Matthew P. Young
We show that a large number of elliptic curve L-functions do not vanish at the central point, conditionally on the generalized Riemann hypothesis and on a hypothesis on the regular distribution of the root number.
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Subspaces spanned by eigenforms with nonvanishing twisted central $L$-values
本文构造了整权本原形式中非零二次扭曲中心 $L$-值子空间与半整权本原形式中特定非零 Fourier 系数子空间的显式生成集，并证明二者经 Shimura 提升同构。
With the method of moments and the mollification method, we study the central $L$-values of GL(2) Maass forms of weight $0$ and level $1$ and establish a positive-proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law. As an application of this result and a formula of Katok--Sarnak, we give a nonvanishing result on the first Fourier coefficients of Maass forms of weight $\frac{1}{2}$ and level $4$ in the Kohnen plus space.
Averages and nonvanishing of central values of triple product $L$-functions
本文结合Ichino周期公式与相对迹公式，证明了三重积$L$-函数中心值的精确平均值，并给出了固定一个形式时非零中心$L$-值数量的下界。
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 study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto $x$ which do not vanish $\bmod \ell$. We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients $\bmod \ell $ of a Siegel modular form with integral algebraic Fourier coefficients provi…
On the coefficients of the Taylor expansion of $L$-functions of elliptic curves
本文研究了椭圆曲线复L函数泰勒展开系数，在GRH下证明了当判别式足够大时，二次扭家族中系数非零。无条件地，通过模形式L函数二次扭中心值导数的矩结果，获得非零系数的下界。
A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=\frac{k}{2}.$ It was shown by Kohnen that there exists a Hecke eigenform $f$ of weight $k$ such that $L^*(f,s) \neq 0$ for sufficiently large $k$ and any point on the line segments $Im(s)=t_0, \frac{k-1}{2} < Re(s) < \frac{k}{2}-ε, \frac{k }{2}+ε< Re(s) < \frac{k+1}{2},$ for any given real number $t_0$ and a positive real number $ε.$ This paper concerns the non-vanishing of the product $L^*(f,s…
We show that for at least 3/8 of the primitive Dirichlet characters $χ$ of large prime modulus, the central value $L(1/2,χ)$ does not vanish.
We prove that for $d \in \{ 2,3,5,7,13 \}$ and $K$ a quadratic (or rational) field of discriminant $D$ and Dirichlet character $χ$, if a prime $p$ is large enough compared to $D$, there is a newform $f \in S_2(Γ_0(dp^2))$ with sign $(+1)$ with respect to the Atkin-Lehner involution $w_{p^2}$ such that $L(f \otimes χ,1) \neq 0$. This result is obtained through an estimate of a weighted sum of twists of $L$-functions which generalises a result of Ellenberg. It relies on the approximate functional equation for the $L$-functions $L(f \otimes χ, \cdot)$ and a Petersson trace formula restricted to A…
A quadratic twist of the L-function associated with a modular form is known to satisfy a functional equation, which may be even or odd. A result due to Gross and Zagier explicitly computes the central value of the L-function or its derivative. In prime level when the functional equation is even, Michel and Ramakrishnan have used an averaging method to prove several consequences of the Gross-Zagier formulae, including a non-vanishing result. The present research concerns L-functions arising from newforms in prime-squared level, which necessarily have odd functional equations. Such an L-function…
We show that for at least $\frac{5}{13}$ of the primitive Dirichlet characters $χ$ of large prime modulus, the central value $L(\frac{1}{2},χ)$ does not vanish, improving on the previous best known result of $\frac{3}{8}$.
We describe algorithms for computing central values of twists of $L$-functions associated to Hilbert modular forms, carry out such computations for a number of examples, and compare the results of these computations to some heuristics and predictions from random matrix theory.
