J. Brian Conrey, Jonathan Keating, Michael Rubinstein, Nina Snaith
We present theoretical and numerical evidence for a random matrix theoretic approach to a conjecture about vanishings of quadratic twists of certain L-functions
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
On the existence of Siegel modular forms with extra twists
该研究聚焦带额外扭的Siegel模形式的存在性问题，给出了这类模形式存在额外扭时水平与亏格的必要条件，还构造了不同于复共轭的具体实例，推进了相关数论问题研究。
We prove an asymptotic for the second moment of quadratic twists of a modular $L$-function. This was previously known conditionally on GRH by the work of Soundararajan and Young.
Non-vanishing theorems for prime twists of some modular $L$-functions
本文研究了特定椭圆模形式的模 $L$-函数经虚二次域素数扭曲后的中心值非零性问题，并证明在根数为正时这些中心值恒不为零。
Bounds for Moments of Twisted Fourier coefficients of Modular Forms
本文在广义黎曼假设下，为模L函数的移位矩及二次扭矩建立上界，并据此推导出模形式傅里叶系数与狄利克雷字符扭和之矩的界。
In this paper, we study central values of the family of quadratic twists of modular $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish a positive proportion non-vanishing result for the corresponding $L$-values.
Zagier proved that the generating series for the traces of singular moduli is a \textit{weakly holomorphic} modular form of weight 3/2 on $Γ_0(4)$. Bruinier and Funke extended the results of Zagier to modular curves of arbitrary genus. Zagier also showed that the twisted traces of singular moduli are generated by a weakly holomorphic modular form of weight 3/2. In this paper, we study the extension of Zagier's result for the twisted traces of singular moduli to congruence subgroups $Γ_0(N)$. As an application, we study congruences for the twisted traces of singular moduli of weakly holomorphic…
We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.
Cheng constructed product expansions from twists of elliptic genera of symmetric powers of K3 surfaces that are related to M_24 moonshine. We study which of them are Siegel modular forms. If the predicted level is non-composite, they are modular, and their powers can be represented as products of rescaled Borcherds products.
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.
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.
In this article, we give some results on simultaneous non-vanishing and simultaneous sign-changes for the Fourier coefficients of two modular forms. More precisely, given two modular forms $f$ and $g$ with Fourier coefficients $a_n$ and $b_n$ respectively, we consider the following questions: existence of infinitely many primes $p$ such that $a_p b_p\neq 0$; simultaneous non-vanishing in the short intervals and in arithmetic progressions; simultaneous sign changes in short intervals.
We prove a non-vanishing result of modular L-values with quadratic twists, where the quadratic discriminants are in a short interval. Using this fact and Waldspurger's theorem, we improve the results of Balog-Ono[The chebotarev density theorem in short intervals and some questions of Serre, Journal of number theory. 91(2):356-371(2001)] on the non-vanishing of Fourier coefficients of half-integral weight eigenform.
Twisted traces of modular functions on hyperbolic $3$-space
本文计算了双曲3-空间上调和模函数与爱森斯坦级数的扭曲迹，证明前者由j-不变量的傅里叶系数给出且为整数，后者可用狄利克雷L-函数与除数和表示。
We study the second moment of the central values of quadratic twists of a modular $L$-function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.
Non-Vanishing of L-Functions of Vector-Valued Modular Forms
本文证明了全模群上向量值模形式尖点形式空间正交基的L-函数平均值非消没，并在特定条件下存在不消没的基元素。
We investigate non-correlation of Fourier coefficients of Maass forms against a class of real oscillatory functions, in analogy to known results with Frobenius trace functions. We also establish an equidistribution result for twisted horocycles as a consequence of our non-correlation result.
We construct Stickelberger elements for Hilbert modular cusp forms of parallel weight 2 and use recent results of Dasgupta and Spiess to bound their order of vanishing from below. As a special case the vanishing part of Mazur and Tate's refined "Birch and Swinnerton-Dyer type"-conjecture for elliptic curves of rank 0 follows.
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.
Let $K$ be an imaginary quadratic field where $p$ splits. We study signed Selmer groups for non-ordinary modular forms over the anticyclotomic $\mathbf{Z}_p$-extension of $K$, showing that their $μ$-invariants vanish. This generalizes and gives a new proof of a recent result of Matar on the vanishing of the $μ$-invariants of plus and minus signed Selmer groups for elliptic curves.
