Sandro Bettin
We compute the first moment of twisted Hecke $L$-functions of prime power level going to infinity, uniformly in the conductor of the twist and in the vertical shift.
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Suppose $π_1$ and $π_2$ are two Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ such that for all primitive character $χ$ we have $$ L(\tfrac{1}{2},π_1\otimesχ)=L(\tfrac{1}{2},π_2\otimesχ). $$ Then we show that $π_1=π_2$.
Let $f_1,...,f_d$ be an orthogonal basis for the space of cusp forms of even weight $2k$ on $Γ_0(N)$. Let $L(f_i,s)$ and $L(f_i,χ,s)$ denote the $L$-function of $f_i$ and its twist by a Dirichlet character $χ$, respectively. In this note, we obtain a ``trace formula'' for the values $L(f_i,χ,m)\overline{L(f_i,n)}$ at integers $m$ and $n$ with $0<m,n<2k$ and proper parity. In the case N=1 or N=2, the formula gives us a convenient way to evaluate precisly the value of the ratio $L(f,χ,m)/L(f,n)$ for a Hecke eigenform $f$.
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。
In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is $\sqrt{n}$ at prime arguments.
A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke eigenvalue. Direct corollaries include subconvexity bounds for L-functions in the level aspect and a short proof of an upper bound for the fifth moment of automorphic L-functions.
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。
Let $q \in \mathbb{Z} [i]$ be prime and $χ$ be the primitive quadratic Hecke character modulo $q$. Let $π$ be a self-dual Hecke automorphic cusp form for $\mathrm{SL}_3 (\mathbb{Z} [i] )$ and $f$ be a Hecke cusp form for $Γ_0 (q) \subset \mathrm{SL}_2 (\mathbb{Z} [i])$. Consider the twisted $L$-functions $ L (s, π\otimes f \otimes χ) $ and $L (s, π\otimes χ)$ on $\mathrm{GL}_3 \times \mathrm{GL}_2$ and $\mathrm{GL}_3$. We prove the subconvexity bounds \begin{equation*} L \big(\tfrac 1 2, π\otimes f \otimes χ\big) \ll_{\, \varepsilon, π, f } \mathrm{N} (q)^{5/4 + \varepsilon}, L \big(\tfrac 1 2…
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。
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.
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 $q$ be a large prime, and $χ$ the quadratic character modulo $q$. Let $ϕ$ be a self-dual Hecke--Maass cusp form for $SL(3,\mathbb{Z})$, and $u_j$ a Hecke--Maass cusp form for $Γ_0(q)\subseteq SL(2,\mathbb{Z})$ with spectral parameter $t_j$. We prove the hybrid subconvexity bounds for the twisted $L$-functions \[ L(1/2,ϕ\times u_j\timesχ)\ll_{ϕ,\varepsilon} (qt_j)^{3/2-θ+\varepsilon},\quad L(1/2+it,ϕ\timesχ)\ll_{ϕ,\varepsilon} (qt)^{3/4-θ/2+\varepsilon}, \] for any $\varepsilon>0$, where $θ=1/23$ is admissible.
Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average behavior of the square of the central value of this $L$-function? Stefanicki proved an asymptotic valid only for $q$ having very few prime factors, and we extend this to almost all $q$.
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.
In this paper we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residue characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy-Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semi-simple descend in the twisted space according to Waldspurger's fundamental work "L'endoscopie tordue n'est pas si tordue".
Hecke Equivariance of Divisor Lifting with respect to Sesquiharmonic Maass Forms
本文研究了Sesquiharmonic Maass形式的Hecke等变性，通过除数提升和Borcherds同构的结合，扩展了Matsusaka关于扭曲迹的研究。
We prove the transfer congruence between $p$-adic Hecke $L$-functions for CM fields over cyclotomic extensions, which is a non-abelian generalization of the Kummer's congruence. The ingredients of the proof include the comparison between Hilbert modular varieties, the $q$-expansion principle, and some modification of Hsieh's Whittaker model for Katz' Eisenstein series. As a first application, we prove explicit congruence between special values of Hasse-Weil $L$-function of a CM elliptic curve twisted by Artin representations. As a second application, we prove the existence of a non-commutative…
In this paper we prove a version of Deligne's conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.
