In this paper, we investigate the non-modular solutions to the Schwarz differential equation $\{f,τ\}=sE_4(τ)$ where $E_4(τ)$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we provide explicit solutions for each $s=2π^2(n/6)^2$ with $n\equiv 1\mod 12$. These solutions are obtained as integrals of meromorphic weight 2 modular forms. As a consequence, we find explicit solutions to the differential equation $\displaystyle y''+\frac{π^2n^2}{36}\,E_4\,y=0$ for each $n\equiv 1\mod 12$ generalizing the work of Hurwitz and Klein on the case $n=1$. Our investigation relies on the theory of equivariant functions on the complex upper half-plane. This paper supplements a previous work where we determine all the parameters $s$ for which the above Schwarzian equation has a modular solution.
This paper concerns the study of the Schwarz differential equation $\{h,τ\}=s\,E_4(τ)$ where $E_4$ is the weight 4 Eisenstein series and $s$ is a complex parameter. In particular, we determine all the values of $s$ for which the solutions $h$ are modular functions for a finite index subgroup of $\mbox{SL}_2(\mathbb Z)$. We do so using the theory of equivariant functions on the complex upper-half plane as well as an analysis of the representation theory of $\mbox{SL}_2(\mathbb Z)$. This also leads to the solutions to the Fuchsian differential equation $y''+s\,E_4\,y=0$.
Let $X$ be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on $X$ in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation $$_2F_1(1/24,7/24,5/6, -\frac{2^{10}\cdot3^3\cdot5}{11^4})=\sqrt6 \sqrt[6]{\frac{11}{5^5}} $$ and other similar identities.
Marvin Knopp developed the theory of automorphic integrals, which generalize automorphic forms; each automorphic integral has an additional period function in its automorphic relation. The period functions satisfy relations that arise from the underlying group relations. Knopp showed that entire automorphic integrals with rational period functions satisfy a Hecke correspondence theorem, provided the rational period functions have poles only at 0 or infinity. For other automorphic integrals each corresponding Dirichlet series has a functional equation with a remainder term that arises from the …
Higher order automorphic forms have recently been introduced to study important questions in number theory and mathematical physics. We investigate the connection between these functions and Chen's iterated integrals. Then using Chen's theory, we prove a structure theorem for automorphic forms of all orders. This allows us to define an analogue of a mixed Hodge structure on a space of higher order automorphic forms.
A general Voronoï summation formula for the (metaplectic) double cover of $\text{GL}_2$ is derived via the representation theoretic framework à la Ichino--Templier. The identity is also formulated classically and used to establish Voronoï summation formulae for half-integral weight modular forms and Maaß forms.
This paper exposes the underlying mechanism for obtaining second integral moments of $GL_2$ automorphic $L$--functions over an arbitrary number field. Here, moments for $GL_2$ are presented in a form enabling application of the structure of adele groups and their representation theory. To the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic terms, distinguishing global and local issues, and allowing uniform application to number fields. When specialized to the field of rational numbers $\Bbb{Q}$, we recover the classical results.
Building on the works of S. Bochner on equivalence of modular relation with functional equation associated to the Dirichlet series, K. Chandrasekharan and R. Narasimhan obtained new equivalences between the functional equation and some arithmetical identities. Sister Ann M. Heath considered the functional equation in the Hawkins and Knopp context and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we use techniques of Chandrasekharan and Narasimhan and extend the re…
We obtain uniform lower bounds, true for all automorphic L-functions L(s) associated to cuspidal representations of GL(m,A) where A denotes the adeles of the rationals Q, of the integral on the vertical line (Re(s)=1/2) of the absolute value squared of L(s)/s; and also of L(s)/(s-s_0) when s_0 is a zero of the L-function on the critical line. Several variants are also obtained in small degrees m, for the vertical integrals at different abscissas in the critical strip. For the estimates required to prove convergence, we are led to generalise a result of Friedlander-Iwaniec (Can. J. Math. 57,200…
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. Knopp generalized this to real weights. We show that for weights that are not an integer at least 2 the generalized Eichler integral gives an injection into the first cohomology group with values in a module of holomorphic functions, and characterize the image. We impose no condition on the growth of the automorphic forms at the cusps. For real weights that are not an int…
The Bohr-Jessen limit theorem is a probabilistic limit theorem on the value-distribution of the Riemann zeta-function in the critical strip. Moreover their limit measure can be written as an integral involving a certaindensity function. The existence of the limit measure is now known for a quite general class of zeta-functions, but the integral expression has been proved only for some special cases (such as Dedekind zeta-functions). In this paper we give an alternative proof of the existence of the limit measure for a general setting, and then prove the integral expression, with an explicitly …
We investigate the relations for $L$-functions satisfying certain functional equation, summationa formulas of Voronoi-Ferrar type and Maass forms of integral and half-integral weight. Summation formulas of Voronoi-Ferrar type can be viewed as an automorphic property of distribution vectors of non-unitary principal series representations of the double covering group of $SL(2)$. Our goal is converse theorems for automorphic distributions and Maass forms of level $N$ characterizing them by analytic properties of the associated $L$-functions. As an application of our converse theorems, we construc…
We convolve a theta function on an $n$-fold cover of $GL_3$ with an automorphic form on an $n'$-fold cover of $GL_2$ for suitable $n,n'$. To do so, we induce the theta function to the $n$-fold cover of $GL_4$ and use a Shalika integral. We show that in particular when $n=n'=3$ this construction gives a new Eulerian integral for an automorphic form on the 3-fold cover of $GL_2$ (the first such integral was given by Bump and Hoffstein), and when $n=4$, $n'=2$, it gives a Dirichlet series with analytic continuation and functional equation that involves both the Fourier coefficients of an automorp…
Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta functions attached to q satisfies certain symmetries. As an application, we prove the symmetries for automorphic forms on the orthogonal group of q closely related to Heegener divisors (Borcherds lifts and automorphic Green functions).
