Alexandre Peyrot
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.
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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.
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。
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。
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.
We present theoretical and numerical evidence for a random matrix theoretic approach to a conjecture about vanishings of quadratic twists of certain L-functions
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。
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.
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.
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。
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.
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 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 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.
We construct isomorphisms between spaces of vector-valued modular forms for the dual Weil representation and certain spaces of scalar-valued modular forms in the case that the underlying finite quadratic module $A$ has order $p$ or $2p$, where $p$ is an odd prime. The isomorphisms are given by twisted sums of the components of vector-valued modular forms. Our results generalize work of Bruinier and Bundschuh to the case that the components $F_γ$ of the vector-valued modular form are antisymmetric in the sense that $F_γ = -F_{-γ}$ for all $γ\in A$. As an application, we compute restrictions of …
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。
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。
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。
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.
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。
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.
Twisted Kronecker series and periods of modular forms on $Γ_0(N)$
本文构造特征扭曲的 Kronecker 级数族，并在 N 无平方因子时用其乘积给出 $Γ_0(N)$ 上 Hecke 特征形式及其扭曲周期多项式之和的闭式公式，推广了 Zagier 恒等式。
The purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can for each odd prime $p$, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is Steinberg, Principal Series or Supercuspidal at $p$ by analyzing the change of sign under a suitable twist. We also explain the case $p=2$, where twisting is not enough in general.
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。
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。
Lower Bounds on High Moments of Twisted Fourier coefficients of Modular Forms
本文研究了模形式扭曲Fourier系数的高次矩问题，针对大素数q和实数k≥2，证明了2k次矩的下界，并在广义Riemann假设下证明了该界在常数因子范围内是精确的。
Let $\overlineρ: G_{\mathbf{Q}} \rightarrow {\rm GSp}_4(\mathbf{F}_3)$ be a continuous Galois representation with cyclotomic similitude character -- or, what turns out to be equivalent, the Galois representation associated to the $3$-torsion of a principally polarized abelian surface $A/\mathbf{Q}$. We prove that the moduli space $\mathcal{A}_2(\overlineρ)$ of principally polarized abelian surfaces $B/\mathbf{Q}$ admitting a symplectic isomorphism $B[3] \simeq \overlineρ$ of Galois representations is never rational over $\mathbf{Q}$ when $\overlineρ$ is surjective, even though it is both ratio…
We form real-analytic Eisenstein series twisted by Manin's noncommutative modular symbols. After developing their basic properties, these series are shown to have meromorphic continuations to the entire complex plane and satisfy functional equations in some cases. This theory neatly contains and generalizes earlier work in the literature on the properties of Eisenstein series twisted by classical modular symbols.
Computing Euler products and coefficients of classical modular forms for twisted L-functions
提出一种快速计算经典模形式系数及欧拉乘积的算法，并在大导子三重积L-函数上验证其效率。
Let $\cal A$ be a maximal (or more generally a hereditary) order in a central simple algebra over a global field $F$ of positive characteristic. We study the reduction of the modular scheme of $\cal A$-elliptic sheaves at all places of $F$. We show that some of these varieties - for different $\cal A$ - are twists of each other. This result can be viewed as a global analogue of the Cherednik-Drinfeld theorem for these varieties.
For every point on the Jacobian of the modular curve $X_0(l)$ we define and study certain twisted holomorphic Eisenstein series. These are particular cases of a more general notion of twisted modular forms which correspond to sections on the modular curve $X_1(l)$ of the degree zero twists of line bundles of usual modular forms. We conjecture that a point on the Jacobian is rational if and only if the ratios of these twisted Eisenstein series of the same weights have rational coefficients.
