Subconvexity for twisted GL(3) L-functions

Uniform subconvexity and symmetry breaking reciprocity

A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the l-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level l and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.

Subconvexity for twisted $L$-functions on $\mathrm{GL}_3$ over the Gaussian number field

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…

GL₂×GL₂ L-函数扭曲的混合次凸性界

Hybrid subconvexity bounds for twists of $\rm GL_2\times\rm GL_2$ $L$-functions

本文针对模素数幂的本原Dirichlet特征χ所扭曲的GL₂×GL₂ Rankin–Selberg L-函数，在t和深度两个方面证明了混合次凸性界。

GL(2)×GL(1) 的中心 L 值平均、混合次凸性与同时非零

Average of Central L-values for GL(2)$\times$GL(1), Hybrid Subconvexity, and Simultaneous Nonvanishing

本文利用正则化相对迹公式研究数域上扭转 L 函数的二阶矩，得到 Hecke 与扭转 L 函数的混合次凸界，并应用于同时非零问题。

Hilbert模形式的中心L-值及其在次凸性中的应用

Averages of central L-values of Hilbert modular forms with an application to subconvexity

利用相对迹公式，得到了一类扭二次基变L-函数在固定权和水平的Hilbert模形式上平均的中心值的精确公式.我们将这些公式应用于这些L-函数的次凸性问题。我们还建立了由这些L值加权的Hecke特征值的等分布结果。

GL(3)×GL(2) L-函数扭曲的混合次凸性界

Hybrid subconvexity bounds for twists of $\rm GL (3)\times GL(2)$ $L$-functions

本文解决了由模素数 M 的原初 Dirichlet 特征扭曲的 GL(3)×GL(2) L-函数在 M 和 t 方面的混合次凸性问题，并改进了 GL(3) L-函数扭曲的混合次凸性界。

关于\(\mathrm{GL}(2)\)的对称谱互反性与一致次凸性

Symmetric Spectral Reciprocity for $\mathrm{GL}(2)$ and Uniform Subconvexity

构建\(\mathrm{GL}_2\)在数域\(F\)上Eisenstein级数的Petersson范数恒等式新解析正则化，导出\(\mathrm{GL}_2\)\(L\)-函数扭曲四阶矩的对称谱互反公式，获\(\mathrm{GL}_2/F\)新一致次凸性界等成果。

Central values of twisted base change $L$-functions associated to Hilbert modular forms

We use relative trace formula to prove a non-vanishing result and a subconvexity result for the twisted base change $L$-functions associated to Hilbert modular forms whose local components at ramified places are some supercuspidal representations. This generalizes the work of Feigon and Whitehouse.

强混合次凸性 for 自对偶 GL_3 L-函数

Strong Hybrid Subconvexity for Twisted Selfdual $\mathrm{GL}_3$ $L$-Functions

本文研究自对偶 GL_3 基本形式的 L-函数在 q 和 t 方面的强混合次凸界，并探讨 GL_3 × GL_2 Rankin-Selberg L-函数中心值的类似界。

Twisted moments of L-functions and spectral reciprocity

A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.

$t$-aspect subconvexity for $GL(2) \times GL(2)$ $L$-function

In this paper we shall prove a subconvexity bound for $GL(2) \times GL(2)$ $L$-function in $t$-aspect by using a $GL(1)$ circle method.

Subconvexity for $\rm{GL}(3)$ L-functions

We improve on the subconvexity bound for self-dual $\rm{GL}(3)$ $L$-functions in the $t$-aspect. Previous results were obtained by Li and by Mckee, Sun and Ye.

$SL(3,Z)$的自同构$L$ -函数的一个次凸性界

A Subconvexity Bound for Automorphic $L$-functions for $SL(3,Z)$

本文给出了与$SL(3,Z)$的Maass形式相关联的Godement-Jacquet $L$ -函数的条件次凸性界.

Subconvexity for $L$-Functions on $\mathrm{GL}_3$ over Number Fields

In this paper, over an arbitrary number field, we prove subconvexity bounds for self-dual $\mathrm{GL}_3$ $L$-functions in the $t$-aspect and for self-dual $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions in the $\mathrm{GL}_2$ Archimedean aspect.