In this paper, we introduce a new family of period integrals attached to irreducible cuspidal automorphic representations $σ$ of symplectic groups $\mathrm{Sp}_{2n}(\mathbb{A})$, which detects the right-most pole of the $L$-function $L(s,σ\timesχ)$ for some character $χ$ of $F^\times\backslash\mathbb{A}^\times$ of order at most $2$, and hence the occurrence of a simple global Arthur parameter $(χ,b)$ in the global Arthur parameter $ψ$ attached to $σ$. We also give a characterisation of first occurrences of theta correspondence by (regularised) period integrals of residues of certain Eisenstein…
We present a pairing of automorphic distributions that applies in situations where a Lie group acts with an open orbit on a product of generalized flag varieties. The pairing gives meaning to an integral of products of automorphic distributions on these varieties. This generalizes classical integral representations or "Rankin-Selberg integrals" of L-functions, and gives new constructions and analytic continuations of automorphic L-functions. Keywords: Automorphic forms, invariant pairings, automorphic distributions, L-functions, analytic continuation, rapid decay.
We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type. More precisely, we first build on Kisin's work to construct integral canonical models over rings of integers of number fields with finitely many primes inverted for Shimura varieties of abelian type with hyperspecial level at all primes we do not invert, compatible with Kisin's construction. We then define a notion of an integral canonical model for the standard principal bundles lying over Shimura varieties and proceed to construct them in the abelian type case. With these …
In the framework of Kontsevich-Zagier periods, we derive integral representations for weight-$k$ automorphic Green's functions invariant under modular transformations in $\varGamma_0(N)$ ($N\in\mathbb Z_{\geq1} $), provided that there are no cusp forms on the respective Hecke congruence groups with an even integer weight $k\geq4$. These Kontsevich-Zagier integral representations for automorphic Green's functions give explicit formulae for certain Eichler-Shimura maps connecting Eichler cohomology to Maaß cusp forms. We construct integral representations for weight-4 Gross-Zagier renormalized G…
Based on Furusawa's theory, we present an integral representation for the L-function L(s,π\times τ), where πis a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel modular form, and where τis an arbitrary cuspidal automorphic representation on GL(2). As an application, a special value result for this L-function in the spirit of Deligne's conjecture is proved.
We generalize Zagiers work on regularized integral to the singular case in the adelic setting. We develop necessary tools of treating various singular cases of regularized triple product formulas, which appear naturally in the work of Michel and Venkatsh on the subconvexity problem for $GL_2$.
We introduce interaction entropies, which can be represented as logarithmic couplings of certain cycles on a class of algebraic curves of arithmetic interest. In particular, via interaction entropies for Legendre-Ramanujan curves $ Y^n=(1-X)^{n-1}X(1-αX)$ ($ n\in\{6,4,3,2\}$), we reformulate the Kontsevich-Zagier integral representations of weight-4 automorphic Green's functions $ G_2^{\mathfrak H/\overline{\varGamma}_0(N)}(z_1,z_2)$ ($N=4\sin^2(π/n )\in\{1,2,3,4\}$), in a geometric context. These geometric entropies allow us to establish algebraic relations between certain weight-4 automorphi…
We prove an integrality result for the value at s=1 of the adjoint L-function associated to a cohomological cuspidal automorphic representation on GL(n) over any number field. We then show that primes (outside an exceptional set) dividing this special value give rise to congruences between automorphic forms. We also prove a non-vanishing property at infinity for the relevant Rankin-Selberg L-functions on GL(n) x GL(n).
