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 the adelic setting.
We prove a refinement of the global Gan-Gross-Prasad conjecture proposed by Ichino-Ikeda and N. Harris for unitary groups under some local conditions. We need to assume some expected properties of L-packets and some part of the local Gan-Gross-Prasad conjecture.
Let $\mathbb{A}$ be the adele ring of a totally real algebraic number field $F$. We push forward an explicit computation of a relative trace formula for periods of automorphic forms along a split torus in $GL(2)$ from a square free level case done by Masao Tsuzuki, to an arbitrary level case. By using a relative trace formula, we study central values of automorphic $L$-functions for cuspidal automorphic representations of $GL(2, \mathbb{A})$ corresponding to Maass forms with arbitrary level.
The zeros and poles of standard automorphic $L$-functions attached to representations of classical groups are linked to the nonvanishing of lifts in the theory of the theta correspondence. The results of this paper show that when a cuspidal representation $σ$ of a symplectic group $G$ lifts to a cuspidal representation $π= θ_ψ(σ)$ of an orthogonal group $H$ attached to a quadratic space $V$ of dimension $m$, the Fourier coefficients of automorphic forms in $σ$ are linked to periods of automorphic forms in $π$. Consequently, when our results are combined with the Rallis inner product formula in…
We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character χmodulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.χ,s) do not vanish at s=1/2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.χ,s), and from that derive an upper bound for the average analytic rank of the studied L-functions.
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.
We prove Deligne's conjecture for central critical values of certain automorphic $L$-functions for ${\rm GL}(3)\times {\rm GL}(2)$. The proof is base on rationality results for central critical values of triple product $L$-functions, which follow from establishing explicit Ichino's formulae for trilinear period integrals for Hilbert cusp forms on totally real etale cubic algebras over $\mathbb{Q}$.
In this paper we investigate arithmetic properties of automorphic forms on the group G' = GL_m/D, for a central division-algebra D over an arbitrary number field F. The results of this article are generalizations of results in the split case, i.e., D=F, by Shimura, Harder, Waldspurger and Clozel for square-integrable automorphic forms and also by Franke and Franke-Schwermer for general automorphic representations. We also compare our theorems on automorphic forms of the group G' to statements on automorphic forms of its split form using the global Jacquet-Langlands correspondence developed by …
Let $F$ be a global field. Let $G$ and $H$ be two connected reductive group defined over $F$ endowed with an $F$-morphism $f: H\rightarrow G$ such that the induced morphism $H_{der}\rightarrow G_{der}$ on the derived groups is a central isogeny. Our main results yield in particular the following theorem: Given any irreducible cuspidal representation $π$ of $G(\mathbb A_F)$ its restriction to $H(\mathbb A_F)$ contains a cuspidal representation $σ$ of $H(\mathbb A_F)$. Conversely, assuming moreover that $f$ is an injection, any irreducible cuspidal representation $σ$ of $H(\mathbb A_F)$ appears …
Let $π$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $π$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(\frac{1}{2},π\timesχ)$, with a certain automorphic character $χ$ of ${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general ${\mathfrak n}\geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results…
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…
In this paper we prove a version of Deligne's conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.
In this paper, we give an asymptotic formula for the second moment of Dirichlet twists of an automorphic $L$-function $L(s, π)$ on the critical line averaged over characters and conductors, where $π$ denotes an irreducible tempered cuspidal automorphic representation of $\textrm{GL}_{4}$ with unitary central character. We give some hybrid bound for the error term with respect to the size of conductors of Dirichlet characters and that of the automorphic representation.
Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $ϕ(s)$ denote the automorphic scattering determinant. From the known functional equation $ϕ(s)ϕ(1-s)=1$ one concludes that $ϕ(1/2)^{2} = 1$. However, except for the relatively few instances when $ϕ(s)$ is explicitly computable, one does not know $ϕ(1/2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $ϕ(s)$ in $(1/2,\infty)$, counted with multiplicities. Let $d(1)$ be the coefficient of …
In this paper we generalize the work of Harris-Soudry-Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on GL_2 over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semisimplification.
In this article, we establish polynomial-growth bound for the sequence of Fourier coefficients associated to even integer weight vector-valued automorphic forms of Fuchsian groups of the first kind. At the end, their $L$-functions and exponential sums have been discussed.
In this paper we explore some properties of periods attached to automorphic representations of unitary groups over CM fields and the critical values of their $L$-functions. We prove a formula expressing the critical values in the range of absolute convergence in terms of Petersson norms of holomorphic automorphic forms. On the other hand, we express the Deligne period of a related motive as a product of quadratic periods and compare the two expressions by means of Deligne's conjecture.
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).
We consider invariant hyperfunctions associated to automorphic forms on the upper half plane. We give two interpretations of the period function of Maass forms introduced by Lewis. The first interpretation shows that the period function arises from the explicit description of a representative of the hyperfunction associated to the Maass form. Under certain conditions, automorphic forms determine cohomology classes in a cohomology group with values in the hyperfunctions with bounded support on the line. A map from hyperfunctions to holomorphic functions leads to a second, cohomological, interpr…
We prove a sharp bound for the average value of the triple product of modular functions for the Hecke subgroup Γ_0(N). Our result is an extension of the main result in {Bernstein&Reznikov-2004} to a fixed cuspidal representation of the adele group PGL(2,A).
