In this paper, we will give a certain formula for the Riemann zeta function that expresses the Riemann zeta function by an infinte series consisting of $K$-Bessel functions. Such an infinite series expression can be regarded as an analogue of the Chowla-Selberg formula. Roughly speaking, the Chowla-Selberg formula is the formula that expresses the Epstein zeta-function by an infinite series consisting of $K$-Bessel functions. In addition, we also give certain analogues of the Chowla-Selberg formula for Dirichlet $L$-functions and $L$-functions associated with holomorphic cusp forms. Moreover, we introduce a two variable function which is analogous to the real analytic Eisenstein series and give a certain limit formula for this one. Such a limit formula can be regarded as an analogue of Kronecker's limit formula.
We obtain a first moment formula for Rankin-Selberg convolution $L$-series of holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an orthonormal basis of Maass forms. One consequence is the best result to date, uniform in level, spectral value and weight, for the equality of two Maass or holomorphic cusp forms if their Rankin-Selberg convolutions with the orthonormal basis of Mass forms $u_j$ is equal at the center of the critical strip for sufficiently many $u_j$. The main novelty of our approach is the new way the error terms are treated. They are brought into an exa…
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. …
For Kleinian groups acting on hyperbolic three-space, we prove factorization formulas for both the Selberg zeta-function and the automorphic scattering matrix. We extend results of Venkov and Zograf from Fuchsian groups, to Kleinian groups, and we give a proof that is simple and extendable to more general groups.
This paper describes our method of pairing automorphic distributions. This represents a third technique for obtaining the analytic properties of automorphic L-functions, in addition to the existing methods of integral representations (Rankin-Selberg) and Fourier coefficients of Eisenstein series (Langlands-Shahidi). We recently used this technique to establish new cases of the full analytic continuation of the exterior square L-functions. The paper here gives an exposition of our method in two special, yet representative cases: the Rankin-Selberg tensor product L-functions for PGL(2,Z), as wel…
We prove Siegel-Walfisz type theorems (over long and short intervals) for the Fourier coefficients of certain automorphic $L$-functions and Rankin-Selberg $L$-functions over number fields.
We establish the universality theorem for the first four symmetric power L-functions of automorphic forms and their associated Rankin-Selberg L-functions. This generalizes some results of Laurincikas & Matsumoto and Matsumoto respectively.
We consider certain Dirichlet series of Selberg type, constructed from periods of automorphic forms. We study analytic properties of these Dirichlet series and show that they have analytic continuation to the whole complex plane.
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.
We prove that for a compact locally symmetric Riemannian space $M$ of rank 1 there exist infinitely many automorphic Tate motives $f$ such that the generalized Selberg zeta function $Z_{M(f)}(s)$ satisfies a simple functional equation in the sense that it has no gamma factors.
Using Hecke triangle surfaces of finite and infinite area as examples, we present techniques for thermodynamic formalism approaches to Selberg zeta functions with unitary finite-dimensional representations $(V,χ)$ for hyperbolic surfaces (orbifolds) $Γ\backslash\mathbb{H}$ as well as transfer operator techniques to develop period-like functions for $(Γ,χ)$-automorphic cusp forms. This leads to several natural conjectures. We further show how to extend these results to the billiard flow on the underlying triangle surfaces, and study the convergence of transfer operators along sequences of Hecke…
By generalizing the classical Selberg-Chowla formula, we establish the analytic continuation and functional equation for a large class of Epstein zeta functions. This continuation is studied in order to provide new classes of theorems regarding the distribution of zeros of Dirichlet series in their critical lines and to produce a new method for the study of these problems. Due to the symmetries provided by the representation via the Selberg-Chowla formula, some generalizations of well-known formulas in analytic number theory are also deduced as examples.
In the 80s, Zagier and Jacquet-Zagier tried to derive the Selberg trace formula by applying the Rankin-Selberg method to the automorphic kernel function. Their derivation was incomplete due to a puzzle of the computation of a residue. We solve this puzzle and complete the derivation. The main input is an extension of the theory of regularized integrals invented by Zagier, which is of independent interest.
We give a simple proof of a standard zero-free region in the $t$-aspect for the Rankin--Selberg $L$-function $L(s,π\times \widetildeπ)$ for any unitary cuspidal automorphic representation $π$ of $\mathrm{GL}_n(\mathbb{A}_F)$ that is tempered at every nonarchimedean place outside a set of Dirichlet density zero.
In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $λ_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $Re(λ_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involvin…
We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a p-adic distribution interpolating the special values of the twisted Rankin-Selberg L-function attached to cuspidal automorphic representations of GL(n) and GL(n-1) over number fields. If the representations are ordinary at p, our distribution is bounded and yields analyticity of the associated p-adic L-function.