Let pi be a cuspidal, automorphic representation of GSp(4) attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,pi x tau), where tau runs through certain cuspidal, automorphic representation of GL(2). Our calculations include the case of square-free level for the p-adic components of tau, and a wide class of archimedean types including Maass forms. As an application we obtain a special value result for L(s,pi x tau).
In the present paper we find explicit formulas for the degrees of Heegner divisors on arithmetic quotients of the orthogonal group $\Orth(2,p)$ and for the integrals of certain automorphic Green's functions associated with Heegner divisors. The latter quantities are important in the study of the arithmetic degrees of Heegner divisors in the context of Arakelov geometry. In particular, we obtain a different proof and a generalization of results of Kudla relating these quantities to the Fourier coefficients of certain non-holomorphic Eisenstein series of weight $1+p/2$ for the metaplectic group …
We present a regularization procedure of period integrals of automorphic forms on a group $G$ over an arbitrary reductive subgroup $G' \subset G$. As a consequence we obtain an explicit $G'(\mathbb{A})$-invariant functional on the space of automorphic forms on $G$ whose exponents avoid certain prescribed hyperplanes. We also provide a necessary and sufficient condition for convergence of period integrals of automorphic forms in terms of their exponents.
The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that it decomposes into a space span…
We give constraints on the existence of $(χ,b)$-factors in the global $A$-parameter of a genuine cuspidal automorphic representation $σ$ of a metaplectic group in terms of the invariant, lowest occurrence index, of theta lifts to odd orthogonal groups. We also give a refined result that relates the invariant, first occurrence index, to non-vanishing of period integrals of residues of Eisenstein series associated to the cuspidal datum $χ\otimesσ$. This complements our previous results for symplectic groups.
We classify all primes appearing in the denominators of the Hauptmodul $J$ and modular forms for non-arithmetic triangle groups with a cusp. These primes have a congruence condition in terms of the order of the generators of the group. As a corollary we show that for the Hecke group of type $(2,m,\infty)$, the prime $p$ does not appear in the denominator of $J$ if and only if $p\equiv \pm 1\pmod m$.
In this paper we consider some global constructions of liftings of automorphic representations attached to some commuting pairs in the exceptional group $F_4$. We consider two families of integrals. The first uses the minimal representation on the double cover of $F_4$, and in the second we consider examples of integrals of descent type associated with unipotent orbits of $F_4$.
We prove a cohomological formula for non-critical residues of degree eight automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky's integral representation of the $L$-functions.
Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations $σ$ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter $(χ,b)$ in the global Arthur parameter $ψ_σ$ associated to $σ$, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the $(χ,b)$-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of c…
We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger's formula in terms of equality between global distributions. As applications we generalize the Kohnen-Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindelof hypothesis for integral weight forms. We also study the Kohnen space in t…
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…
We use results about Fourier coefficients appearing in [T] (and some more obtained here), to obtain information for certain among the integrals of the form $$I=\int_{GL_n(\kkk)Z_n(\A)\s GL_n(\A)}φ(g)ϕ(g)\F(E)(\tj(g))dg$$ where: $\A$ is the adele ring of a number field $\kkk$; $φ$ is a $GL_n(\A)$-cuspidal automorphic form; $ϕ$ is a $GL_n(\A)$-automorphic function (even the trivial for some results); $E$ is a $GL_{N}(\A)$-automorphic form for a multiple $N$ of $n$; $\F(E)$ is a Fourier coefficient of $E$ for certain choices of additive functions $\F$ in a set $\BBnk[N]$ which we defined in $[T];…
Given a maximal even-integral lattice $\cL$ of signature $(m+, 2-)$ with an odd $m\geq 3$, we consider the holomorphic cusp forms $F$ of weight $l$ on the bounded symmetric domain of type IV of dimension $m$ with respect to the discriminant subgroup of the orthogonal group $O(\cL)$ defined by $\cL$. Under a non-negativity assumption on the central $L$-values, we prove an equidistribution result of Satake parameters in an ensemble constructed from the central values of standard $L$-functions and the square of the Whittaker-Bessel periods.
Let L(s) = L(s, π) be the standard L-function of a cuspidal representation πof GL(m,A) where A denotes the adèles of the field of rationals. We consider the integral, on the real line Re(s)= 1/2, of the squared absolute value of L(s)/s. In an earlier paper, partly with P. Sarnak (arxiv:2203.12475) we obtained a universal lower bound on this integral, independently of m. In this paper, for m fixed, we first obtain a universal lower bound for the integral on an interval [-A logC, A log C] where C is the analytic conductor of π; this bound is of order c(log C)^{-1/2} ; A, c are absolute positive …
Let $\mathbb{E}$ be a quadratic extension of a number field $\mathbb{F}$. Let $E(g, s)$ be an Eisenstein series on $GL_2(\mathbb{E})$, and let $F$ be a cuspidal automorphic form on $GL_2(\mathbb{F})$. We will consider in this paper the following automorphic integral: $$\int_{Z_{A}GL_{2}(\mathbb{F})\backslash GL_{2}(\mathbb{A}_{\mathbb{F}})} F(g)E(g,s) dg.$$ This is in some sense the complementary case to the well-known Rankin-Selberg integral and the triple product formula. We will approach this integral by Waldspurger's formula. We will discuss when the integral is automatically zero, and oth…
In this paper, we consider the $\SL(2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\GL(2)$: one due to Harder-Langlands-Rapoport, and the other due to Waldspurger.