In this paper, we show that, on average, the derivatives of $L$-functions of cuspidal Hilbert modular forms with sufficiently large weight $k$ do not vanish on the line segments $\Im(s)=t_{0}$, $\Re(s)\in(\frac{k-1}{2},\frac{k}{2}-ε)\cup(\frac{k}{2}+ε,\frac{k+1}{2})$. This is analogous to the case of classical modular forms.
Bounds for integral moments of quadratic twists of Fourier coefficients of modular forms
本文在广义黎曼猜想下给出模形式傅里叶系数二次扭特征和的积分矩上界，并证明其最优性。
In this paper, we show that, by applying some results on modular symbols, for a family of certain elliptic curves defined over $\mathbb Q$, there is a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s=1$, and for which the $2$-part of Birch-Swinnerton-Dyer conjecture holds.
Moment of Derivatives of Quadratic Twists of Modular $L$-Functions
本文研究了两个不同模$L$-函数二次扭导数矩的渐进行为，首次在不依赖GRH条件下给出证明。
Lower bounds for moments of quadratic twists of modular $L$-functions
该论文证明了模$L$-函数二次扭曲在中心点的$2k$-阶矩（$k \geq 1/2$）的尖锐下界。
Ratios conjecture for quadratic twist of modular $L$-functions
本文在广义黎曼假设下，利用多重狄利克雷级数，为模形式 $L$-函数的二次扭曲族在特定范围内建立了含单平移的 $L$-函数比值猜想。
In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4-th twisted Kloosterman sheaf sum. To prove these identities we express certain character sum over finite field in terms of special values of $p$-adic hypergeometric functions. One conjecture of Evans behaves as a bridge to connect $p$-adic hypergeometric functions with Kloosterman sheaf sum. We also connect $p$-adic hypergeometric functions with Fourier coefficients of certain modular forms.
Given elliptic modular forms f and g satisfying certain conditions on their weights and levels, we prove (a quantitative version of the statement) that there exist infinitely many imaginary quadratic fields K and characters chi of the ideal class group Cl_K such that L(1/2, f_K \times chi) \neq 0 and L(1/2, g_K \times chi) \neq 0. The proof is based on a non-vanishing result for Fourier coefficients of Siegel modular forms combined with the theory of Yoshida liftings.
The second moment of derivatives of quadratic twists of modular $L$-functions
本文证明了二次模形式 $L$-函数二次扭曲导数的二阶矩渐进行式，包含三个主导项，改进了 Kumar 等人的先前结果，方法基于 Li 建立的大筛不等式及大位移扭曲 $L$-函数渐进行式计算。
We introduce and study certain deformations of Drinfeld quasi-modular forms by using rigid analytic trivialisations of corresponding Anderson's t-motives. We show that a sub-algebra of these deformations has a natural graduation by the group Z^2 x Z/(q-1)Z and an homogeneous automorphism, and we deduce from this and other properties multiplicity estimates.
We study sums of additively twisted Fourier coefficients of a holomorphic cusp form, a Maass cusp form, and the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform with respect to both the form and the terms of the additive twist. We also improve a bound on a sum of additively twisted Fourier coefficients of a holomorphic cusp form that have also been twisted by a nonlinear exponential term. This last bound is uniform with respect to the terms of the additive twist but it is not uniform with respect to the form.
We prove the non-vanishing of special $L$-values of cuspidal automorphic forms on GL(2) twisted by Hecke characters of prime power orders and totally split prime power conductors. Main ingredients of the proof are estimating the Galois averages of fast convergent series expressions of special $L$-values and considering Shintani cone decomposition.
In this paper, we study the non-vanishing of the central values of the Rankin-Selberg $L$-function of two adèlic Hilbert primitive forms ${\bf f}$ and ${\bf g}$, both of which have varying weight parameter $k$. We prove that, for sufficiently large $k$, there are at least $\frac{k}{(\log k)^{c}}$ adèlic Hilbert primitive forms ${\bf f}$ of weight $k$ for which $L(\frac12, {\bf f}\otimes{\bf g})$ are nonzero.
We consider the question of the correlation of Fourier coefficients of modular forms with functions of algebraic origin. We establish the absence of correlation in considerable generality (with a power saving of Burgess type) and a corresponding equidistribution property for twisted Hecke orbits. This is done by exploiting the amplification method and the Riemann Hypothesis over finite fields, relying in particular on the ell-adic Fourier transform introduced by Deligne and studied by Katz and Laumon.