We describe a new method to obtain weak subconvexity bounds for $L$-functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic $L$-functions and (with mild restrictions) the Rankin-Selberg $L$-functions attached to two automorphic representations. The proof relies on a new unconditional log-free zero density estimate for Rankin-Selberg $L$-functions.
This is a report on the global aspects of the Langlands-Shahidi method which in conjunction with converse theorems of Cogdell and Piatetski-Shapiro has recently been instrumental in establishing a significant number of new and surprising cases of Langlands Functoriality Conjecture over number fields. They have led to striking new estimates towards Ramanujan and Selberg conjectures.
Given unitary automorphic cuspidal representations $π$ and $π'$ defined on $GL_n(\mathbb{A}_E)$ and $GL_m(\mathbb{A}_F)$, respectively, with $E$ and $F$ solvable algebraic number fields we deduce a prime number theorem for the Rankin-Selberg L-function $L(s,AI_{E/\mathbb{Q}}(π)\times AI_{F/\mathbb{Q}}(π'))$ under a self-contragredient assumption and a suitable Galois invariance condition on the representations, where $AI_{K/\mathbb{Q}}$ denotes the automorphic induction functor for any number field $K/\mathbb{Q}$.
Let $F$ be a number field, and let $π_1$ and $π_2$ be distinct unitary cuspidal automorphic representations of $\operatorname{GL}_{n_1}(\mathbb{A}_F)$ and $\operatorname{GL}_{n_2}(\mathbb{A}_F)$ respectively. In this paper, we derive new lower bounds for the Rankin-Selberg $L$-function $L(s, π_1 \times \widetildeπ_2)$ along the edge $\Re s = 1$ of the critical strip in the $t$-aspect. The corresponding zero-free region for $L(s, π_1 \times \widetildeπ_2)$ is also determined.
Let $π$ and $π'$ be cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ with unitary central characters. We establish a new zero-free region for all $\mathrm{GL}(1)$-twists of the Rankin-Selberg $L$-function $L(s,π\timesπ')$, generalizing Siegel's celebrated work on Dirichlet $L$-functions. As an application, we prove the first unconditional Siegel-Walfisz theorem for the Dirichlet coefficients of $-L'(s,π\timesπ')/L(s,π\timesπ')$. Also, for $n\leq 8$, we extend the region of holomorphy and nonvanishing for the twisted symmetric power $L$-functions $L(s,π,\mathrm{Sym…
We show that a class of Dirichlet series ${\mathfrak{A}}^{\#}$ that is much larger than the extended Selberg class ${\mathscr{S}}^{\#}$, and also contains the standard as well as the tensor product, exterior square and symmetric square $L$-functions of automorphic $L$-functions of $GL_n$ over number fields, does not have any elements of degrees between $1$ and $2$. The proof of our more general theorem is very different from the proof of Kaczorowski and Perelli for the class ${\mathscr{S}}^{\#}$, and is much shorter and simpler even in that case.
We prove an exact formula for the second moment of Rankin-Selberg $L$-functions $L(1/2,f \times g)$ twisted by $λ_f(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The formula is a reciprocity relation that exchanges the twist parameter $p$ and the level $q$. The method involves the Bruggeman/Kuznetsov trace formula on both ends; finally the reciprocity relation is established by an identity of sums of Kloosterman sums.
We prove an asymptotic expansion of the second moment of the central values of the $\mathrm{GL}(n)\times\mathrm{GL}(n)$ Rankin--Selberg $L$-functions $L(1/2,π\otimesπ_0)$, for a fixed cuspidal automorphic representation $π_0$, over the family of $π$ with analytic conductors bounded by a quantity which is tending off to infinity. Our proof uses the integral representations of the $L$-functions, period with regularized Eisenstein series, and the invariance properties of the analytic newvectors.
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).
In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two methods of proof. One uses Kronecker's limit formula and the other uses the geometry of a family of abelian varieties. We discuss a generalization of this formula, which was proposed by Colmez, as well as some explicit Hodge cycles which appear in the geometric proof.
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…
Let $F(s)=\sum_{n=1}^{\infty}\frac{a_n}{n^s}$ be a Dirichlet series in the axiomatically defined class ${\mathfrak A}^{\#}$ . The class ${\mathfrak A}^{\#}$ is known to contain the extended Selberg class ${\mathcal S}^{\#}$, as well as all the $L$-functions of automorphic forms on $GL_n/K$, where $K$ is a number field. Let $d$ be the degree of $F(s)$. We show that $\sum_{n<X}|a_n|=Ω(X^{\frac{1}{2}+\frac{1}{2d}})$, and hence, that the abscissa of absolute convergence of $σ_a$ of $F(s)$ must satisfy $σ_a\ge 1/2+1/2d$.
