K. Soundararajan
We describe a general method to obtain weak subconvexity bounds for many classes of $L$-functions. This has applications to a conjecture of Rudnick and Sarnak for the mass equidistribution of Hecke eigenforms (see arxiv.org:math/0809.1636).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove hybrid subconvexity bounds twisted L-functions $L(s,f\times χ)$ at the central point using a fourth moment estimate, including a new instance of the Burgess subconvexity bound.
We prove hybrid subconvexity bounds for a wide class of twisted L-functions $L(s,f\times χ)$ at the central point, including a new instance of the Weyl subconvexity bound.
We use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms of a fixed weight and level. We apply these formulas to the subconvexity problem for these L-functions. We also establish an equidistribution result for the Hecke eigenvalues weighted by these L-values.
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.
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.
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 $f\in S^*_k(N)$.
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。
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.
In this paper we generalized Venkatesh and Woodbury's work on the subconvexity bound of triple product L-function in level aspect, allowing joint ramifications, higher ramifications, general unitary central characters and general special values of local epsilon factors. In particular we derived a nice general formula for the local integrals whenever one of the representations has sufficiently higher level than the other two.
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.
Second Moment of Central Values of Half-Integral Weight Modular Forms and Subconvexity
本文通过相对迹公式和放大方法，首次在权重方面获得半整权重模形式的次凸性界，并证明了L函数中心值的同时非消失结果。
Effective correlation and decorrelation for newforms, and weak subconvexity for $L$-functions
本文研究了新形式的相关与去相关问题，提出有效界并改进了 $L$-函数的弱次凸性结果，扩展了 Huang 的去相关结论。
We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the twisting parameter. A similar hybrid bound holds for quadratic Dirichlet $L$-functions, improving on a result of Heath-Brown. As a consequence of these new bounds, we obtain explicit estimates for the number of Heegner points of large odd discriminant in shrinking sets.
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…
We derive a Motohashi-type formula for the cubic moment of central values of $L$-functions of level $q$ cusp forms twisted by quadratic characters of conductor $q$, previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for $L$-functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.
In this paper we investigate some consequences of the Gross/Zagier types of formulae which were introduced by Gross and Zagier, and were then generalized in various directions by Hatcher, Zhang, Kudla and several other people. Working in the classical context of central values of L-series of holomorphic forms of prime level, we deduce an exact average formula for suitable twists of such L-values, with a relation to the class number of associated imaginary quadratic fieds, thereby strengthening a result of Duke. One also obtains a stability result, as well as subconvexity (in this setting), and…
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 …
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.
We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6, breaking the long-standing Weyl exponent barrier. In fact, we develop a general new theory of estimation of short exponential sums involving p-adically analytic phases, which can be naturally seen as a p-adic analogue of the method of exponent pairs. This new method is presented in a ready-to-use form and applies to a wide class of well-behaved phases including…
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。
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…
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。
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。
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.
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。
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。
Rankin--Selberg Subconvexity via Spectral Reciprocity
本文通过改进 Michel 和 Venkatesh 的谱互反框架，建立了数域上 GL_2 酉尖自守表示对的 Rankin-Selberg L-函数中心值的显式次凸界，改进了所有已知结果，并应用于多个算术问题。
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。
Subconvexity for $GL(1)$ twists of Rankin-Selberg $L$-functions
本文研究$SL(2,\mathbb{Z})$的本原尖点形式对应的带Dirichlet特征扭的Rankin-Selberg $L$-函数中心值次凸界，通过约化为移位卷积和问题，将指数界从$1-1/1324$改进至$1-1/28$。
We prove explicit Harris-Kudla type formulas for triples of Maass forms, holomorphic forms, and combinations thereof, on the hyperbolic plane modulo congruence groups and co-compact lattices arising from Eichler orders of quaternion algebras. These formulas relate the central value of the corresponding Rankin triple product L-function to a squared trilinear period integral. Assuming subconvexity estimates for these L-values, we prove Quantum Unique Ergodicity on such quotients; the relevant Lindelof hypotheses imply a quantitative form of QUE, with an optimal rate. In connection with the Berry…
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。
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。
Let K be a number field containing the n-th roots of unity for some n > 2. We prove a uniform subconvexity result for a family of double Dirichlet series built out of central values of Hecke L-functions of n-th order characters of K. The main new ingredient, possibly of independent interest, is a large sieve for n-th order characters. As further applications of this tool, we derive several results concerning L(s,χ) for n-th order Hecke characters: an estimate of the second moment on the critical line, a non-vanishing result at the central point, and a zero-density theorem.
The cubic moment of Hecke--Maass cusp forms and moments of $L$-functions
该研究证明Hecke–Maass尖形式的光滑三次矩多项式衰减（随机波猜想新案例），通过估计中心$L$-值矩，得到$\rm GL(3)\times GL(2)$等$L$-函数的矩界与新次凸界。
Uniform approximate functional equation for principal L-functions
本文证明了数域上 GL(m) 不可约尖点自守表示的 L-级数中心值的一致近似函数方程，利用解析导子研究衰减率，并基于 Ramanujan--Selberg 猜想进展探讨其在幂均值界限与次凸性问题中的应用。
Zero-free regions for the Riemann zeta function
本文利用作者关于ζ(σ+it)增长的显式界及一种关联函数竖直线增长与竖条零点加权计数的Jensen公式，改进了黎曼ζ函数无零点区域的Vinogradov-Korobov型显式界（含大高度t和所有t的情况）。
We give a formula for the values of automorphic Green functions on the special rational 0-cycles (big CM points) attached to certain maximal tori in the Shimura varieties associated to rational quadratic spaces of signature (2d,2). Our approach depends on the fact that the Green functions in question are constructed as regularized theta lifts of harmonic weak Mass forms, and it involves the Siegel-Weil formula and the central derivatives of incoherent Eisenstein series for totally real fields. In the case of a weakly holomorphic form, the formula is an explicit combination of quantities obtain…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。