短区间k自由指数和的次凸性

Subconvexity of Short $k$-Free Exponential Sums

该研究针对短区间k自由整数上的指数和，证明了其s阶矩的紧界，由此得到带Möbius扭转的指数和的$L^1$均值下界，并指出改进相关$\ell^2$估计可进一步提升结果。

两层Rankin-Selberg L函数的次凸性

Subconvexity for the Rankin-Selberg L-function in both levels

本文得到了Rankin-Selberg L函数在两个层次上的一个次凸性结果。该结果的新特点是将Duke-Friedlander-Iwaniec的放大方法应用于双Peterson-Kuznetsov迹公式。当迹公式覆盖整个$GL_2$谱时，这个次凸性结果是Rankin-Selberg L函数中Hecke特征形选取的无条件条件.

Subconvexity for $GL(3) \times GL(2)$ $L$-functions in the depth aspect

In this article, we will prove subconvex bounds for $GL(3) \times GL(2)$ $L$-functions in the depth aspect.

Hybrid Bounds on Twisted L-Functions Associated to Modular Forms

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.

Subconvexity for L-functions of non-spherical cusp forms on GL(3)

Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll T^{3/4 - 1/140000}$ is proved.

半整权$L$ -函数的次凸性

Subconvexity for Half Integral Weight $L$-functions

我们证明了$L(s,f,χ)$在导体方面的一个次凸性界，其中$f$是半整数权模形式。这个$L$ -函数有解析延拓和函数方程，但没有欧拉积.由于缺乏欧拉积，人们不期望半整数权模形式的黎曼假设。然而，人们可以推测一个林德洛夫型假设，目前的次凸性结果是对它的真理的指示。

关于GL上L函数的次凸性问题（3）

On the subconvexity problem for L-functions on GL(3)

设f是群SL（3，Z）的尖点形式，其参数为μ，L-函数为L（s，f）.如果μ处于一般位置，即远离Weyl腔壁和远离自对偶形式，我们证明了次凸界L（1/2，f）<<||亩||^{3/4 - 1/120000}.

扭曲对称方形$L$函数的边界- III

Bounds for twisted symmetric square $L$-functions - III

设$f$是一个新形式，$χ$是导体$q^{\ell}$的一个本原特征标.假设$q$是一个奇素数。本文证明了任意$\varepsilon>0$的次凸界$$ L(\t1/2,\Sym f\otimesχ)\ll_{f,q,\varepsilon} q^{3\ell(1/4-1/36+\varepsilon)} $$.这可以与最近建立的对称平方$L$ -函数的$t$ -方面次凸性进行比较。

Subconvexity of Shintani's zeta function

Enumerating integral orbits in prehomogeneous vector spaces plays an important role in arithmetic statistics. We describe a method of proving subconvexity of the zeta function enumerating the integral orbits, illustrated by proving a subconvex estimate for the Shintani $ζ$ function enumerating class numbers of binary cubic forms.

Hybrid subconvexity and the partition function

We give an upper bound for the error term in the Hardy-Ramanujan-Rademacher formula for the partition function. The main input is a new hybrid subconvexity bound for the central value $L(\tfrac 12,f\times (\tfrac{q}{\cdot}))$ in the $q$ and spectral parameter aspects, where $f$ is a Hecke-Maass cusp form for $Γ_0(N)$ and $q$ is a fundamental discriminant.

三重积L函数在水平方向上的次凸界

The subconvexity bound for triple product L-function in level aspect

本文在水平方面推广了Venkatesh和Woodbury关于三重积L函数的次凸性界的工作，允许联合分歧、高阶分歧、一般酉中心特征标和局部扩张因子的一般特殊值.特别是，我们推导出一个很好的一般公式的局部积分时，表示之一有足够高的水平比其他两个。

Epstein zeta-functions, subconvexity, and the purity conjecture

Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.