High moments of random multiplicative functions twisted by Fourier coefficients of modular forms
在广义黎曼假设下，研究了由模形式傅里叶系数扭曲的Steinhaus或Rademacher随机乘性函数的2q阶矩的量级，其中q的范围为1到c log x / log log x。
Generation of Hecke fields by squares of cyclotomic twists of modular $L$-values
本文证明非CM椭圆新形式及其分圆扭曲的Hecke域复合可由中心L值绝对值平方生成，方法基于单幂混合猜想解决技术。
In this paper, we study twisted arithmetic divisors on the modular curve X_0(N) with N square-free. For each pair (Δ, r) where Δ>0 and Δ\equiv r^2 \mod 4N, we constructed a twisted arithmetic theta function ϕ_{Δ, r}(τ) which is a generating function of arithmetic twisted Heegner divisors. We prove the modularity of ϕ_{Δ, r}(τ), along the way, we also identify the arithmetic pairing \langle ϕ_{Δ, r}(τ),\widehatω_N \rangle with special value of some Eisenstein series, where \widehatω_N is a normalized metric Hodge line bundle.
Twisted periods of modular forms
本文研究了模形式的扭周期，证明了不同扭或索引的周期线性无关，并应用该方法证明了卷积和恒等式和Maeda猜想的非消失结果。
$p$-adic Asai and twisted triple product $L$-functions for finite slope families
本文利用Hilbert模曲面的近过度收敛高阶Coleman理论，定义了有限斜率Hilbert模形式族的二元$p$-进Asai及扭曲三重积$L$-函数，推广了已有构造并拓展了经典$L$-值的插值范围。
Iwasawa theory of twists of elliptic modular forms over imaginary quadratic fields at inert primes
该文研究惯性素数处虚二次域上相关自守形式半普通族的岩泽理论，借助优化分解程序与特殊元素证明岩泽主猜想的整除性结果，为三变量岩泽主猜想的整除关系论证提供关键支撑。
Upper Bounds for low moments of twisted Fourier coefficients of modular forms
本文研究了模形式扭傅里叶系数低矩的上界，通过证明一个关于 $2k$-阶矩的上界，得出当 $x$ 和 $q/x$ 趋于无穷大时，相关求和的渐进行为。
On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters
本文研究模形式的Tamagawa数猜想，针对满足特定条件的正规化新形式、虚二次域与反循环Hecke特征，证明临界值的Tamagawa数猜想可由Bertolini-Darmon-Prasanna $p$进$L$函数的Iwasawa主猜想推出。
We construct a natural family of rational functions $\tildeΨ_m$ on a Hilbert modular surface from the classical $j$-invariant and its Hecke translates. These functions are obtained by means of a multiplicative analogue of the Doi-Naganuma lifting and can be viewed as twisted Borcherds products. We then study when the value of $\tildeΨ_m$ at a CM point associated to a non-biquadratic quartic CM field generates the `CM class field' of the reflex field. For the real quadratic field $\Q(\sqrt{5})$, we factorize the norm of some of these CM values to $\Q(\sqrt 5)$ numerically.
In this note, we work out the dimension of the subspace of base-change forms and their twists inside the space of Bianchi modular forms with fixed (Galois stable) level and weight.
Twisted first moment of quadratic and quadratic twist $L$-functions
本文计算了二次Dirichlet $L$-函数与模$L$-函数的二次扭曲乘积的中心值的扭曲一阶矩。
The Asai (or twisted tensor) $L$-function of a Bianchi modular form $Ψ$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. In this paper, when $Ψ$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the A…
We construct a meromorphic function on the eigencurve that interpolates a square root of the ratio of the central values of two quadratic twists of the $L$-function at classical points.
Values of quaternionic modular forms are related to twisted central $L$-values via periods and a theorem of Waldspurger. In particular, certain twisted $L$-values must be non-vanishing for forms with no zeroes. Here we study, theoretically and computationally, zeroes of definite quaternionic modular forms of trivial weight. Local sign conditions force certain forms to have trivial zeroes, but we conjecture that almost all forms have no nontrivial zeroes. In particular, almost all forms with appropriate local signs should have no zeroes. We show these conjectures follow from a conjecture on the…
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