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 of Waldspurger's theorem relating the central critical value of the $D$-th twist ($D < 0$ a fundamental discriminant) of the $L$-function of a cusp form $f$ of even weight $2k$ to the square of the $|D|$-th Fourier coefficient of a form of half-integral weight $k+1/2$ associated to $f$ under the Shimura correspondence.
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $χ$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(Γ_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes χ,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes χ, \cdot)$ and a Petersson trace formula restricted to A…
We derive a Motohashi-type formula for the cubic moment of central values of $L$-functions of level $q$ cusp forms twisted by quadratic characters of conductor $q$, previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for $L$-functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.
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…
Recently, Juyal, Maji, and Sathyanarayana have studied a Lambert series associated with a cusp form over the full modular group and the Möbius function. In this paper, we investigate the Lambert series $\sum_{n=1}^{\infty}[a_f(n)ψ(n)*μ(n)ψ'(n)]\exp(-ny),$ where $a_f(n)$ is the $n$th Fourier coefficient of a cusp form $f$ over any congruence subgroup, and $ψ$ and $ ψ'$ are primitive Dirichlet characters. This extends the earlier work to the case of higher level subgroups and also gives a character analogue.
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 $q$.
We study sums of additively twisted Fourier coefficients of a holomorphic cusp form, a Maass cusp form, and the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform with respect to both the form and the terms of the additive twist. We also improve a bound on a sum of additively twisted Fourier coefficients of a holomorphic cusp form that have also been twisted by a nonlinear exponential term. This last bound is uniform with respect to the terms of the additive twist but it is not uniform with respect to the form.
Let $Γ$ be a geometrically finite Fuchsian group and suppose that $χ\colonΓ\to\mathrm{GL}(V)$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $Γ$ with twist $χ$ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.
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 obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients of holomorphic cusp forms twisted by rational additive twists with small denominators.
Let $\mathfrak{q}>2$ be a prime number, $χ$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$ L(1/2,f\otimes χ)\ll \mathfrak{q}^{1/2-1/12+\varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifying trivial delta method developed in [1].
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 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.
Let $q$ be a large prime, and $χ$ the quadratic character modulo $q$. Let $ϕ$ be a self-dual Hecke--Maass cusp form for $SL(3,\mathbb{Z})$, and $u_j$ a Hecke--Maass cusp form for $Γ_0(q)\subseteq SL(2,\mathbb{Z})$ with spectral parameter $t_j$. We prove the hybrid subconvexity bounds for the twisted $L$-functions \[ L(1/2,ϕ\times u_j\timesχ)\ll_{ϕ,\varepsilon} (qt_j)^{3/2-θ+\varepsilon},\quad L(1/2+it,ϕ\timesχ)\ll_{ϕ,\varepsilon} (qt)^{3/4-θ/2+\varepsilon}, \] for any $\varepsilon>0$, where $θ=1/23$ is admissible.
We establish a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and construct distributions interpolating these L-values. Using the method of abstract Kummer congruences, we then outline the main steps needed to show that these distributions come from a measure.
We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.
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 …
We study the asymptotic behaviour of the twisted first moment of central $L$-values associated to cusp forms in weight aspect on average. Our estimate of the error term allows extending the logarithmic length of mollifier $Δ$ up to 2. The best previously known result, due to Iwaniec and Sarnak, was $Δ<1$. The proof is based on a representation formula for the error in terms of Legendre polynomials.
Suppose $π_1$ and $π_2$ are two Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ such that for all primitive character $χ$ we have $$ L(\tfrac{1}{2},π_1\otimesχ)=L(\tfrac{1}{2},π_2\otimesχ). $$ Then we show that $π_1=π_2$.
Let $q \in \mathbb{Z} [i]$ be prime and $χ$ be the primitive quadratic Hecke character modulo $q$. Let $π$ be a self-dual Hecke automorphic cusp form for $\mathrm{SL}_3 (\mathbb{Z} [i] )$ and $f$ be a Hecke cusp form for $Γ_0 (q) \subset \mathrm{SL}_2 (\mathbb{Z} [i])$. Consider the twisted $L$-functions $ L (s, π\otimes f \otimes χ) $ and $L (s, π\otimes χ)$ on $\mathrm{GL}_3 \times \mathrm{GL}_2$ and $\mathrm{GL}_3$. We prove the subconvexity bounds \begin{equation*} L \big(\tfrac 1 2, π\otimes f \otimes χ\big) \ll_{\, \varepsilon, π, f } \mathrm{N} (q)^{5/4 + \varepsilon}, L \big(\tfrac 1 2…
We prove the non-vanishing of special $L$-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special $L$-values and considering Shintani cone decomposition.
In this paper, we prove that if the Fourier coefficients of a $\mathrm{SL}(3,\mathbb{Z})$ Hecke--Maaß cusp form $π$ are not too correlated with additive characters, then there exists infinitely many Dirichlet characters such that \begin{align*} L\left(\frac{1}{2},π\otimesχ\right)L\left(\frac{1}{2},χ\right)\neq 0. \end{align*} To prove this result, we compute the first twisted moment of these $L$ function averaged over a well chosen set of conductors.
