A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=\frac{k}{2}.$ It was shown by Kohnen that there exists a Hecke eigenform $f$ of weight $k$ such that $L^*(f,s) \neq 0$ for sufficiently large $k$ and any point on the line segments $Im(s)=t_0, \frac{k-1}{2} < Re(s) < \frac{k}{2}-ε, \frac{k }{2}+ε< Re(s) < \frac{k+1}{2},$ for any given real number $t_0$ and a positive real number $ε.$ This paper concerns the non-vanishing of the product $L^*(f,s)L^*(f,w)$ $(s,w\in \mathbb{C})$ on average.
Let $f$ be a holomorphic cusp form of integral weight $k \geq 3$ for $Γ_{0}(N)$ with nebentypus character $ψ$. Generalising work of Kohnen and Raghuram we construct a kernel function for the $L$-function $L(f,χ,s)$ of $f$ twisted by a primitive Dirichlet character $χ$ and use it to show that the average $\sum_{f \in S_{k}(N,ψ)}\frac{L(f,χ,s)}{<f,f>}\bar{a_{f}(1)}$ over an orthogonal basis of $S_{k}(N,ψ)$ does not vanish on certain line segments inside the critical strip if the weight $k$ or the level $N$ is big enough. As another application of the kernel function we prove an averaged version …
We prove that given a Hecke-Maass form $f$ for $\text{SL}(2, \mathbb{Z})$ and a sufficiently large prime $q$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that the $L$-values $L(\tfrac{1}{2}, f \otimes χ)$ and $L(\tfrac{1}{2}, χ)$ do not vanish. We expect the same method to work for any large integer $q$.
In a previous paper \cite{BorGunn}, we defined the space of toric forms $\TTT(l)$, and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group $Γ_1(l)$. In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that $L(f,1) \not = 0$. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols.
In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $χ$ of conductor $q$ such that $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)\neq 0$. To prove this, we establish asymptotic formulas of $L\left(\frac{1}{2},f\otimes χ\right)L\left(\frac{1}{2},χ\right)$ over the family of even primitive Dirichlet characters $χ$ of conductor $q$ for more general $q$.
Let $ϕ= \sum_{r^{2} \leq 4mn}c(n,r)q^{n}ζ^{r}$ be a Jacobi form of weight $k$ (with $k > 2$ if $ϕ$ is not a cusp form) and index $m$ with integral algebraic coefficients which is an eigenfunction of all Hecke operators $T_{p}, (p,m) = 1,$ and which has at least one nonvanishing coefficient $c(n,r)$ with $r$ prime to $m$. We prove that for almost all primes $\ell$ there are infinitely many fundamental discriminants $D = r^{2}-4mn < 0$ prime to $m$ with $ν_{\ell}(c(n,r)) = 0$, where $ν_{\ell}$ denotes a continuation of the $\ell$-adic valuation on $\mathbb{Q}$ to an algebraic closure. As applica…
We prove under mild hypotheses the three-variable Iwasawa main conjecture for $p$-ordinary modular forms in the indefinite setting. Our result is in a setting complementary to that in the work of Skinner-Urban, and it has applications to Greenberg's nonvanishing conjecture for the first derivatives at the center of $p$-adic $L$-functions of cusp forms in Hida families with root number $-1$ and to Howard's horizontal nonvanishing conjecture.
We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.
In this article, we study the simultaneous sign changes of the Fourier coefficients of two Hilbert cusp forms of different integral weights. We also study the simultaneous non-vanishing of Fourier coefficients, of two distinct non-zero primitive Hilbert cuspidal non-CM eigenforms of integral weights, at powers of a fixed prime ideal.
We study the simultaneous sign change of Fourier coefficients of a pair of distinct normalized newforms of integral weight supported on primes power indices, we also prove some equidistribution results. Finally, we consider an analogous question for Fourier coefficients of a pair of half-integral weight Hecke eigenforms.
Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. In this note we compare the signs of the…
We study simultaneous non-vanishing of $L(\tfrac{1}{2},\di)$ and $L(\tfrac{1}{2},g\otimes \di)$, when $\di$ runs over an orthogonal basis of the space of Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ and $g$ is a fixed $SL(2,\mathbb{Z})$ Hecke cusp form of weight $k\equiv 0 \pmod 4$.
We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.
Let $F$ and $G$ be Siegel cusp forms for $\Sp_4(\Z)$ and weights $k_1, k_2$ respectively. Also let $F$ and $G$ be Hecke eigenforms lying in distinct eigen spaces. Further suppose that neither $F$ nor $G$ is a Saito-Kurokawa lift. In this article, we study simultaneous arithmetic behaviour of Hecke eigenvalues of these Hecke eigenforms.
We develop a reciprocity formula for a spectral sum over central values of L-functions on GL(4)xGL(2). As an application we show that for any self-dual cusp form Pi for SL(4,Z), there exists a Maass form pi for SL(2,Z) such that L(1/2, Pi x pi) is nonvanishing. An important ingredient is a "balanced" Voronoi summation formula involving Kloosterman sums on both sides, which can also be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL(4) Hecke eigenvalues.
