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

$t$-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 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.

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。

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 $$ L\left(\tfrac{1}{2}+it,π\times f\right)\ll_{π,f,\varepsilon} (1+|t|)^{\frac{3}{2}-\frac{1}{42}+\varepsilon}. $$

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.

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*}

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 $ε> 0.$

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

Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $π$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $μ$ in generic position i.e. away from Weyl chamber walls and away from self dual forms. We study an amplified second moment $\sum_{j} A(π_j)|L(1/2,π_j\times f)|^2$ and deduce the subconvexity bound \begin{equation*} L(1/2,π\times f)\ll_{f,ε} \|μ\|^{3/2-1/2022+ε}. \end{equation*} As a corollary, when $f=E(z,1/2)$, we also obtain the subconvexity bound \begin{equation*} L(1/2,π)\ll_ε \|μ\|^{3/4-1/4044+ε}. \end{equation*}

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.

Hybrid Level Aspect Subconvexity for $GL(2)\times GL(1)$ Rankin-Selberg $L$-Functions

Let $M$ be a squarefree positive integer and $P$ a prime number coprime to $M$ such that $P\sim M^η$ with $0 < η< 2/5$. We simplify the proof of subconvexity bounds for $L(\frac{1}{2},f\otimesχ)$ when $f$ is a primitive holomorphic cusp form of level $P$ and $χ$ is a primitive Dirichlet character modulo $M$. These bounds are attained through an unamplified second moment method using a modified version of the delta method due to R. Munshi. The technique is similar to that used by Duke-Friedlander-Iwaniec save for the modification of the delta method.

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.

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.

Subconvexity for twisted GL(3) L-functions

Using the circle method, we obtain subconvex bounds for GL(3) L-functions twisted by a character modulo a prime p, hybrid in the t and p-aspects.

Sup-norms of eigenfunctions in the level aspect for compact arithmetic surfaces, II: newforms and subconvexity

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$ where $D$ is an indefinite quaternion division algebra over $\mathbb{Q}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times θ_χ)$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $θ_χ$ is an (essentially fixed) automorphic form on $\mathrm{GL}_2$ obtained as the theta lift of a Hecke character $χ$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-na…

$GL(2)\ imes GL(2)$ $L$-函数在谱方面的均匀次凸性界

Uniform subconvexity bounds for $GL(2)\times GL(2)$ $L$-functions in the spectral aspect

本文研究GL(2)×GL(2) L函数在谱方面的二次矩，得到均匀次凸性界，并应用于二面体Maass新形式的QUE收缩结果。

First moment of Rankin-Selberg central L-values and subconvexity in the level aspect

Let $1\le N<M$ with $N$ and $M$ coprime and square-free. Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\times g) $ where $f\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and trivial nebentypus and $g$ is a given form of level $M$. As a result, we recover the bound $ L(1/2,f\times g) \ll_\varepsilon (N + \sqrt{M}) N^\varepsilon M^\varepsilon $ when $g$ is dihedral. The first moment method also applies to the special derivative $L'(1/2,f\times g)$ under the assumption that it is non-negative for all $f\in S^*_k(N)$.

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(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。

虚二次域上的混合Weyl次凸性

Hybrid Weyl Subconvexity over Imaginary Quadratic Fields

该研究在虚二次域上建立GL₂贝塞尔δ方法，证明GL₂×GL₁扭曲L函数的混合Weyl型次凸性界，推动数论中L函数次凸性问题的进展。

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.

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.

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

Fix $g$ a self-dual Hecke-Maass form for $SL_3(\mathbb{Z})$. Let $f$ be a holomorphic newform of prime level $q$ and fixed weight. Conditional on a lower bound for a short sum of squares of Fourier coefficients of $f$, we prove a subconvexity bound in the $q$ aspect for $L(s, g\times f)$ at the central point.

GL(3)×GL(2) L-函数的混合次凸界：t 与 level 方面

Hybrid subconvexity bound for $GL(3)\times GL(2)$ $L$-functions: t and level aspect

本文利用 delta 符号方法，在 GL(2) 形式的 level 与 t-方面的混合设置下，证明了 GL(3)×GL(2) 六次 Rankin-Selberg L-函数中心值的非平凡次凸估计。

GL_3数域上的子凸性问题：扭转方面

Subconvexity Problem on $\operatorname{GL}_3$ over number fields: the twist aspect

该研究通过分析GL_3数域上的自守表示与特征标的扭合L函数，证明了在特定条件下，该函数的值随素理想范数的增长而递减。

Subconvexity for Half Integral Weight $L$-functions

We prove a subconvexity bound in the conductor aspect for $L(s,f,χ)$ where $f$ is a half integer weight modular form. This $L$-function has analytic continuation and functional equation, but no Euler product. Due to the lack of an Euler product, one does not expect a Riemann hypothesis for half integer weight modular forms. Nevertheless one may speculate a Lindelof-type hypothesis, and this current subconvexity result is an indication towards its truth.