$L$ -函数的圆方法和界- III：$t$ -GL（3）$L$ -函数的方面次凸性

The circle method and bounds for $L$-functions - III: $t$-aspect subconvexity for GL(3) $L$-functions

设$π$是$SL(3,\mathbb Z)$的Hecke-Maass尖点形式。本文将证明下列次凸界$$ L(\tfrac{1}{2}+it,π)\ll_{π,\varepsilon} (1+|t|)^{3/4-1/16+\varepsilon}. $$

Rankin-Selberg $L$ -函数的水平方面次凸性

Level Aspect Subconvexity For Rankin-Selberg $L$-functions

设$M$是一个无平方数的整数，且$P$是一个不整除$M$的素数，使得$P \sim M^η$与$0<η<2/21$。当$f$和$g$分别是水平$P$和$M$的两个本原全纯尖点形式时，证明了$L(\tfrac{1}{2}, f \otimes g)$的次凸性界.这些界限是通过未放大的二阶矩方法实现的。

关于$GL(3)\times GL(2)$ $L$ -函数的次凸性问题

On the subconvexity problem for $GL(3)\times GL(2)$ $L$-functions

将$g$修正为$SL_3(\mathbb{Z})$的自对偶Hecke-Maass形式。设$f$是素水平$q$和固定权的全纯新形式.在$f$的傅里叶系数的短平方和的下界的条件下，我们证明了在$q$方面$L(s, g\times f)$在中心点的次凸性界。

Adelic Voronoi Summation and Subconvexity for GL(2) L-Functions in the depth Aspect

In this paper we establish a very flexible and explicit Voronoi summation formula. This is then used to prove an almost Weyl strength subconvexity result for automorphic $L$-functions of degree two in the depth aspect. That is, looking at twists by characters of prime power conductor. This is the natural $p$-adic analogue to the well studied $t$-aspect.

Strong subconvexity for self-dual $\mathrm{GL} (3)$ $L$-functions

In this paper, we prove strong subconvexity bounds for self-dual $\mathrm{GL}(3)$ $L$-functions in the $t$-aspect and for $\mathrm{GL}(3)\times\mathrm{GL}(2)$ $L$-functions in the $\mathrm{GL}(2)$-spectral aspect. The bounds are strong in the sense that they are the natural limit of the moment method pioneered by Xiaoqing Li, modulo current knowledge on estimate for the second moment of $\rm GL(3)$ $L$-functions on the critical line.

黎曼ζ函数的次凸性与除数问题

Subconvexity for the Riemann zeta-function and the divisor problem

给出了黎曼ζ函数的经典次凸性界$ζ(1/2+it) \ll_εt^{1/6+ε}$的一个简单证明，并讨论了用更精细的技巧进行估计的问题.分析了Dirichlet除数问题与$|ζ(1/2+it)|$均方之间的联系。

Subconvexity for $GL_{3}(R)$ $L$-Functions: The Key Identity via Integral Representations

We study the subconvexity problem for $GL_{3}(R)$ $L$-functions in the t-aspect using integral representations by combining techniques employed by Michel-Venkatesh in their study of the corresponding problem for $GL_{2}$ with ideas from recent works of Munshi, Holowinsky-Nelson and Lin. Our main objective is to give - from the perspective of integral representations of $L$-functions and automorphic representation theory - a possible explanation of the origin of the "key identity" arising in the latter series of works.

水平方面标准 L 函数的次凸性界

The subconvexity bound for standard L-function in level aspect

针对 GL_n 酉尖点自守表示的标准 L 函数，本文在可变有限大导子位集合且局部参数满足一致增长条件时，建立了新的次凸性界。

Weak subconvexity without a Ramanujan hypothesis

We describe a new method to obtain weak subconvexity bounds for $L$-functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic $L$-functions and (with mild restrictions) the Rankin-Selberg $L$-functions attached to two automorphic representations. The proof relies on a new unconditional log-free zero density estimate for Rankin-Selberg $L$-functions.

$t$-aspect subconvexity for $GL(2)$ $L$-functions

Let $f$ be a holomorphic cusp form for $SL_2(\mathbb{Z})$ of weight $k>1$. In these notes, we follow Munshi to prove the Burgess bound $$ L(1/2+it,f)\ll_{f,\varepsilon} (1+|t|)^{1/2-1/8+\varepsilon}. $$

Subconvexity for symmetric square $L$-functions

Let $f$ be a holomorphic modular form of prime level $p$ and trivial nebentypus. We show that there exists a computable $δ>0$, such that $$ L\left(\tfrac{1}{2},\mathrm{Sym}^2 f\right)\ll p^{\tfrac{1}{2}-δ}, $$ with the implied constant depending only on $δ$ and the weight of $f$.

