Related papers: New bounds for automorphic L-functions

The classical Kloosterman sums give rise to a Galois representation of the function field unramfied outside 0 and $\infty$. We study the local monodromy of this representation at $\infty$ using $l$-adic method based on the work of Deligne…

Let $f$ be a Hecke cusp form of weight $k$ for the full modular group, and let $\{\lambda_f(n)\}_{n\geq 1}$ be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of $\lambda_f(n)$, we…

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$,…

In this paper, we focus on the strong subconvexity bounds for triple product L-functions in the cubic level aspect. Our proof on the Weyl-type bound synthesizes techniques from classical analytic number theory with methods in automorphic…

Let $f$ be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus and denote by $\lambda_f(n)$ its $n$-th Hecke eigenvalue. Let $$ r(n)=\#\left\{(n_1,n_2)\in \mathbb{Z}^2:n_1^2+n_2^2=n\right\}. $$ In this paper, we…

We obtain a spectral decomposition of shifted convolution sums in Hecke eigenvalues of holomorphic or Maass cusp forms.

The authors study the central values of L-functions in certain families; in particular they bound the sum of the cubes of these values.Contents:

We prove for L-function attached to an automorphic cusp form for the Hecke congruence group $\Gamma_0(D)$, which is also an eigenfunction of all the Hecke operators, that a positive proportion of its non-trivial zeros lie on the critical…

We construct a $p$-adic $L$-function for $P$-ordinary Hida families of cuspidal automorphic representations on a unitary group $G$. The main new idea of our work is to incorporate the theory of Schneider-Zink types for the Levi quotient of…

We prove a generalization to the totally real field case of the Waldspurger's formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on…

We explicitly construct cusp forms on the orthogonal group of signature $(1,8n+1)$ for an arbitrary natural number $n$ as liftings from Maass cusp forms of level one. In our previous works, the fundamental tool to show the automorphy of the…

The infinity symmetric power $L$-functions play a fundamental role in Wan's groundbreaking work on Dwork's conjecture[16]. Building upon this foundation, Haessig[8] established the $p$-adic estimates for these $L$-functions in the case of…

Let $f$ be a Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $1/4+\mu_f^2$, $\mu_f>0$. Let $g$ be an arbitrary but fixed holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$. In this paper, we establish the following…

In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic…

Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a…

Given unitary automorphic cuspidal representations $\pi$ and $\pi'$ 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…

In this paper, we estimate the shifted convolution sum \[\sum_{n\geqslant1}\lambda_1(1,n)\lambda_2(n+h)V\Big(\frac{n}{X}\Big),\] where $V$ is a smooth function with support in $[1,2]$, $1\leqslant|h|\leqslant X$, $\lambda_1(1,n)$ and…

In this paper, we consider M\"obius functions associated with two types of $L$-functions: Rankin-Selberg $L$-functions of symmetric powers of distinct holomorphic cusp forms and $L$-functions of Maass cusp forms. We show that these M\"obius…

In this paper we recall the method of Greenberg and Stevens to calculate derivatives of p-adic L-functions using deformations of Galois representation and we apply it to the symmetric square of a modular form Steinberg at p. Under certain…

Let $f$ be a primitive Maass cusp form for a congruence subgroup $\Gamma_0(D) \subset $ SL($2,\mathbb{Z}$) and $\lambda_f(n)$ its $n$-th Fourier coefficient. In this paper it is shown that with knowledge of only finitely many $\lambda_f(n)$…