We prove a new case of spectral reciprocity formulae for the product of $GL(n+1) \times GL(n)$ and $GL(n) \times GL(n-1)$ Rankin-Selberg $L$-functions ($n \geq 3$), which are first developed by Blomer and Khan in \cite{BK17} for degree 8 $L$-functions ($n=2$ case, the product of $GL(3) \times GL(2)$ and $GL(2) \times GL(1)$ Rankin-Selberg $L$-functions). Our result can be viewed as a generalization of Blomer and Khan's work to higher rank case. We will mainly follow the method developed in \cite{Nun20}. We will use the integral representations of Rankin-Selberg $L$-functions generalized by Ich…
We show that Selberg's eigenvalue conjecture concerning small eigenvalues of the automorphic Laplacian for congruence groups is equivalent to a conjecture about the non-existence of residual eigenvalues for a perturbed system. We prove this using a combination of methods from asymptotic perturbation theory and number theory.
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 derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 \textless{} d \textless{} 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [1…
Let $π$ and $π_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,π\timesπ_0)$, where $π$ varies in a given family and $π_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion…
We extend the Fourier expansion of automorphic functions of Piateski-Shapiro and Shalika in a way that the Euclidean algorithm appears nontrivially. Then, we discuss a potential application in the Rankin-Selberg method.
If C is a smooth projective curve over a number field k, then, under fair hypotheses, its L-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is X-mean-periodic for some appropriate functional space X. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such L-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of GL_{2g}-automorphic representations…
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.
Over the last few years Pohl (partly jointly with coauthors) developed dual `slow/fast' transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $Γ\backslash\mathbb{H}$ with cusps and all finite-dimensional unitary representations $χ$ of $Γ$. The eigenfunctions with eigenvalue $1$ of the fast transfer operators determine the zeros of the Selberg zeta function for $(Γ,χ)$. Further, if $Γ$ is cofinite and $χ$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue $1$ of the s…
In this paper we consider Dirichlet series absolutely converging for $σ>1$ with an Euler product, natural bounds on the coefficients and satisfying orthogonality relations of Selberg type. Let $N\geq 1$, $F_1(s),...,F_N(s)$ be as above and $P(X_1,...,X_N)$ be a non-monomial polynomial with coefficients in the ring of $p$-finite Dirichlet series absolutely converging for $σ\geq 1$; then $P(F_1(s),\ldots,F_N(s))$ has infinitely many zeros for $σ>1$. Our result in particular applies to Artin $L$-functions, automorphic $L$-functions under the Ramanujan conjecture, and the elements of the Selberg c…
The Rankin-Selberg integral of Kohnen and Skoruppa produces the Spin $L$-function for holomorphic Siegel modular forms of genus two. In this paper, we reinterpret and extend their integral to apply to arbitrary cuspidal automorphic representations of $\mathrm{PGSp}_4$. We show that the integral is related to a non-unique model and analyze it using the approach of Piatetski-Shapiro and Rallis.
Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg's later proofs.
Let M be an imaginary quadratic field, f a Hecke eigenform on GL2(Q) and πthe unitary base-change to M of the automorphic representation associated to f. Take a unitary arithmetic Hecke character χof M inducing the inverse of the central character of f. The celebrated formula of Waldspurger relates the square of an integral L(f) of f and χover the idele class group of M to the central critical value L(1/2,π\otimesχ). In this paper, we present a construction of a new p-adic L-function that interpolates L(f) over arithmetic characters for a cusp form f in the spirit of the landmark result of Kat…
Let $F$ be a number field and let $\mathbb{A}_F$ be its ring of adeles. Let $B$ be a quaternion algebra over $F$ and let $ν:B \to F$ be the reduced norm. Consider the reductive monoid $M$ over $F$ whose points in an $F$-algebra $R$ are given by \begin{align*} M(R):=\{(γ_1,γ_2) \in (B \otimes_F R)^{2}:ν(γ_1)=ν(γ_2)\}. \end{align*} Motivated by an influential conjecture of Braverman and Kazhdan we prove a summation formula analogous to the Poisson summation formula for certain spaces of functions on the monoid. As an application, we define new zeta integrals for the Rankin-Selberg $L$-function a…
We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the congruence subgroups $Γ_0(p)$ and $Γ_0(q)$ (resp) with trivial nebentypus, then for all $ε>0$ we show that there exists an $A >0$ such that $$ L\left(\frac{1}{2}+it, f \times g \right) \ll_{ε,μ_f, μ_g}(1+|t|)^A \frac{(pq)^{1/2+ε}}{\max\{p,q \}^{\frac{1}{64}}}. $$ The dependence on $μ_f$ and $μ_g$, the parameters at infinity for $f$ and $g$ respectively, is poly…
