In a 2005 paper, Yang constructed families of Hilbert Eisenstein series, which when restricted to the diagonal are conjectured to span the underlying space of elliptic modular forms. One approach to these conjectures is to show the non-vanishing of an inner product of elliptic eigenforms with the restrictions of Eisenstein series. In this paper, we compute this inner product locally by using explicit values of new vectors in the Waldspurger model.
Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n', generalizing the methods and results of previous work with Guenter Harder where the base field was totally real. In contrast to the totally real situation, the internal structure of the totally imaginary base field has a delicate effect on the rationality results.
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.
In this paper we make a systematic study of certain motivic cohomology classes ("Rankin-Eisenstein classes") attached to the Rankin--Selberg convolution of two modular forms of weight $\ge 2$. The main result is the computation of the $p$-adic syntomic regulators of these classes. As a consequence we prove many cases of the Perrin-Riou conjecture for Rankin--Selberg convolutions of cusp forms.
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 show that the Euler system associated to Rankin--Selberg convolutions of modular forms, introduced in our earlier works with Lei and Kings, varies analytically as the modular forms vary in $p$-adic Coleman families. We prove an explicit reciprocity law for these families, and use this to prove cases of the Bloch--Kato conjecture for Rankin--Selberg convolutions.
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.
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.
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.
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.
In this article, we compute the Gan-Gross-Prasad period integral of Klingen Eisenstein series over the unitary group $\mathrm{U}(m+1, n+1)$ with a cuspidal automorphic form over $\mathrm{U}(m+1, n)$, and show that it is related to certain special Rankin-Selberg $L$-values. We $p$-adically interpolate these Gan-Gross-Prasad period integrals as the Klingen Eisenstein series and the cuspidal automorphic form vary in Hida families. As a byproduct, we obtain a $p$-adic $L$-function of Rankin-Selberg type over $\mathrm{U}(m,n) \times \mathrm{U}(m+1, n)$. The ultimate motivation is to show the $p$-pr…
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.
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.
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.
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.
We prove new congruences between special values of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$. As an application, we prove strong non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.
We give effective lower bounds for $L(1,χ)$ via Eisenstein series on $Γ_0(q) \backslash \mathbb{H}$. The proof uses the Maass-Selberg relation for truncated Eisenstein series and sieve theory in the form of the Brun-Titchmarsh inequality. The method follows closely the work of Sarnak in using Eisenstein series to find effective lower bounds for $ζ(1+it)$.
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.
We prove a functional equation for the three-variable $p$-adic $L$-function attached to the Rankin-Selberg convolution of a Coleman family and a CM Hida family, which was studied by Loeffler and Zerbes. Consequentially, we deduce that an anticyclotomic $p$-adic $L$-function attached to a $p$-non-ordinary modular form vanishes identically in the indefinite setting.
Let $Δ_1(x;ϕ)$ be the error term of the first Riesz means of the Rankin-Selberg zeta function. We study the higher power moments of $Δ_1(x;ϕ)$ and derive an asymptotic formula for 3-rd, 4-th and 5-th power moments by using Ivić 's large value arguments.
In this paper, we study the non-vanishing of the central values of the Rankin-Selberg $L$-function of two adèlic Hilbert primitive forms ${\bf f}$ and ${\bf g}$, both of which have varying weight parameter $k$. We prove that, for sufficiently large $k$, there are at least $\frac{k}{(\log k)^{c}}$ adèlic Hilbert primitive forms ${\bf f}$ of weight $k$ for which $L(\frac12, {\bf f}\otimes{\bf g})$ are nonzero.
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.
Let $f$ be a Maass form for $SL(3, \mathbb{Z})$ which is fixed and $u_j$ be an orthonormal basis of even Maass forms for $SL(2, \mathbb{Z}),$ we prove an asymptotic formula for the average of the product of the Rankin-Selberg $L$-function of $f$ and $u_j$ and the $L$-function of $u_j$ at the central value 1/2. This implies simultaneous nonvanishing results of these $L$-functions at $1/2.$
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…
If $$ Δ(x) := \sum_{n\le x}c_n - Cx $$ denotes the error term in the classical Rankin-Selberg problem, then it is proved that $$ \int_0^X Δ^4(x)\d x \ll_εX^{3+ε},\quad \int_0^X Δ_1^4(x)\d x \ll_εX^{11/2+ε}, $$ where $Δ_1(x) = \int_0^xΔ(u) du$. The latter bound is, up to `$ε$', best possible.
In this work, we establish a zero density result for the Rankin-Selberg $L$-functions. As an application, we apply it to distinguish the holomorphic Hecke eigenforms for $\operatorname{SL}_2(\mathbb{Z}).$
In this paper, we prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs through all primitive Hilbert cusp forms of level $\mathfrak{q}$ for infinitely many prime ideals $\mathfrak{q}$. This result is a generalization of a theorem of Luo to the setting of totally real number fields.
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…
We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.
The conjectural newform theory for generic representations of $p$-adic ${\rm SO}_{2n+1}$ was formulated by P.-Y. Tsai in her thesis in which Tsai also verified the conjecture when the representations are supercuspidal. The main purpose of this work is to compute the Rankin-Selberg integrals for ${\rm SO}_{2n+1}\times{\rm GL}_r$ with $1\le r\le n$ attached to newforms and also oldforms under the validity of the conjecture.
As shown by Michel-Ramakrishan (2007) and later generalized by Feigon-Whitehouse (2008), there are "stable" formulas for the average central L-value of the Rankin-Selberg convolutions of some holomorphic forms of fixed even weight and large level against a fixed imaginary quadratic theta series. We obtain exact finite formulas for twisted first moments of Rankin-Selberg L-values in much greater generality and prove analogous "stable" formulas when one considers either arbitrary modular twists of large prime power level or real dihedral twists of odd type associated to a Hecke character of mixe…
We prove non-vanishing modulo p, for a prime $\ell$ different from p, of central critical Rankin-Selberg L-values with anticyclotomic twists of $\ell$-power conductor. The L-function is Rankin product of a cusp form and a theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with the case when the weight of Hecke character is greater than that of cusp form, so the L-value is essentially different in nature from the one in the landmark work of Vatsal and Cornut-Vatsal on the same theme.
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 $F$ be a non-Archimedean locally compact field. Let $σ$ and $τ$ be finite-dimensional semisimple representations of the Weil-Deligne group of $F$. We give strong upper and lower bounds for the Artin and Swan exponents of $σ\otimesτ$ in terms of those of $σ$ and $τ$. We give a different lower bound in terms of $σ\otimes\checkσ$ and $τ\otimes\checkτ$. Using the Langlands correspondence, we obtain the bounds for Rankin-Selberg exponents.
We consider the Rankin-Selberg L-functions associated with a fixed modular form of full level and holomorphic cuspidal newforms of large even weight, fixed level and fixed primitive nebentypus. We compute the second moment of this family in fairly general ranges, and obtain an asymptotic formula with a power saving error term. A special case treats the fourth moment of L-functions associated with holomorphic cusp forms.
In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.