Let $f$ be a holomorphic cusp form of integral weight $k \geq 3$ for $Γ_{0}(N)$ with nebentypus character $ψ$. Generalising work of Kohnen and Raghuram we construct a kernel function for the $L$-function $L(f,χ,s)$ of $f$ twisted by a primitive Dirichlet character $χ$ and use it to show that the average $\sum_{f \in S_{k}(N,ψ)}\frac{L(f,χ,s)}{<f,f>}\bar{a_{f}(1)}$ over an orthogonal basis of $S_{k}(N,ψ)$ does not vanish on certain line segments inside the critical strip if the weight $k$ or the level $N$ is big enough. As another application of the kernel function we prove an averaged version …
We show that the central value of class group L-functions of CM fields can be expressed in terms of derivatives of real-analytic Hilbert Eisenstein series at CM points. Then, following an idea of Iwaniec and Kowalski we obtain a conditional explicit lower bound of class numbers of CM fields under a weaker assumption. Some results in the proof lead to an effective nonvanishing result for class group L-functions of general CM fields, generalizing the only known ineffective results.
Periods and nonvanishing of central L-values for GL(2n)
本文证明Guo-Jacquet关于GL(2n)中心L值非消失性与GL(n,E)周期非消失性关系的猜想，并应用于超尖点表示，部分证实Prasad-Takloo-Bighash猜想。
Nonvanishing of ray class $L$-functions
本文研究固定虚二次域上射线类 L-函数中心值的非零性，通过改进磨光器方法，证明当模的范数趋于无穷时，至少 1/3 比例的本原射线类特征满足 L(1/2,χ)≠0。
Nonvanishing of generalised Kato classes and Iwasawa main conjectures
本文针对正偶数解析秩椭圆曲线的广义Kato类非零性猜想，给出κ_p(E)≠0蕴含Selmer群二维的新证明，并证明逆蕴含成立当且仅当局部化映射非零，将作者在CM情形的方法推广到非CM情形。
Applications of nonarchimedean developments to archimedean nonvanishing results for twisted $L$-functions
本文通过结合伊万萨瓦理论中的非阿基米德方法与新机制，证明了特定无限族中除有限个酉特征外所有扭曲的某类自守 $L$-函数的扭曲中心临界值非零。
Let $ψ$ be a real primitive character modulo $D$. If the $L$-function $L(s,ψ)$ has a real zero close to $s=1$, known as a Landau-Siegel zero, then we say the character $ψ$ is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values $L(1/2,χ)$ of the Dirichlet $L$-functions $L(s,χ)$ are nonzero, where $χ$ ranges over primitive characters modulo $q$ and $q$ is a large prime of size $D^{O(1)}$. Under the same hypothesis we also show that, for almost all $χ$, the function $L(s,χ)$ has at most a simple zero at $s = 1/2$.
In Iwaniec-Sarnak [IS] the percentages of nonvanishing of central values of families of GL_2 automorphic L-functions was investigated. In this paper we examine the distribution of zeros which are at or neat s=1/2 (that is the central point) for such families of L-functions. Unlike [IS], most of the results in this paper are conditional, depending on the Generalized Riemann Hypothesis (GRH). It is by no means obvious, but on the other hand not surprising, that this allows us to obtain sharper results on nonvanishing.
Let F(z) be a newform of weight 2k and level one with a trivial character, and assume that F(z) is a non-zero eigenform of all Hecke operators. In this paper, we study nonvanishing for central values of twisted modular L-function of F.
Nonvanishing of self-dual $L$-values via spectral decomposition of shifted convolution sums
本文通过移位卷积和的谱分解，在广义Lindelöf和Ramanujan假设的最佳逼近下，证明自对合Rankin-Selberg L函数中心值的非消失性，并应用于界定椭圆曲线的Mordell-Weil秩。
We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Roh…
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。
Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.