Sparse equidistribution problems, period bounds, and subconvexity

We introduce a ``geometric'' method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier coefficients of automorphic forms.

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 …

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…

Relative trace formulas and subconvexity estimates of L-functions for Hilbert modular forms

We elaborate an explicit version of the relative trace formula on $\PGL(2)$ over a totally real number field for the toral periods of Hilbert cusp forms along the diagonal split torus. As an application, we prove (i) a spectral equidistribution result in the level aspect for Satake parameters of holomorphic Hilbert cusp forms weighted by central $L$-values, and (ii) a bound of quadratic base change $L$-functions for Hilbert cusp forms with a subconvex exponent in the weight aspect.

$\rm GL_2 \times GL_2$ $L$-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。

对称平方L-函数在水平方向的次凸性问题

The subconvexity problem for symmetric square $L$-functions in level aspect

本文针对GL₂(Q)尖点自守表示的对称平方L-函数，在素数p处有指定局部分歧的条件下，利用δ符号法等工具，首次得到GL₃(Q)自守表示L-函数的水平方向次凸界。

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.

The subconvexity problem for $\GL_{2}$

Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of $\GL_{1}$ and $\GL_{2}$ automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino--Ikeda.

GL(2)×GL(2) L-函数的水平次凸界

Level aspect subconvexity for $\textrm{GL(2)}\times \textrm{GL(2)}$ $\textrm{L}$-functions

该论文证明了 GL(2)×GL(2) L-函数在素数水平下的次凸界，改进了 Harcos 和 Michel 的结果，方法基于 Kloosterman 分数双线性形式的非平凡界。

Bounds for $GL(3)\times GL(2)$ $L$-functions and GL(3) $L$-functions

In this paper, we will give the subconvexity bounds for self dual GL(3) $L-$functions in the $t$ aspect as well as subconvexity bounds for self dual $GL(3)\times GL(2)$ $L-$functions in the GL(2) spectral aspect.

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

Let $M,N$ be coprime square-free integers. Let $f$ be a holomorphic cusp form of level $N$ and $g$ be either a holomorphic or a Maaß form with level $M$. Using a large sieve inequality, we establish a bound of the form $\sum_{g}\left|L^{(j)}\left(1/2+it,f \otimes g\right)\right|^2 \ll_t M+M^{2/3-β}N^{4/3}$ where $β\approx 1/500$. As a consequence, we obtain subconvexity bounds for $L^{(j)}\left(1/2+it,f \otimes g\right)\ll (MN)^{1/2 - α}$ for any $N<M$ satisfying the conditions above without using amplification methods. Moreover, by the symmetry, we establish a level aspect hybrid subconvexity…

Hybrid subconvexity bounds for $L \left(\tfrac{1}{2}, \text{Sym}^2 f \otimes g\right)$

Fix an integer $κ\geqslant 2$. Let $P$ be prime and let $k> κ$ be an even integer. For $f$ a holomorphic cusp form of weight $k$ and full level and $g$ a primitive holomorphic cusp form of weight $2 κ$ and level $P$, we prove hybrid subconvexity bounds for $L \left(\tfrac{1}{2}, \text{Sym}^2 f \otimes g\right)$ in the $k$ and $P$ aspects when $P^{\frac {13} {64} + δ} < k < P^{\frac 3 8 - δ}$ for any $0 < δ< \frac {11} {128}$. These bounds are achieved through a first moment method (with amplification when $P^{\frac {13} {64}} < k \leqslant P^{\frac 4 {13}}$).

