This document describes the authors' current research project: the evaluation of a tower of Rankin-Selberg integrals on the group E_6. We recall the notion of a tower, and two known towers, making observations about how the integrals within a tower may be related to one another via formal manipulations, and offering a heuristic for how the L-functions should be related to one another when the integrals are related in this way. A detailed description of the E_6 tower is then given.
We prove a division algebra analogue of an ultrametric inequality of Bushnell--Henniart for Rankin--Selberg conductors. Under the Jacquet--Langlands correspondence, the two versions are equivalent.
We formulate and prove the archimedean period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$. As a consequence, we prove the period relations for critical values of the Rankin-Selberg L-functions for $\mathrm{GL}(n)\times \mathrm{GL}(n-1)$ over arbitrary number fields.
We study the critical numbers of the Rankin-Selberg convolution of arbitrary pairs of cohomological cuspidal automorphic representations and we parametrize these critical numbers by certain 1-dimensional subrepresentations attached to the corresponding pair of finite-dimensional representations of the related general linear groups.
In this paper, we establish bounds of the Rankin-Selberg $L$-function for $SL(2)$ using the supnorm of the Eisenstein series and a purely representation theoretic index over the real group. Consequently, we obtain a subconvexity bound $L(\frac{1}{2}+ it, f_1 \times f_2) \leq C (1+ |t|)^{\frac{5}{6}+ε}$ for two Maass cusp forms of $SL(2, \mathbb Z)$.
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
For a $p$-adic local field $F$ of characteristic 0, with residue field $\mathfrak{f}$, we prove that the Rankin-Selberg gamma factor of a pair of level zero representations of linear general groups over $F$ is equal to a gamma factor of a pair of corresponding cuspidal representations of linear general groups over $\mathfrak{f}$. Our results can be used to prove a variant of Jacquet's conjecture.
Let $F$ be a $p$-adic field and $E/F$ be a quadratic extension. In this paper, we prove the stability of Rankin-Selberg gamma factors for $\textrm{Sp}_{2n}(F)$, $\widetilde {\textrm{Sp}}_{2n}(F)$ and $\textrm{U}_{E/F}(n,n)$ when the characteristic of the residue field of $F$ is not $2$.
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.
We use the uniqueness of various invariant functionals on irreducible unitary representations of PGL(2,R) in order to deduce the classical Rankin-Selberg identity for the sum of Fourier coefficients of Maass cusp forms and its new anisotropic analog. We deduce from these formulas non-trivial bounds for the corresponding unipotent and spherical Fourier coefficients of Maass forms. As an application we obtain a subconvexity bound for certain L-functions. Our main tool is the notion of Gelfand pair.
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 $K/\mathbb{Q}$.
Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\mathrm{GSp}_4$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGL}_4$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\mathrm{SL}_4(\mathbf{C})$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGU}(2…
We study the local zeta integrals attached to a pair of generic representations $(π,τ)$ of $GL_n\times GL_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $π$ and $τ$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.