A zero-density estimate for $L$-functions associated with $\rm GL(3)$ Hecke--Maass cusp forms
研究通过建立 $\rm SL(3,\mathbb{Z})$ Hecke–Maass 尖点形式对应 $L$-函数的扭曲二阶矩渐近公式，得到其谱方面的加权零点密度估计。
We use a relative trace formula on GL(2) to compute a sum of twisted modular L-functions anywhere in the critical strip, weighted by a Fourier coefficient and a Hecke eigenvalue. When the weight k or level N is sufficiently large, the sum is nonzero. Specializing to the central point, we show in some cases that the resulting bound for the average is as good as that predicted by the Lindelof hypothesis in the k and N aspects.
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.
Distribution of central values of Hecke $L$-functions of $n$-th order characters
该研究针对n≥3的n阶Hecke特征稀疏族，证明了log|L(1/2,χ)|的单侧中心极限定理，结合相关方法与渐近公式，还得到0<k<1时2k阶矩的猜想量级上界。
Generation of cyclotomic Hecke fields by $L$-values of cusp forms on $\mathrm{GL}(2)$ with certain $\mathbb{Z}_p$ twist
本文证明在数域 F 上 GL(2) 的代数自守新形式 f 可由其被特定 Z_p-扩张的 Galois 特征扭曲的 L-值唯一确定，并在全实或 CM 域情形下证明 Hecke 域与分圆域的合成由这些代数 L-值生成。
Towards Keating-Snaith's conjecture for cubic Hecke $L$-functions over the Eisenstein field
本文基于Radziwill和Soundararajan的工作，在广义黎曼假设下通过扩展David和Güloğlu的三次Hecke L函数零点1级密度扭曲估计，得到艾森斯坦域上稀疏三次Hecke L函数族的Keating-Snaith猜想条件下界。
On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters
该研究针对r≥3的Hecke L-函数，利用多重Dirichlet级数理论，推导其在含2r次单位根的整体域上的扭曲矩渐近公式，证明其中心值存在正比例非零值。
Let $\mathfrak{q}>2$ be a prime number, $χ$ a primitive Dirichlet character modulo $\mathfrak{q}$ and $f$ a primitive holomorphic cusp form or a Hecke-Maass cusp form of level $\mathfrak{q}$ and trivial nebentypus. We prove the subconvex bound $$ L(1/2,f\otimes χ)\ll \mathfrak{q}^{1/2-1/12+\varepsilon}, $$ where the implicit constant depends only on the archimedean parameter of $f$ and $\varepsilon$. The main input is a modifying trivial delta method developed in [1].
We establish the first moment bound $$ \sum_φ L(φ\otimes φ\otimes Ψ, \tfrac{1}{2}) \ll_{\varepsilon} p^{5/4+\varepsilon} $$ for triple product $L$-functions, where $Ψ$ is a fixed Hecke-Maass form on $\operatorname{SL}_2(\mathbb{Z})$ and $φ$ runs over the Hecke-Maass newforms on $Γ_0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows …
We study the algebraicity of the central critical values of twisted triple product $L$-functions associated to motivic Hilbert cusp forms over a totally real étale cubic algebra in the totally unbalanced case. The algebraicity is expressed in terms of the cohomological period constructed via the theory of coherent cohomology on quaternionic Shimura varieties developed by Harris. As an application, we generalize our previous result on Deligne's conjecture for certain automorphic $L$-functions for ${\rm GL}_3 \times {\rm GL}_2$. We also establish a relation for the cohomological periods under tw…
We obtain the formula for the twisted harmonic second moment of the $L$-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean value theorem is reminiscent of recent results of Conrey and Young on the reciprocity formula for the twisted second moment of Dirichlet $L$-functions.