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。

Subconvexity Bound for Hecke character $L$-Functions of Imaginary quadratic Number fields

Let $K=\mathbb{Q}(\sqrt{-D})$ be an imaginary number field, $(p)=\mathfrak{p}\mathfrak{p}'$ be a split odd prime and $ψ$ be a Hecke character of conductor $\mathfrak{p}$. Let $L(s,ψ)$ be the associated $L$-function. We prove the Burgess bound in $t$-aspect and a hybrid bound in conductor aspect, \begin{equation*} L(1/2+it,ψ)\ll_{D,\varepsilon} (1+|t|)^{3/8+\varepsilon}p^{1/8} \end{equation*} for $p\ll t$. In Appendix A, we present the ideas for an elementary proof of Voronoi summation formula for holomorphic cusp forms with CM and squarefree level. This is done by exploiting the lattice struct…

对称平方L函数的矩与混合次凸性

Moments and hybrid subconvexity for symmetric-square L-functions

本文研究对称平方L函数的二阶矩，在中心点获得更短的区间$T^{1/5}$，并在临界线上导出次凸性界，逼近谱方面的次凸性问题。

深度层面下 ${\rm U}(n) \times {\rm U}(n+1)$ 上 $L$-函数的次凸性

Subconvexity for $L$-functions on ${\rm U}(n) \times {\rm U}(n+1)$ in the depth aspect

本文研究深度层面下酉 Gan–Gross–Prasad 对的中心 L 函数次凸性；利用 Ichino–Ikeda 周期公式将 L 值转化为自守周期，并通过 Iwaniec–Sarnak 放大方法估计周期，证明了 GL_{n+1}×GL_n 基变换 L 函数的幂次次凸界。

相对迹公式、次凸性与Rankin-Selberg $L$-函数在$\mathrm{GL}(n+1)\times\mathrm{GL}(n)$上的定量非消没

Relative Trace Formula, Subconvexity and Quantitative Nonvanishing of Rankin-Selberg $L$-functions for $\mathrm{GL}(n+1)\times\mathrm{GL}(n)$

本文通过放大正则化相对迹公式，建立了GL(n+1)×GL(n)上Rankin-Selberg L函数的混合次凸性界及高秩定量同时非消没结果，改进了Nelson的指数。

分析解析新向量的稳态相位及其在子凸性问题中的应用

Stationary phase analysis for analytic newvectors and application to subconvexity problems

本文通过分析解析新向量的稳态相位，建立了三重积和Rankin-Selberg L-函数的子凸性界，并应用于局部三重积周期的测试向量猜想。

带简单超尖分歧的GL₂的Weyl次凸性

Weyl Subconvexity for $\mathrm{GL}_2$ with Simple Supercuspidal Ramification

本文针对数域F上GL₂的简单超尖分歧尖自守表示，建立了Weyl型次凸性界，证明了相关四阶矩估计，推导得出Weyl型界，且该界适用于三次矩方法无法处理的真正非自对偶族。

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。

Real zeros and size of Rankin-Selberg L-functions in the level aspect

In this paper, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg L-functions. One of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average. A first consequence is a new subconvexity bound for Rankin-Selberg L-functions in the level aspect. Moreover, infinitely many Rankin-Selberg L-functions having at most eight non-trivial real zeros are produced and some new non-trivial estimates for the analytic rank of the family…

New error term for the fourth moment of automorphic L functions

We improve the error term in the asymptotic formula for the twisted fourth moment of automorphic L functions of prime level and weight two proved by Kowalski, Michel and Vanderkam. As a consequence, we obtain a new subconvexity bound in the level aspect and improve the lower bound on proportion of simultaneous non-vanishing.

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。

Rankin–Selberg L 函数在特殊点处的次凸性

Subconvexity for Rankin Selberg L-Functions at Special Points

本文研究两个 Hecke–Maass 尖点形式的 GL(2)×GL(2) Rankin–Selberg L 函数，在谱参数相加或相减形成的特殊点处建立分段次凸界。

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…

Second moments of Rankin-Selberg convolution and shifted Dirichlet series

In this paper we work over $Γ_0(N)$, for any $N$ and write the spectral moment of a product of two distinct Rankin-Selberg convolutions at a general point on the critical line $\frac{1}{2}+it$ as a main term plus a sharp error term in the $t$ aspect and the spectral aspect. As a result we obtain hybrid Weyl type subconvexity results in the $t$ and spectral aspects. Also, for fixed modular forms $f$, $g$ of even weight $k\geq 4$ we show there exists a Maass cusp form $u_j$ such that $L(1/2, f\times u_j)$, $L(1/2, g\times u_j)$ are simultaneous non-zero.

Bounds for Rankin--Selberg integrals and quantum unique ergodicity for powerful levels

Let f be a classical holomorphic newform of level q and even weight k. We show that the pushforward to the full level modular curve of the mass of f equidistributes as qk -> infinity. This generalizes known results in the case that q is squarefree. We obtain a power savings in the rate of equidistribution as q becomes sufficiently "powerful" (far away from being squarefree), and in particular in the "depth aspect" as q traverses the powers of a fixed prime. We compare the difficulty of such equidistribution problems to that of corresponding subconvexity problems by deriving explicit extensions…

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.