Let $π_1,π_2$ be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$ over a non-archimedean local field $F$ of residual characteristic $p$, different to $\ell$. Whenever the local Rankin-Selberg $L$-factor $L(X,π_1,π_2)$ is nontrivial, we exhibit explicit test vectors in the Whittaker models of $π_1$ and $π_2$ such that the local Rankin-Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_F^n$ is equal to $L(X,π_1,π_2)$. We give an initial application of the test vectors to reduction modulo $\ell$ of $\…
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…
Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$
Institute for Advanced Study in Mathematics, Zhejiang University（浙江大学数学高等研究院）；School of Mathematical Sciences, Zhejiang University（浙江大学数学系）；Institute for Advanced Study in Mathematics and New Cornerstone Science Laboratory, Zhejiang University（浙江大学数学高等研究院与新Cornerstone科学实验室）
In this paper, we partially complete the local Rankin-Selberg theory of Asai $L$-functions and $ε$-factors as introduced by Flicker and Kable. In particular, we establish the relevant local functional equation at Archimedean places and prove the equality between Rankin-Selberg's and Langlands-Shahidi's $ε$-factors at every place. Our proofs work uniformly for any characteristic zero local field and use as only input the global functional equation and a globalization result for a dense subset of tempered representations that we infer from work of Finis-Lapid-Müller. These results are used in an…
Let $π$ and $τ$ be a smooth generic representation of ${\rm SO}_5$ and ${\rm GL}_2$ respectively over a non-archimedean local field. Assume that $π$ is irreducible and $τ$ is irreducible or induced of Langlands' type. We show that the $L$- and $ε$-factors attached to $π\timesτ$ defined by the Rankin-Selberg integrals and the associated Weil-Deligne representation coincide. Similar compatibility results are also obtained for the local factors defined by the Novodvorsky's local zeta integrals attached to generic representations of ${\rm GSp}_4\times{\rm GL}_2$.
We construct analogues of Rankin--Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet--Piatetski-Shapiro--Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models in…
Recently Atobe-Oi-Yasuda established the newform theory for irreducible tempered generic representations of unramified ${\rm U}_{2n+1}$ over non-archimedean local fields. In this paper we extend their result to every irreducible generic representations and compute the dimensions of the spaces of oldforms. We also compute the Rankin-Selberg integrals attached to newforms and oldforms under a natural assumption on the $γ$-factors defined by these integrals.
Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.
The aim of this article is to study rank-one Eisenstein cohomology for the group GL(N)/F, where F is a totally real field extension of Q. This is then used to prove rationality results for ratios of successive critical values for Rankin-Selberg L-functions for GL(n) x GL(n') over F with the parity condition that nn' is even. The key idea is to interpret Langlands's constant term theorem in terms of Eisenstein cohomology.
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 $π$ be an irreducible admissible (complex) representation of $GL(2)$ over a non-archimedean characteristic zero local field with odd residual characteristic. In this paper we prove the equality between the local symmetric square $L$-function associated to $π$ arising from integral representations and the corresponding Artin $L$-function for its Langlands parameter through the local Langlands correspondence. With this in hand, we show the stability of local symmetric $γ$-factors attached to $π$ under highly ramified twists.
We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,π\timesπ')$ over any Galois extension with $π$ and $π'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at least one of the representations subject to a self-contragredient assumption. We then extend these results to two representations $π$ defined on $GL_n/E$ and $π'$ defined on $GL_m/F$ with $E$ and $F$ cylic algebraic number fields of coprime degree where $π$ and $π'$ admit a base change lift from $\mathbb{Q}$ again given a self-contragredient assumption on at least one…
This is an announcement of results on rank-one Eisenstein cohomology of GL(N), with N > 1 an odd integer, and algebraicity theorems for ratios of successive critical values of certain Rankin-Selberg L-functions for GL(n) x GL(n') when n is even and n' is odd.
We unconditionally construct cyclotomic p-adic L-functions for Rankin-Selberg convolutions for GL(n+1) x GL(n) over arbitrary number fields, and show that they satisfy an expected functional equation.
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.
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.
An overview of the classical Rankin-Selberg problem involving the asymptotic formula for sums of coefficients of holomorphic cusp forms is given. We also study the function $Δ(x;ξ) (0\leξ\le1)$, the error term in the Rankin-Selberg problem weighted by $ξ$-th power of the logarithm. Mean square estimates for $Δ(x;ξ)$ are proved.