In this paper, we study the Drinfeld cusp forms for $Γ_1(T)$ and $Γ(T)$ using Teitelbaum's interpretation as harmonic cocycles. We obtain explicit eigenvalues of Hecke operators associated to degree one prime ideals acting on the cusp forms for $Γ_1(T)$ of small weights and conclude that these Hecke operators are simultaneously diagonalizable. We also show that the Hecke operators are not diagonalizable in general for $Γ_1(T)$ of large weights, and not for $Γ(T)$ even of small weights. The Hecke eigenvalues on cusp forms for $Γ(T)$ with small weights are determined and the eigenspaces characte…
Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error term of O(N^{-1/2} log N). The bound here is used in the author's paper "Galois representations attached to Q-curves and the generalized Fermat equation A^4 + B^2 = C^p," (to appear, Amer. J. Math.) to show that certain spaces of cuspforms arising there contain forms whose L-functions have nonvanishing special value. Version of May 2005: Nathan Ng found an er…
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 construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet-Langlands correspondence, generalizing works of Bertolini-Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger type formula, as well as a formula for the dihedral mu-invariant. We also make a note of Howard's nonvanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certai…
In this paper, we study the average behaviour of the coefficients of triple product L-functions and some related L-functions corresponding to normalized primitive holomorphic cusp form $f(z)$ of weight $k$ for the full modular group $SL(2,\ \mathbb{Z}).$ Here we call $f(z)$ a primitive cusp form if it is an eighenfunction of all Hecke operators simultaneously.
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 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…
Let $f$ and $g$ be two Hecke-Maass cusp forms of weight zero for $SL_2(\mathbb Z)$ with Laplacian eigenvalues $\frac{1}{4}+u^2$ and $\frac{1}{4}+v^2$, respectively. Then both have real Fourier coefficients say, $λ_f(n)$ and $λ_g(n)$, and we may normalize $f$ and $g$ so that $λ_f(1)=1=λ_g(1)$. In this article, we first prove that the sequence $\{λ_f(n)λ_g(n)\}_{n \in \mathbb{N}}$ has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of $f$ and $g$.
We consider a new integral representation for $L(s_1, Π\times τ_1) L(s_2, Π\times τ_2),$ where $Π$ is a globally generic cuspidal representation of $GSp_4,$ and $τ_1$ and $τ_2$ are two cuspidal representations of $GL_2$ having the same central character. As and application, we find a new period condition for two such $L$ functions to have a pole simultaneously. This points to an intriguing connection between a Fourier coefficient of a residual representation on $GSO(12)$ and a theta function on $\widetilde{Sp}(16).$ A similar integral on $GSO(18)$ fails to unfold completely, but in a way that …
The Galois representations associated to weight $1$ newforms over $\bar{\mathbb{F}}_p$ are remarkable in that they are unramified at $p$, but the computation of weight $1$ modular forms has proven to be difficult. One complication in this setting is that a weight $1$ cusp form over $\bar{\mathbb{F}}_p$ need not arise from reducing a weight $1$ cusp form over $\bar{\mathbb{Q}}$. In this article we propose a unified "Hecke stability method" for computing spaces of weight $1$ modular forms of a given level in all characteristics simultaneously. Our main theorems outline conditions under which a f…
In this article, we distinguish Siegel cuspidal eigenforms of degree two on the full symplectic group from the signs of their Hecke eigenvalues. To establish our theorem, we obtain a result towards simultaneous sign changes of eigenvalues of two Siegel eigenforms. In the course of the proof, we also prove that the Satake $p$-parameters of two different Siegel eigenforms are distinct for a set of primes $p$ of density 1. The main ingredient to prove the latter result is the theory of Galois representations attached to Siegel eigenforms.
We establish a relative trace formula on $\mathrm{GL}(n+1)$ weighted by cusp forms on $\mathrm{GL}(n)$ over number fields. The spectral side is a weighted average of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ over the \textit{full spectrum}, and the geometric side consists of Rankin-Selberg $L$-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n),$ and certain explicit holomorphic functions. The formula yields new results towards central $L$-values for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$ (over number fields): the second moment evaluation, and simultaneous nonvnaish…
In this paper we work over $Γ_0(N)$, for any $N$ and write the spectral moment of a product of two distinct Rankin-Selberg convolutions at a general point on the critical line $\frac{1}{2}+it$ as a main term plus a sharp error term in the $t$ aspect and the spectral aspect. As a result we obtain hybrid Weyl type subconvexity results in the $t$ and spectral aspects. Also, for fixed modular forms $f$, $g$ of even weight $k\geq 4$ we show there exists a Maass cusp form $u_j$ such that $L(1/2, f\times u_j)$, $L(1/2, g\times u_j)$ are simultaneous non-zero.
This paper is a continuation of the author's previous wotk. We supplement four results on a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$. First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $GSp_4$ under a conjecture in local-global compatibility and Arthur's classification for $GSp_4$. Second, we prove simultaneous vertical Sato-Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many…
The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their decompensation on the prime numbers. Chapter 2 is devoted to the results known concerning the first problem in the case of classical modular forms and to explain our motivation of research. In Chapter 3, we treated the case of primitive Maass forms and obtained an estimate of the subconvexity-type for the least integer n such that the n-th coefficient is negat…