On converse theorems for Hilbert modular forms assuming unramified twists
本文证明了次数大于1的全实域上 Hilbert 模形式的两个逆定理，分别通过非分歧 Hecke 特征扭曲的 L-级数函数方程，以及结合 Euler 乘积恢复出预期水平的模形式，推广了 GL2 逆定理并完全去除了对分歧扭曲的假设。
The first moment of quadratic twists of modular $L$-functions
本文基于 Young 的工作，研究了模 $L$ 函数二次扭曲及其一阶导数的平滑第一矩，并给出了误差项为 $O(X^{\frac{1}{2} + \varepsilon})$ 的渐近公式。
Modular forms with non-vanishing central values and linear independence of Fourier coefficients
本文通过将VanderKam定理推广到高权模符号，证明对大素数p及D²≪p，Hecke算子T₁,…,T_D在权2k尖点模符号空间的winding元素上线性无关，从而给出中心临界值非零的新形式数量界及其模l约化的线性无关性。
Let f be a newform of weight two, prime level p. If D is a fundamental discriminant, define the twisted L-function L(f,D,s) to be the L-function associated to the twist of f by the quadratic character of conductor D. In this paper we consider the question of computing the family of twisted central values {L(f,D,1) : |D| <= x} for some x, by using an explicit version of Waldspurger's formula relating the central values L(f,D,1) to the |D|-th Fourier coefficient of weigth 3/2 modular forms in Shimura correspondence with f.
In this article, we investigate the variance of local $\varepsilon$-factor for a modular form with arbitrary nebentypus with respect to twisting by a quadratic character. We detect the type of the supercuspidal representation from that. For modular forms with trivial nebentypus, similar results are proved by Pacetti. Our method however is completely different from that of Pacetti and we use representation theory crucially. For ramified principal series (with $\infdiv{p}{N}$ and $p$ odd) and unramified supercuspidal representations of level zero, we relate these numbers with the Morita's $p$-ad…
Twisted second moment of modular $L$-functions to a fixed modulus
本文渐近研究了模 $L$ 函数族关于固定模的扭曲二阶矩，并以此为基础，为这些函数在临界线上的 $2k$ 阶矩确立了特定范围内的精确上下界。
We prove non-vanishing modulo p, for a prime $\ell$ different from p, of central critical Rankin-Selberg L-values with anticyclotomic twists of $\ell$-power conductor. The L-function is Rankin product of a cusp form and a theta series of arithmetic Hecke character of an imaginary quadratic field. The paper is concerned with the case when the weight of Hecke character is greater than that of cusp form, so the L-value is essentially different in nature from the one in the landmark work of Vatsal and Cornut-Vatsal on the same theme.
In this paper, we prove that if the Fourier coefficients of a $\mathrm{SL}(3,\mathbb{Z})$ Hecke--Maaß cusp form $π$ are not too correlated with additive characters, then there exists infinitely many Dirichlet characters such that \begin{align*} L\left(\frac{1}{2},π\otimesχ\right)L\left(\frac{1}{2},χ\right)\neq 0. \end{align*} To prove this result, we compute the first twisted moment of these $L$ function averaged over a well chosen set of conductors.
Mean squares of quadratic twists of the Fourier coefficients of modular forms
本文渐近求值了给定模形式傅里叶系数在二次扭曲下的平滑求和版本，涉及Kronecker符号及正奇无平方整数求和。
Low moments of random multiplicative functions twisted by Fourier coefficients of modular forms
研究随机乘法函数与模形式傅里叶系数的矩性质，确定在特定范围内的矩阶大小。
Let $\mathscr{S}_k^+(\cn,Φ)$ denote the space generated by Hilbert modular newforms (over a fixed totally real field $K$) of weight $k$, level $\cn$ and Hecke character $Φ$. We show how to decompose $\mathscr{S}_k^+(\cn,Φ)$ into direct sums of twists of other spaces of newforms. This sheds light on the behavior of a newform under a character twist: the exact level of the twist of a newform, when such a twist is itself a newform, and when a newform may be realized as the twist of a primitive newform. These results were proven for elliptic modular forms by Hijikata, Pizer and Shemanske by employ…
Descent for projective twists of modular curves
本研究针对数域F与有理素数p≥7，给出了刻画p级主模曲线射影扭可定义在ℚ上的射影伽罗瓦表示的下降判别准则，并给出循环数域上椭圆曲线对应表示的更具体结果。
We investigate in this paper the vanishing at $s=1$ of the twisted $L$-functions of elliptic curves $E$ defined over the rational function field $\mathbb{F}_q(t)$ (where $\mathbb{F}_q$ is a finite field of $q$ elements and characteristic $\geq 5$) for twists by Dirichlet characters of prime order $\ell \geq 3$, from both a theoretical and numerical point of view. In the case of number fields, it is predicted that such vanishing is a very rare event, and our numerical data seems to indicate that this is also the case over function fields for non-constant curves. For constant curves, we adapt th…
These are notes for four lectures given at the 2010 CIMPA Research School "Automorphic Forms and L-functions" in Weihai, China. The lectures focused on a Burgess-like subconvex bound for twisted Hilbert modular L-functions published jointly with Valentin Blomer in the same year. They discussed the proof in some detail, especially how spectral theory can be used to estimate the relevant shifted convolution sums efficiently. They also discussed briefly an application for the number of representations by a totally positive ternary quadratic form over a totally real number field.