Let $F$ be a $p$-adic field and $π$ be an irreducible smooth representation of $\textrm{Sp}_{2n}(F)$. In this paper, we show that if $π$ and $π^κ$ are both generic for a common generic character of the maximal unipotent of a fixed Borel, then $π\cong π^κ$, where $π^κ$ is the representation induced by the conjugation action of an element $κ\in \textrm{GSp}_{2n}(F)$. This result is a consequence of the standard local Langlands conjecture and local Gan-Gross-Prasad conjecture. As a consequence, we extend the dependence relation of the local Rankin-Selberg gamma factors for $\textrm{Sp}_{2n}\times…
In this paper, we obtain a subconvexity result for the Rankin-Selberg L-function in both levels. The new feature in this result is applying an amplification method of Duke-Friedlander-Iwaniec to a double Petersson-Kuznetsov trace formula. As the trace formula ranges over all the $GL_2$ spectrum, this subconvexity result is unconditional of which Hecke eigenforms are chosen in the Rankin-Selberg L-function.
In this paper, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg L-functions. One of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. A first consequence is a new subconvexity bound for Rankin-Selberg L-functions in the level aspect. Moreover, infinitely many Rankin-Selberg L-functions having at most eight non-trivial real zeros are produced and some new non-trivial estimates for the analytic rank of the family…
We derive integral representations for the Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2) by a process of unipotent averaging at archimedean places. A key feature of our result is that it allows one to fix the choice of test vector at finite places, irrespective of ramification. This enables a new proof of the functional equation for GL(3) x GL(2) Rankin-Selberg L-functions in many cases.
We try to understand the poles of L-functions via taking a limit in a trace formula. This technique avoids endoscopic and Kim-Shahidi methods. In particular, we investigate the poles of the Rankin-Selberg L-function. Using analytic number theory techniques to take this limit, we essentially get a new proof of the analyticity of the Rankin-Selberg L-function at $s=1.$ Along the way we discover the convolution operation for Bessel transforms.
We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blas…
A standard zero free region is obtained for Rankin Selberg L-functions $L(s, f\times \widetilde{f})$ where $f$ is an almost everywhere tempered Maass form on $GL(n)$ and $f$ is not necessarily self dual. The method is based on the theory of Eisenstein series generalizing a work of Sarnak.
In this paper, we solve the Rankin--Selberg problem. That is, we break the well known Rankin--Selberg's bound on the error term of the second moment of Fourier coefficients of a $\mathrm{GL}(2)$ cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions.
We present a Rankin-Selberg integral on the exceptional group $G_2$ which represents the L-function for generic cuspidal representations of $\widetilde{SL}_2\times GL_2$. As an application, we show that certain Fourier-Jacobi type periods on $G_2$ are non-vanishing.
We prove a version of the Extra-zero conjecture formulated by the first named author for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong evidence in support of this conjecture in the non-critical case, which remained essentially unstudied.
We address the problem of identifying a Hecke-Maass cusp form $f$ of full level from the central values of the Rankin-Selberg $L$-functions $L(1/2,f\otimes h)$ where $h$ runs through the set of Hecke-Maass eigenforms of full level. We prove a quantitative result in this direction.
In this article, we prove an explicit Waldspurger formula for the toric Hilbert modular forms. As an application, we construct a class of anticyclotomic p-adic Rankin-Selberg L-functions for Hilbert modular forms, generalizing the construction of Bertolini, Darmon and Prasanna in the elliptic case. Moreover, building on works of Hida, we give a necessary and sufficient condition when the Iwasawa mu-invariant of this p-adic L-function vanishes and prove a result on the non-vanishing modulo $p$ of central Rankin-Selberg L-values with anticyclotomic twists.
By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic representations. By the multiplicity-one theorems due to Sun-Zhu and Chen-Sun such invariant forms are unique up to scalar multiples and can therefore be related to invariant forms on equivalent principal series representations. We construct meromorphic families of such invariant forms for spherical principal series representations of ${\rm GL}(n,\mathbb{R})$ an…
We compute the second moment of a certain family of Rankin-Selberg $L$-functions L(f x g, 1/2) where f and g are Hecke-Maass cusp forms on GL(n). Our bound is as strong as the Lindelöf hypothesis on average, and recovers individually the convexity bound. This result is new even in the classical case n=2.
