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

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 $ε> 0.$

Subconvexity and equidistribution of Heegner points in the level aspect

Let q be a prime and -D < -4 be an odd fundamental discriminant such that q splits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q and h the weight one theta series of level D corresponding to an ideal class group character of Q(\sqrt{-D}), we establish a hybrid subconvexity bound for L(f \times h,s) at the central point when q = D^η for 0 < η< 1. With this circle of ideas, we show that the Heegner points of level q and discriminant D become equidistributed, in a natural sense, as q, D become large with q < D^{1/20-\varepsilon}. Our approach to these problems is connected t…

$\rm GL_2 \times GL_2$ $L$-函数在深度方面的次凸性

Subconvexity for $\rm GL_2 \times GL_2$ $L$-functions in the depth aspect

本文针对 $\rm GL_2 \times GL_2$ $L$-函数在深度方面建立次凸性界，通过 DFI 圆法结合导子降低、Voronoi 求和及 $p$-进 van der Corput 方法估计特征和，得到 $q^{9/10+\varepsilon}$ 的界。

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

Let $f$ be a newform, and let $χ$ be a primitive character of conductor $q^{\ell}$. Assume that $q$ is an odd prime. In this paper we prove the subconvex bound $$ L(\t1/2,\Sym f\otimesχ)\ll_{f,q,\varepsilon} q^{3\ell(1/4-1/36+\varepsilon)} $$ for any $\varepsilon>0$. This can be compared with the recently established $t$-aspect subconvexity of the symmetric square $L$-functions.

Twists of $GL(3)$ $L$-functions

Let $π$ be a $SL(3,\mathbb Z)$ Hecke-Maass cusp form, and let $χ$ be a primitive Dirichlet character modulo $M$, which we assume to be prime. In this note we revisit the subconvexity problem addressed in `The circle method and bounds for $L$-functions IV' and establish the following unconditional bound \begin{align*} L\left(\tfrac{1}{2},π\otimesχ\right)\ll M^{3/4-1/308+\varepsilon}. \end{align*}

三次层面中三重积 L-函数的 Weyl 界

The Weyl bound for triple product L-functions in the cubic level

本文在三次层面方面研究三重积 L-函数的强次凸性界，通过结合 Petersson 迹公式、Voronoi 求和、Jutila 圆法、Kuznetsov 迹公式和谱大筛不等式等工具证明了 Weyl 型界。

Bounds for $GL_3$ $L$-functions in depth aspect

Let $f$ be a Hecke-Maass cusp form for $SL_3(\mathbb{Z})$ and $χ$ a primitive Dirichlet character of prime power conductor $\mathfrak{q}=p^κ$ with $p$ prime and $κ\geq 10$. We prove a subconvexity bound $$ L\left(\frac{1}{2},π\otimes χ\right)\ll_{p,π,\varepsilon} \mathfrak{q}^{3/4-3/40+\varepsilon} $$ for any $\varepsilon>0$, where the dependence of the implied constant on $p$ is explicit and polynomial. We obtain this result by applying the circle method of Kloosterman's version, summation formulas of Poisson and Voronoi's type and a conductor lowering mechanism introduced by Munshi [14]. The…

三次二次型的定量强近似III

Quantitative strong approximation for ternary quadratic forms III

本文通过Hardy-Littlewood圆方法的δ变体，研究三次二次型的整数点计数问题，并将其与Brauer-Manin障碍联系起来。

族中 Rankin-Selberg L-函数的零点

Zeros of Rankin-Selberg $L$-functions in families

本文对固定 $π'$ 的 Rankin-Selberg $L$-函数族证明了首个无条件零点密度估计，并由此得到几乎全部成员的次凸性界、有效重数一的平均形式及正分布水平。

Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions

This paper gives $n$-dimensional analogues of the Apollonian circle packings in parts I and II. We work in the space $\sM_{\dd}^n$ of all $n$-dimensional oriented Descartes configurations parametrized in a coordinate system, ACC-coordinates, as those $(n+2) \times (n+2)$ real matrices $\bW$ with $\bW^T \bQ_{D,n} \bW = \bQ_{W,n}$ where $Q_{D,n} = x_1^2 +... + x_{n+2}^2 - \frac{1}{n}(x_1 +... + x_{n+2})^2$ is the $n$-dimensional Descartes quadratic form, $Q_{W,n} = -8x_1x_2 + 2x_3^2 + ... + 2x_{n+2}^2$, and $\bQ_{D,n}$ and $\bQ_{W,n}$ are their corresponding symmetric matrices. There are natural…