Subconvexity for a double Dirichlet series and non-vanishing of $L$-functions

We study a double Dirichlet series of the form $\sum_{d}L(s,χ_{d}χ)χ'(d)d^{-w}$, where $χ$ and $χ'$ are quadratic Dirichlet characters with prime conductors $N$ and $M$ respectively. A functional equation group isomorphic to the dihedral group of order 6 continues the function meromorphically to $\mathbb{C}^{2}$. A convexity bound at the central point is established to be $(MN)^{3/8+\varepsilon}$ and a subconvexity bound of $(MN(M+N))^{1/6+\varepsilon}$ is proven. The developed theory is used to prove an upper bound for the smallest positive integer $d$ such that $L(1/2,χ_{dN})$ does not vanis…

GL（2）上自守L函数的深度次凸性界

Subconvexity bounds in depth-aspect for automorphic L-functions on GL(2)

本文从谱恒等式出发，得到了任意数域上GL（2）的自守L函数族的二阶积分矩的带误差项的渐近性，并根据在固定有限位置的任意分支的特征标的扭转得到了这些渐近性.误差项中的功率节省在该非阿基米德位置处打破了凸性。

高斯域上双重狄利克雷级数的次凸性

Subconvexity of a double Dirichlet series over the Gaussian field

该研究针对高斯域上与二次赫克L-函数相关的双重狄利克雷级数，建立了其次凸性界，推进了数论领域相关问题的研究。

Subconvexity for modular form L-functions in the t aspect

Modifying a method of Jutila, we prove a t aspect subconvexity estimate for L-functions associated to primitive holomorphic cusp forms of arbitrary level that is of comparable strength to Good's bound for the full modular group, thus resolving a problem that has been open for 35 years. A key innovation in our proof is a general form of Voronoi summation that applies to all fractions, even when the level is not squarefree.

Subconvexity for $GL(3)\times GL(2)$ $L$-functions in $t$-aspect

Let $π$ be a Hecke-Maass cusp form for $SL(3,\mathbb Z)$ and $f$ be a holomorphic (or Maass) Hecke form for $SL(2,\mathbb{Z})$. In this paper we prove the following subconvex bound $$ L\left(\tfrac{1}{2}+it,π\times f\right)\ll_{π,f,\varepsilon} (1+|t|)^{\frac{3}{2}-\frac{1}{42}+\varepsilon}. $$

Hybrid subconvexity bounds for twists of $\rm GL(3)$ $L$-functions

Let $π$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form and $χ$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^k$ with $p$ prime. In this paper we will prove the following subconvexity bound $$ L\left(\frac{1}{2}+it,π\times χ\right)\ll_{π,\varepsilon} p^{3/4}\big(\mathfrak{q}(1+|t|)\big)^{3/4-3/40+\varepsilon}, $$ for any $\varepsilon >0$ and $t \in \mathbb{R}$.

Subconvexity bound for $\mathrm{GL(3)} \times \mathrm{GL(2)}$ $L$-functions in $\mathrm{GL(2)}$ spectral aspect

Let $π$ be a Hecke-Maass cusp form for $\mathrm{SL(3, \mathbb{Z})}$ and $f$ be a holomorphic cusp form for $\mathrm{SL(2,\mathbb{Z})}$ of weight $k$ or a Hecke-Maass cusp form corresponding to the Laplacian eigenvalue $1/4+k^2$, $k\geq 1$, for $\mathrm{SL(2,\mathbb{Z})}$. In this paper, we prove the following subconvexity bound \begin{align*} L\left({1}/{2},\ π\times f\right) \ll_{π,\,ε} k^{\frac{3}{2}-\frac{1}{51}+ε}. \end{align*}

Subconvexity for Symmetric Square L-functions off-centre

Let $p$ be a prime. Let $f$ be a holomorphic modular form of level $p$ with trivial nebentypus. We prove the bound $L\left(\text{sym}^2f, \frac{1}{2} + it\right) \ll_{f,ε} p^{1/2+ε}t^{3/4-1/12 + ε}$. This bound is subconvex in the $t$-aspect and almost convex in the level aspect simultaneously.