Let L(E/Q,s) be the L-function of an elliptic curve E defined over the rational field Q. We examine the vanishing and non-vanishing of the central values L(E,1,χ) of the twisted L-function as χranges over Dirichlet characters of given order.
We use vanishing results for sheaf cohomology on Siegel modular varieties to study two lifting problems: (a) When can Siegel modular forms (mod p) be lifted to characteristic zero? This uses and extends previous results for cusp forms by Stroh and Lan-Suh. (b) When is the restriction of Siegel modular forms to the boundary of the moduli space a surjective map? We investigate this question in arbitrary characteristic, generalising analytic results of Weissauer and Arakawa.
Diagonal cycles and anticyclotomic twists of modular forms at inert primes
本文借助对角类重构 Castella–Do 的反环圈 Euler 系统，并结合既有结果，在 $p$ 于虚二次域中惰性的条件下推进解析秩一情形的 Bloch–Kato 猜想。
Let E be an elliptic curve and ρan Artin representation, both defined over the rational numbers. Let p be a prime at which E has good reduction. We prove that there exists an infinite set of Dirichlet characters χ, ramified only at p, such that the Artin-twisted L-values L(E,ρχ,β) are non-zero when βlies in a specified region in the critical strip (assuming the conjectural continuations and functional equations for these L-functions). The new contribution of our paper is that we may choose our characters to be ramified only at one prime, which may divide the conductor of ρ.
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.
Central Values of $L$-Functions of Twisted Modular Forms and Local Polynomials
本文通过局部多项式和局部调和 Maass 形式，研究 twisted 模形式 $L$-函数中心值乘积的零点判定问题。
We prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment of central values of L-functions of any two (possibly equal) fixed cusp forms f, g twisted by all primitive characters modulo q, valid for all sufficiently factorable q including 99.9% of all admissible moduli. The two key ingredients are a careful spectral analysis of a potentially highly unbalanced shifted convolution problem in Hecke eigenvalues and power-saving bounds for sums of products of Kloosterman sums where the length of the sum is below the square-root threshold of the modulus. Appli…
Central values of additive twists of Maaß forms $L$-functions
本文研究了Maass形式L函数加法twists的中心值，结合动力学方法和量子模形式性质，证明了其渐近正态分布并拓展了相关算术应用。
Torsion-vanishing for Siegel modular varieties via supercuspidal congruences
该研究通过 Scholze 等人的同余技术，证明带挠系数的西格尔模簇上同调的一般部分集中在中间次数以上，其方法适配西格尔模簇且可处理任意小特征的系数。
Results on the Non-Vanishing of Derivatives of L-Functions of Vector-Valued Modular Forms
本文证明了向量值模形式L函数导数平均值的非零性，并推广至Kohnen正空间与Jacobi形式。
Igusa's map on modular forms vanishing on the hyperelliptic locus
本文将Igusa映射$ρ$扩展至Siegel上半空间超椭圆轨迹上消失的模形式，利用一般Thomae公式计算其最低非零导数作为向量值模形式，并以亏格$3$和$4$的例子进行说明。
On the Vanishing and Cuspidality of $D_4$ Modular Forms
本文研究了 $D_4$ 模形式的消失与 cuspidality 性质，通过标量傅里叶系数理论发展了判据，并刻画了 Pollack 的四元数 Saito-Kurokawa 子空间。
The aim of this paper is to describe some partial advances in the solution of the following two problems. Find the maximal order of vanishing at infinity of a non-zero Drinfeld quasi-modular form of given weight. Determine differential properties of a Drinfeld quasi-modular form of given weight and depth with maximal order of vanishing at infinity.
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。