Gross and Zagier conjectured that the CM values (of certain Hecke translates) of the automorphic Green function $G_s(z_1,z_2)$ for the elliptic modular group at positive integral spectral parameter $s$ are given by logarithms of algebraic numbers in suitable class fields. We prove a partial average version of this conjecture, where we sum in the first variable $z_1$ over all CM points of a fixed discriminant $d_1$ (twisted by a genus character), and allow in the second variable the evaluation at individual CM points of discriminant $d_2$. This result is deduced from more general statements for…
There are many Rankin-Selberg integrals representing Langlands $L$-functions, and it is not apparent what the limits of the Rankin-Selberg method are. The Dimension Equation is an equality satisfied by many such integrals that suggests a priority for further investigations. However there are also Rankin-Selberg integrals that do not satisfy this equation. Here we propose an extension and reformulation of the dimension equation that includes many additional cases. We explain some of these cases, including the new doubling integrals of the authors, Cai and Kaplan. We then show how this same equa…
In this article, we study the co-period integral attached to an automorphic form on $\GL(2)$ and two exceptional theta series on the cubic Kazhdan-Patterson cover of $\GL(2)$. In the local aspect, we show the $\Hom$-space is always of one dimension and conduct the unramified calculations. In the global aspect, we give the Euler decomposition for the co-period integrals of Eisenstein series and propose an Ichino-Ikeda type conjecture relating the co-period integrals of cuspidal forms to the central critical value of symmetric cube $L$-functions. We also deduce from the local multiplicity one re…
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.
Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general decay statement along sets much larger than Siegel sets, and furthermore state and prove the decay for smooth but not necessarily K-finite cusp forms. We also state a general theorem about the convergence of Rankin-Selberg integrals involving unipotent periods, closing a gap in the literature on L-functions. These properties serve as the analytic basis of a new …
We compute the homotopy groups of spectra associated by a theorem of Lurie to the Shimura curves of discriminants 6, 10, and 14, beginning with a computation of integral rings of automorphic forms on these curves. As an application, we find that a generalized truncated Brown-Peterson spectrum BP<2> is an E_\infty ring spectrum at the prime 3.
In this article, we describe an efficient method for computing Teitelbaum's $p$-adic $\mathcal{L}$-invariant. These invariants are realized as the eigenvalues of the $\mathcal{L}$-operator acting on a space of harmonic cocycles on the Bruhat-Tits tree $\mathcal{T}$, which is computable by the methods of Franc and Masdeu described in [FM14]. The main difficulty in computing the $\mathcal{L}$-operator is the efficient computation of the $p$-adic Coleman integrals in its definition. To solve this problem, we use overconvergent methods, first developed by Darmon, Greenberg, Pollack and Stevens. In…
The purpose of the present paper is to study the number theoretic properties of the special values of the spectral zeta functions of the non-commutative harmonic oscillator (NcHO), especially in relation to modular forms and elliptic curves from the viewpoint of Fuchsian differential equations, and deepen the understanding of the spectrum of the NcHO. We study first the general expression of special values of the spectral zeta function $ζ_Q(s)$ of the NcHO at $s=n$ $(n=2,3,\dots)$ and then the generating and meta-generating functions for Apéry-like numbers defined through the analysis of speci…
Motivated by known examples of global integrals which represent automorphic L-functions, this paper initiates the study of a certain two-dimensional array of global integrals attached to any reductive algebraic group, indexed by maximal parabolic subgroups in one direction and by unipotent conjugacy classes in the other. Fourier coefficients attached to unipotent classes, Gelfand-Kirillov dimension of automorphic representations, and an identity which, empirically, appears to constrain the unfolding process are presented in detail with examples selected from the exceptional groups. Two new Eul…
Let $d \ge 1$. We study a subspace of the space of automorphic forms of $\mathrm{GL}_d$ over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place $\infty$ of $F$, and we consider the subspace $\mathcal{A}_{\mathrm{St}}$ consisting of automorphic forms such that the local component at $\infty$ of the associated automorphic representation is the Steinberg representation (to be made precise in the text). We have two results. One theorem (Theorem 16) describes the constituents of $\mathcal{A}_{\mathrm{St}}$ as automorphic representation an…