Let $\GN\leq\SLR$ be a genus zero Fuchsian group of the first kind with $\infty$ as a cusp, and let $\Ek$ be the holomorphic Eisenstein series of weight $2k$ on $\GN$ that is nonvanishing at $\infty$ and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on $\GN,$ and on a choice of a fundamental domain $\F$, we prove that all but possibly $c(\GN,\F)$ of the non-trivial zeros of $\Ek$ lie on a certain subset of $\{z\in\mathfrak{H} : \JN(z)\in\mathbb{R}\}$. Here $c(\GN,\F)$ is a constant that does not depend on the weight $2k$ and $\JN$ i…
We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ an…
Let F(z) be a newform of weight 2k and level one with a trivial character, and assume that F(z) is a non-zero eigenform of all Hecke operators. In this paper, we study nonvanishing for central values of twisted modular L-function of F.
In a previous paper with Schmid (math.NT/0402382) we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound sum_{n < T} a_n exp(2 pi i n alpha) = O_ε(T^{3/4+ε}) uniformly for alpha real, for a_n the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n,R), and the sizes of certain period integrals. These in turn i…
For $f$ a cuspidal modular form for the group $Γ_0(N)$ of integral or half-integral weight, $N$ a multiple of $4$ in case the weight is half-integral, we study the zeros of the $L$-function attached to $f$ twisted by an additive character $e^{2πi n \frac{p}{q}}$ with $\frac{p}{q}\in \mathbb{Q}$. We prove that for certain $f$ and $\frac{p}{q}\in \mathbb{Q}$, the additively twisted $L$-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses distributions to prove the result.
For $f$ a primitive holomorphic cusp form of even weight $k \geq 4$, level $N$, and $χ$ a Dirichlet character mod $Q$ with $(Q,N)=1$, we establish a new hybrid subconvexity bound for $L(1/2 + it, f_χ)$, which improves upon all known hybrid bounds. This is done via amplification and taking advantage of a shifted convolution sum of two variables defined and analyzed in a recent paper of Hoffstein and Hulse.
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 study the algebraicity of the central critical values of twisted triple product $L$-functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result on Deligne's conjecture for certain automorphic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$. We also establish a relation for the cohomological periods under tw…
We prove that a cuspidal automorphic representation of GL(3) over any number field is determined by the quadratic twists of its central value. In the case of a non-Gelbart-Jacquet lift, the result is conditional on the analytic behavior of a certain Euler product. We deduce the nonvanishing of infinitely many quadratic twists of central values. This generalizes a result of Chinta and Diaconu that was valid only over the field of rational numbers and explored only for Gelbart-Jacquet lifts.
We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Appli…
We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local…
Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients $λ_{f}(n)$. Let $χ$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the following Weyl-type bound: for any $ε>0$, $$\sum_{|n| \ll N}λ_{f}(n)χ(n) \ll_{f,ε}N^{3/4 }p^{1/6}(pN)^ε.$$ We can see an improvement of the range $N > p^{3/4}$ to the range $N > p^{2/3}$ and we get a bound for $S_{f,χ}(N)$ without going into the $L$-function.
Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross-Zagier formula relating the value of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel-Jacobi imag…
Let $f_1,...,f_d$ be an orthogonal basis for the space of cusp forms of even weight $2k$ on $Γ_0(N)$. Let $L(f_i,s)$ and $L(f_i,χ,s)$ denote the $L$-function of $f_i$ and its twist by a Dirichlet character $χ$, respectively. In this note, we obtain a ``trace formula'' for the values $L(f_i,χ,m)\overline{L(f_i,n)}$ at integers $m$ and $n$ with $0<m,n<2k$ and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisly the value of the ratio $L(f,χ,m)/L(f,n)$ for a Hecke eigenform $f$.
Let $f$ be a Hecke-Maass cusp form for the full modular group and let $χ$ be a primitive Dirichlet character modulo a prime $q$. Let $s_0=σ_0+it_0$ with $\frac{1}{2}\leqσ_0<1$. We improve the error term for the first moment of $L(s_0,f\otimesχ)\overline{L(s_0,χ)}$ over the family of even primitive Dirichlet characters. As an application, we show that for any $t\in\mathbb{R}$, there exists a primitive Dirichlet character $χ$ modulo $q$ for which $L(1/2+it,f\otimesχ)L(1/2+it,χ)\neq0$ if the prime $q$ satisfies $q\gg (1+|t|)^{\frac{543}{25}+\varepsilon}$.
We prove an explicit integral representation -- involving the pullback of a suitable Siegel Eisenstein series -- for the twisted standard $L$-function associated to a holomorphic vector-valued Siegel cusp form of degree $n$ and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to $L$-functions of vector-valued Siegel cusp forms. The key new ingredient in our method is a novel choice of local vectors at the archimedean place which allows us to exactly compute the archimedean local…
