The twisted symmetric square $L$-function of $GL(r)$

Bounds for twisted symmetric square $L$-functions

Let $f\in S_k(N,ψ)$ be a newform, and let $χ$ be a primitive character of conductor $q^{\ell}$. Assume that $q$ is a prime and $\ell>1$. In this paper we describe a method to establish convexity breaking bounds of the form $$ L(\tfrac{1}{2},\Sym f\otimesχ)\ll_{f,\varepsilon} q^{3/4\ell-δ_{\ell}+\varepsilon} $$ for some $δ_{\ell}>0$ and any $\varepsilon>0$. In particular, for $\ell=3$ we show that the bound holds with $δ_{\ell}=1/4$.

Local Langlands correspondence for the twisted exterior and symmetric square $ε$-factors of $\textrm{GL}_n$

Let $F$ be a non-Archimedean local field. Let $\mathcal{A}_n(F)$ be the set of equivalence classes of irreducible admissible representations of $\textrm{GL}_n(F)$, and $\mathcal{G}_n(F)$ be the set of equivalence classes of n-dimensional Frobenius semisimple Weil-Deligne representations of $W'_F$. The local Langlands correspondence(LLC) establishes the reciprocity maps $\textrm{Rec}_{n,F}: \mathcal{A}_n(F)\longrightarrow \mathcal{G}_n(F)$ , satisfying some nice properties. An important invariant under this correspondence is the L- and $ε$-factors. This is also expected to be true under paralle…

Square integrability of representations on p-adic symmetric spaces

A symmetric space analogue of Casselman's criterion for square integrability of representations of a p-adic group is established. It is described in terms of exponents of Jacquet modules along parabolic subgroups associated to the symmetric space.

$\mathrm{GL}_n$ 的尖点表示的外平方 gamma 因子：简单超尖点表示

Exterior square gamma factors for cuspidal representations of $\mathrm{GL}_n$: simple supercuspidal representations

本文计算了简单超尖点表示的局部扭曲外平方 gamma 因子，并据此证明了该类表示的局部逆定理。

关于扭曲周期函数与 Dirichlet L-函数在临界线上加权均方均值的注记

On twisted period functions and Moments of a weighted mean square of Dirichlet L-functions on the critical line

本文将周期函数等工具推广至任意本原特征的 Dirichlet L-函数，给出了加权均方测度矩的闭式表达式，完整刻画了临界线上的模性质，并导出了适用于所有本原特征的无限个非平凡正性条件。

Twisted moments of GL(3)xGL(2) L-functions

We compute an asymptotic formula for the twisted moment of GL(3)xGL(2) L-functions and their derivatives. As an application we prove that symmetric-square lifts of GL(2) Maass forms are uniquely determined by the central values of the derivatives of GL(3)xGL(2) L-functions.

The first moment of Maass form symmetric square L-functions

We prove an asymptotic formula for the twisted first moment of Maass form symmetric square L-functions on the critical line and at the critical point. The error term is estimated uniformly with respect to all parameters.

Twisted arithmetic Siegel Weil formula on X0(N)

In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (Δ, r) where Δ>0 and Δ\equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function ϕ_{Δ, r}(τ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of ϕ_{Δ, r}(τ), along the way, we also identify the arithmetic pairing \langle ϕ_{Δ, r}(τ),\widehatω_N \rangle with special value of some Eisenstein series, where \widehatω_N is a normalized metric Hodge line bundle.

原始二次Dirichlet $L$-函数中心值的扭曲一阶矩

Twisted first moment of central values of primitive quadratic Dirichlet $L$-functions

本文利用双Dirichlet级数与递归论证求值原始二次Dirichlet L-函数中心值的扭曲一阶矩，得到误差项大小为主项平方根的渐近公式。

Galois表示与Kloosterman和的扭曲矩的模性

Galois representations and modularity for twisted moments of Kloosterman sums

本文研究Kloosterman和的二次扭曲对称幂矩，构造其有理数域上的动机，并确定低次数情形的模形式。

The mean-square of Dirichlet L-functions

We verify the conjecture of [CFKRS] for the mean square near the critical point of Dirichlet L-functions for a composite modulus q. We also prove a kind of reciprocity formula when the second moment for a prime modulus is twisted by a character evaluated at a different prime.

On the spectral large sieve inequality for symmetric-squares

We improve on the spectral large sieve inequality for symmetric-squares. We also prove a lower bound showing that the most optimistic upper bound is not true for this family.

Spectral decomposition formula and moments of symmetric square L-functions

We prove a spectral decomposition formula for averages of Zagier L-series in terms of moments of symmetric square L-functions associated to Maass and holomorphic cusp forms of levels 4, 16, 64.

Iwasawa theory for the symmetric square of a modular form

We construct a compatible family of global cohomology classes (an Euler system) for the symmetric square of a modular form, and apply this to bounding Selmer groups of the symmetric square Galois representation and its twists.

Bounds for twisted symmetric square $L$-functions via half-integral weight periods

We establish the first moment bound $$ \sum_φ L(φ\otimes φ\otimes Ψ, \tfrac{1}{2}) \ll_{\varepsilon} p^{5/4+\varepsilon} $$ for triple product $L$-functions, where $Ψ$ is a fixed Hecke-Maass form on $\operatorname{SL}_2(\mathbb{Z})$ and $φ$ runs over the Hecke-Maass newforms on $Γ_0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows …

Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions

We study the second moment of Dirichlet $L$-functions to a large prime modulus $q$ twisted by the square of an arbitrary Dirichlet polynomial. We break the $\frac{1}{2}$-barrier in this problem, and obtain an asymptotic formula provided that the length of the Dirichlet polynomial is less than $q^{51/101} = q^{1/2 +1/202}$. As an application, we obtain an upper bound of the correct order of magnitude for the third moment of Dirichlet $L$-functions. We give further results when the coefficients of the Dirichlet polynomial are more specialized.

The second moment of twisted modular L-functions

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…