t-Aspect Subconvexity Bound for $GL(2)$ L-functions

Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it \right) \ll_{f, ε} \left( 1 + |t|\right)^{1/3 + ε}, \] for any $ε> 0.$

Rankin-Selberg 与三重积 L-函数的次凸性问题

The subconvexity problem for Rankin-Selberg and triple product L-functions

本文扩展 Michel-Venkatesh 方法，研究 Rankin-Selberg 与三重积 L-函数的次凸性问题，在允许联合分歧和导子下降的条件下验证局部猜想，获得新的次凸界。

次凸性界对 zeta 矩的含义

Implications of subconvexity bounds for the moments of zeta

本文利用临界带右半部分的泛函分析，探讨了 Riemann zeta 函数矩的上界与次凸性界之间相反方向的影响，并刻画了矩行为的潜在转变。

Subconvexity bound for $GL(2)$ L-functions: \lowercase{t}-aspect

Let $f $ be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L \left( \frac{1}{2} + it, f \right) \ll_{f, ε} \left( 2 + |t|\right)^{1/3 + ε}, \] for any $ε> 0.$

$L$ -函数- IV的圆方法和界：$GL(3)$ $L$ -函数- B的扭曲的次凸性

The circle method and bounds for $L$-functions - IV: Subconvexity for twists of $GL(3)$ $L$-functions - B

设$π$是满足Ramanujan猜想和Selberg-Ramanujan猜想的$SL(3,\mathbb Z)$ Hecke-Maass尖点形式，$χ$是模$M$的本原Dirichlet特征标，为简单起见，我们假设它是素的.我们将证明以下次凸界$$ L\left(\tfrac{1}{2},π\otimesχ\right)\ll_{π,\varepsilon} M^{\frac{3}{4}-\frac{1}{1612}+\varepsilon}. $$

Rankin-Selberg L函数的二阶矩与混合次凸界

The Second Moment of Rankin-Selberg L-function and Hybrid Subconvexity Bound

设$M,N$是互质无平方整数。设$f$是水平$N$的全纯尖点形式，$g$是水平$M$的全纯尖点形式或马氏尖点形式.利用一个大筛不等式，我们建立了一个形式为$\sum_{g}\left|L^{(j)}\left(1/2+it,f \otimes g\right)\right|^2 \ll_t M+M^{2/3-β}N^{4/3}$的界，其中$β\approx 1/500$ .因此，我们得到了次凸界$L^{(j)}\left(1/2+it,f \otimes g\right)\ll (MN)^{1/2 - α}$的任何$N<M$满足上述条件，而无需使用放大方法。此外，当两种形式都是全纯的时，利用对称性，我们建立了一个满值域的水平方面混合次凸性界。

三重积 $L$-函数的谱互反性与混合次凸界

Spectral Reciprocity and Hybrid Subconvexity Bound for triple product $L$-functions

本文利用谱互反公式并结合已有矩估计，研究三重积 $L$-函数在层数方面的次凸性，给出允许联合分歧与导子下降的新显式混合次凸界。

Improved subconvexity bounds for GL(2)xGL(3) and GL(3) L-functions by weighted stationary phase

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the eigenvalue aspect is proved for the central value at $s=1/2$ of the Rankin-Selberg $L$-function $L(s,f\times u)$. Meanwhile, a subconvexity bound $O\big((1+|t|)^{2/3+\varepsilon}\big)$ in the $t$ aspect is proved for $L(1/2+it,f)$. These bounds improved corresponding subconvexity bounds proved by Xiaoqing Li (Annals of Mathematics, 2011). The main technique in …

圆方法与$L$ -函数的界- II：GL（3）$L$ -函数的扭的次凸性

The circle method and bounds for $L$-functions - II: Subconvexity for twists of GL(3) $L$-functions

设$π$是一个$SL(3,\mathbb Z)$自守形式。设$χ=χ_1χ_2$是一个Dirichlet特征标，其基元为$χ_i$模$M_i$ .假设$M_1$、$M_2$是素数，使得$\sqrt{M_2}M^{4δ}<M_1<M_2M^{-3δ}$，其中$M=M_1M_2$和$0<δ<1/28$。本文将证明如下的次凸界$$ L(\t1/2,π\otimesχ)\ll_{π,\varepsilon} M^{3/4}-δ+\varepsilon}. $$