphism groups of various Lorentzian lattices over the Eisenstein, Gaussian, and Hurwitz integers, and in some of them we find reflection groups of finite index. These provide new finite-covolume reflection groups acting on complex and quaternionic hyperbolic spaces. Specifically, we provide groups acting on CH^n for all n<6 and n=7, and on HH^n for n=1,2,3 and 5. We compare our groups to those discovered by Deligne and Mostow and show that our largest examples are new. For many of these Lorentzian lattices we show that the entire symmetry group is generated by reflections…
We study two closely related problems stemming from the random wave conjecture for Maass forms. The first problem is bounding the $L^4$-norm of a Maass form in the large eigenvalue limit; we complete the work of Spinu to show that the $L^4$-norm of an Eisenstein series $E(z,1/2+it_g)$ restricted to compact sets is bounded by $\sqrt{\log t_g}$. The second problem is quantum unique ergodicity in shrinking sets; we show that by averaging over the centre of hyperbolic balls in $Γ\backslash \mathbb{H}$, quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the …
We define the unit circle for global function fields. We demonstrate that this unit circle (endearingly termed the \emph{$q$-unit circle}, after the finite field $\mathbb{F}_q$ of $q$ elements) enjoys all of the properties akin to the classical unit circle: center, curvature, roots of unity in completions, integrality conditions, embedding into a finite-dimensional vector space over the real line, a partition of the ambient space into concentric circles, Möbius transformations, a Dirichlet approximation theorem, a reciprocity law, and much more. We extend the exponential action of Carlitz by p…