In this paper, we study the value-distributions of $L$-functions of holomorphic primitive cusp forms in the level aspect. We associate such automorphic $L$-functions with probabilistic models called the random Euler products. First, we prove the existence of probability density functions attached to the random Euler products. Then various mean values of automorphic $L$-functions are expressed as integrals involving the density functions. Moreover, we estimate the discrepancies between the distributions of values of automorphic $L$-functions and those of the random Euler products.
Constructing vector-valued automorphic forms on unitary groups
University of California, Berkeley（加州大学伯克利分校）；Michigan State University（密歇根州立大学）；University of Oregon（俄勒冈大学）；Boston College（波士顿学院）；University of Arizona（亚利桑那大学）；Stanford University（斯坦福大学）；Columbia University（哥伦比亚大学）
The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. In the first place, we combine the Ichino--Ikeda--Neal-Harris (IINH) formula -- which is now a theorem -- with an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of ${\rm GL}(n)\times{\rm GL}(m)$ over CM …
We apply some ideas of Bombieri and Garrett to construct natural self-adjoint operators on spaces of automorphic forms whose only possible discrete spectrum is $λ_{s}$ for $s$ in a subset of on-line zeros of an $L$-function, appearing as a compact period of cuspidal-data Eisenstein series on $GL_{4}$. These ideas have their origins in results of Hejhal and Colin de Verdiére. In parallel with the $GL(2)$ case, the corresponding pair-correlation and triple-correlation results limit the fraction of on-the-line zeros that can appear in this fashion.
We prove a new (conditional) result towards the subconvexity problem for certain automorphic $L$-functions for $\mathrm{GL}_2 \times \mathrm{GL}_3$. This follows from the computation of new $\mathrm{SL}_2$-period integrals associated with newforms $f$ and $g$ of even weight and odd squarefree level. The same computations also lead to a central value formula for degree $6$ complex $L$-series of the form $L(f \otimes \mathrm{Ad}(g), s)$, extending previous work in arXiv:1804.02352.
Let $d\in\mathbb{N}$, and let $π$ be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L'}{L}(1+it,π\otimesχ_D)$ as $D$ varies over fundamental discriminants. Here, $t$ is a fixed real number and $χ_D$ is the real character associated with $D$. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L'}{L}…
While vector-valued automorphic forms can be defined for an arbitrary Fuchsian group $Γ$ and an arbitrary representation $R$ of $Γ$ in GL$(n,{\mathbb C})$, their existence has been established in the literature only when restrictions are imposed on both $Γ$ and $R$. In this paper, we prove the existence of $n$ linearly independent vector-valued automorphic forms for any Fuchsian group $Γ$ and any $n$-dimensional complex representation $R$ of $Γ$. To this end, we realize these automorphic forms as global sections of a special rank $n$ vector bundle built using solutions to the Riemann-Hilbert p…
For $m\ge 2$, let $π$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_π(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $π$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_π(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $π$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $π$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Se…
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 give a new and representation theoretic construction of $p$-adic interpolation series for central values of self-dual Rankin-Selberg $L$-functions for $\operatorname{GL}_2$ in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real field corresponding to a $\mathfrak{p}$-ordinary Hilbert modular forms of parallel …
We construct a p-adic Eisenstein measure with values in the space of vector-weight p-adic automorphic forms on certain unitary groups. This measure allows us to p-adically interpolate special values of certain vector-weight C-infinity automorphic forms, including Eisenstein series, as their weights vary. We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz's p-adic families of Eisenstein series for Hilbert modular forms.
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…
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 …
Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)} L\left(\frac{1}{2}+it,f\right)^2λ_f(p^j),\qquad j=1,2, $$ where $λ_f(p^j)$ is the $p^j$-th normalized Fourier coefficient of $f$ and $L(s,f)$ is the $L$-function associated to $f$.
This is a survey of recent work on values of Rankin-Selberg $L$-functions of pairs of cohomological automorphic representations that are {\it critical} in Deligne's sense. The base field is assumed to be a CM field. Deligne's conjecture is stated in the language of motives over $\QQ$, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. …
In this article, we study the Chow group of the motive associated to a tempered global $L$-packet $π$ of unitary groups of even rank with respect to a CM extension, whose global root number is $-1$. We show that, under some restrictions on the ramification of $π$, if the central derivative $L'(1/2,π)$ is nonvanishing, then the $π$-nearly isotypic localization of the Chow group of a certain unitary Shimura variety over its reflex field does not vanish. This proves part of the Beilinson--Bloch conjecture for Chow groups and $L$-functions, which generalizes the Birch and Swinnerton-Dyer conjectur…
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…
Let F be a number field, p a prime number. We construct the (multi-variable) p-adic L-function of an automorphic representation of $GL_2(A_F)$ (with certain conditions at places above p and $\infty$), which interpolates the complex L-function at the central critical point. We use this construction to prove that the p-adic L-function of a modular elliptic curve E over F has vanishing order greater or equal to the number of primes above p at which E has split multiplicative reduction, as predicted by the exceptional zero conjecture. This is a generalization of analogous results by Spieß (arXiv:1…
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.
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…
Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\…
A cuspidal automorphic representation πof a group G is said to to be distinguished with respect to a subgroup H if the integral of f along H is nonzero for a cusp form f in the space of π. Such period integrals are related to (non)vanishing of interesting L-values and also to Langlands functoriality. This article discusses a general principle, labelled arithmeticity, which roughly states that "πis H-distinguished if and only if any Galois conjugate of πis H-distinguished." We study this principle via several examples; starting with GL(2) and leading up to more complicated situations where the …