Building on ideas of Vatsal, Cornut proved a conjecture of Mazur asserting the generic nonvanishing of Heegner points on an elliptic curve E as one ascends the anticyclotomic Z_p-extension of a quadratic imaginary extension K/Q. In the present article Cornut's result is extended by replacing the elliptic curve E with the Galois cohomology of Deligne's 2-dimensional l-adic representation attached to a modular form of weight 2k>2, and replacing the family of Heegner points with an analogous family of special cohomology classes.
Let $f$ be a Maass form for $SL(3, \mathbb{Z})$ which is fixed and $u_j$ be an orthonormal basis of even Maass forms for $SL(2, \mathbb{Z}),$ we prove an asymptotic formula for the average of the product of the Rankin-Selberg $L$-function of $f$ and $u_j$ and the $L$-function of $u_j$ at the central value 1/2. This implies simultaneous nonvanishing results of these $L$-functions at $1/2.$
Zeta functions and nonvanishing theorems for toric periods on $\mathrm{GL}_2$
本文研究了四元数代数上尖点自守表示带有周期积分的 zeta 函数，通过推导其显式公式，证明了在 $L(1/2, π)\neq0$ 时存在无限多个二次扩张使得 $π$ 具有非零环面周期。
We prove that a cuspidal automorphic representation of GL(3) over any number field is determined by the quadratic twists of its central value. In the case of a non-Gelbart-Jacquet lift, the result is conditional on the analytic behavior of a certain Euler product. We deduce the nonvanishing of infinitely many quadratic twists of central values. This generalizes a result of Chinta and Diaconu that was valid only over the field of rational numbers and explored only for Gelbart-Jacquet lifts.
We study the nonvanishing of twists of automorphic L-functions at the centre of the critical strip. Given a primitive character χmodulo D satisfying some technical conditions, we prove that the twisted L-functions L(f.χ,s) do not vanish at s=1/2 for a positive proportion of primitive forms of weight 2 and level q, for large prime q. We also investigate the central values of high derivatives of L(f.χ,s), and from that derive an upper bound for the average analytic rank of the studied L-functions.
Let $E_{-D}$ be the elliptic curve $y^2=x^3+Dx$ defined over $K=\mathbb{Q}(i)$ for $D\in K$ which is coprime to 2. In this paper, we give a lower bound for the 2-adic valuation of the algebraic part of the central value of Hecke $L$-function associated to $E_{-D}$.
Nonvanishing of special $L$-values of cusp forms on $\mathrm{GL}(2)$ with totally split prime power twists: without the class number hypothesis
本文证明在完全分裂素数幂扭下，GL(2) 上尖点形式的特殊 L 值在中心临界点非零，无需类数假设，改进了 Kwon 和 Sun 的结果。
Here we study the recently introduced notion of a locally harmonic Maass form and its applications to the theory of $L$-functions. In particular, we find finite formulas for certain twisted central $L$-values of a family of elliptic curves in terms of finite sums over canonical binary quadratic forms. This yields vastly simpler formulas related to work of Birch and Swinnerton-Dyer for such $L$-values, and extends beyond their framework to special non-CM elliptic curves.
Extreme central $L$-values of almost prime quadratic twists of elliptic curves
该研究证明椭圆曲线殆素数二次扭曲的中心 $L$-函数极值，并在 BSD 猜想下推出其二次扭曲族中 Tate--Shafarevich 群的极值。
Small Algebraic Central Values of Twists of Elliptic $L$-Functions
本文研究椭圆曲线$L$-函数经Dirichlet特征扭形后的小的非零代数中心值，提出启发式预测并给出计算证据，进而探讨其对固定次数循环扩张族中Brauer-Siegel定理类比的推论。
The second and third-named authors (arXiv:1705.04115) established a Central Limit Theorem for the error term in the Sato-Tate law for families of modular forms. This method was adapted to families of elliptic curves in by the first and second-named authors (arXiv:1705.09229). In this context, a Central Limit Theorem was established only under a strong hypothesis going beyond the Riemann Hypothesis. In the present paper, we consider a smoothed version of the Sato-Tate conjecture, which allows us to overcome several limitations. In particular, for the smoothed version, we are able to establish a…
Inspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $τ$-function, one may ask whether an odd integer $α$ can be equal to $τ(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the theory of Lucas sequences and Diophantine analysis to characterize non-admissible values of newforms of even weight $k\geq 4$. We use these methods for weight $2$ and $3$ newforms and apply our results to $L$-functions of modular elliptic curves and certain $K3$ surfaces with Picard number $\ge 19$. In particular, for the complete list of weight $3$ newforms $f_λ(z)=\s…
Upper tail distributions of central $L$-values of quadratic twists of elliptic curves at the variance scale
本文研究判别式有界的椭圆曲线二次扭族的$L$-函数中心值在方差量级的大偏差，通过调整黎曼ζ函数大偏差技术得到密度上界，改进了已有结果。
Moments and distribution of central L-values of quadratic twists of elliptic curves
本文研究 L 函数族的矩与中心值分布，通过略高于特定矩的信息推出更低阶矩的数量级上界及单侧中心极限定理，并应用于椭圆曲线二次扭曲，给出 Tate-Shafarevich 群阶猜想的上界。
Let $q$ be any prime $\equiv 7 \mod 16$, $K = \mathbb{Q}(\sqrt{-q})$, and let $H$ be the Hilbert class field of $K$. Let $A/H$ be the Gross elliptic curve defined over $H$ with complex multiplication by the ring of integers of $K$. We prove the existence of a large explicit infinite family of quadratic twists of $A$ whose complex $L$-series does not vanish at $s=1$. This non-vanishing theorem is completely new when $q > 7$. Its proof depends crucially on the results established in our earlier paper for the Iwasawa theory at the prime $p=2$ of the abelian variety $B/K$, which is the restriction…
Central $L$ values of congruent number elliptic curves
本文证明同余数椭圆曲线的中心 L 值可表示为简单 theta 函数的 CM 值平方，推广了 Gauss 公式，兼具理论与计算价值。
The paper uses Iwasawa theory at the prime $p=2$ to prove non-vanishing theorems for the value at $s=1$ of the complex $L$-series of certain quadratic twists of the Gross family of elliptic curves with complex multiplication by the field $K = \BQ(\sqrt{-q})$, where $q$ is any prime $\equiv 7 \mod 8$. Our results establish some broad generalizations of the non-vanishing theorem first proven by D. Rohrlich using complex analytic methods. Such non-vanishing theorems are important because it is known that they imply the finiteness of the Mordell-Weil group and the Tate-Shafarevich group of the cor…
We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family. Furthermore, as arithmetical applications we make a conjecture on the distribution of a_p's amongst all rank 2 …
Non-vanishing of central $L$-values of the Gross family of elliptic curve
研究证明了特定椭圆曲线二次扭曲的L函数中心值非零性，并由此得出相关群的有限性及反向定理。
In a previous article we had proved an algebraicity result for the central critical value for L-functions for GL(n) x GL(n-1) over Q assuming the validity of a nonvanishing hypothesis involving archimedean integrals. The purpose of this article is to generalize that result for all critical values for L-functions for GL(n) x GL(n-1) over any number field F. Binyong Sun has recently proved that nonvanishing hypothesis and so the results of this article are unconditional. Using such results for the case of GL(3) x GL(2), new unconditional algebraicity results for the special values of symmetric c…
Iwasawa theory of elliptic curves over noncommutative extensions has been a fruitful area of research. The central object of this paper is to use Iwasawa theory over the $GL(2)$ extension to study the dual Selmer group over the $PGL(2)$ extension.
We evaluate in closed form, for the first time, certain classes of double series, which are remindful of lattice sums. Elliptic functions, singular moduli, class invariants, and the Rogers--Ramanujan continued fraction play central roles in our evaluations
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