We compute the moments of L-functions of symmetric powers of modular forms at the edge of the critical strip, twisted by the central value of the L-functions of modular forms. We show that, in the case of even powers, it is equivalent to twist by the value at the edge of the critical strip of the symmetric square L-functions. We deduce information on the size of symmetric power L-functions at the edge of the critical strip under conditions. In a second part, we study the distribution of small and large Hecke eigenvalues. We deduce information on the simultaneous extremality conditions on the v…
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…
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.
Power Savings for Counting (Twisted) Abelian Extensions of Number Fields
本文研究了数域阿贝尔扩张的计数问题，证明了误差项具有显著的幂次节省，并在假设广义Lindelöf假设下达到平方根幂次节省，同时揭示了部分未知的低阶项结构。
In the paper we prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are L-functions of Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary quadratic extension of F. The formula generalizes our former result on L-functions twisted by finite characters. It also eliminates the restriction on ramifications of Popa's paper.
We consider modular functions (i.e., the Eisenstein series and Hecke-Maass forms) for the group PSL(2,Z). We fix a quadratic number field E. This gives rise to twisted (by a Hecke character of the field E) periods of a modular function along the torus corresponding to E. We prove meromorphic continuation for a Dirichlet series generated by these twisted periods.
Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients $λ_{f}(n)$. Let $χ$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the following Weyl-type bound: for any $ε>0$, $$\sum_{|n| \ll N}λ_{f}(n)χ(n) \ll_{f,ε}N^{3/4 }p^{1/6}(pN)^ε.$$ We can see an improvement of the range $N > p^{3/4}$ to the range $N > p^{2/3}$ and we get a bound for $S_{f,χ}(N)$ without going into the $L$-function.
For a cuspidal Hecke eigenform $F$ for $Sp_n(Z)$ and a Dirichlet character $χ$ let $L(s,F,χ,St)$ be the standard $L$-function of $F$ twisted by $χ$. Boecherer showed the boundedness of denominators of the algebraic part of $L(m,F,χ,St)$ at a critical point $m$ when $χ$ varies. In this paper, we give a refined version of his result We also prove a similar result for the products of Hecke $L$ functions of primitive forms for $SL_2(Z)$.
Let $f$ be a Hecke-Maass cusp form for the full modular group and let $χ$ be a primitive Dirichlet character modulo a prime $q$. Let $s_0=σ_0+it_0$ with $\frac{1}{2}\leqσ_0<1$. We improve the error term for the first moment of $L(s_0,f\otimesχ)\overline{L(s_0,χ)}$ over the family of even primitive Dirichlet characters. As an application, we show that for any $t\in\mathbb{R}$, there exists a primitive Dirichlet character $χ$ modulo $q$ for which $L(1/2+it,f\otimesχ)L(1/2+it,χ)\neq0$ if the prime $q$ satisfies $q\gg (1+|t|)^{\frac{543}{25}+\varepsilon}$.
A twisted Hecke algebra, then and now, and a Klein bottle of tempered representations
本文研究了扭曲海丘代数及其与温特表示的克莱因瓶模型的联系，并展示了伯恩斯坦谱中点的同构性。
Bounds for moments of twisted quadratic characters of prime modulus
在广义黎曼假设下，研究素数模扭曲二次特征的傅里叶系数和的矩，确定了未平滑m阶矩的正确量级，得到平滑m阶矩的最优上界并匹配偶数m的下界。
On a multiplicative non-Hecke twist of motivic L-functions
本文研究由素理想上乘法特征诱导的非 Hecke 扭转，证明其在 $|α|<1$ 时改善 $L$-函数收敛性并保持 Euler 乘积与亚纯延拓，且可推广到 $p$-adic 情形。
On moments of L-functions over Dirichlet characters
本文为Dirichlet L-函数二阶矩的Heath-Brown完整渐近展开提供了新证明，并将相应渐近展开推广到Hecke-Maass L-函数的扭曲一阶矩。
Hybrid bounds for ${\rm{GL}}(4)\times {\rm{GL}}(1)$ twisted $L$-functions
本文研究 GL(4)×GL(1) 扭曲 L-函数的混合次凸性问题，在水平与导子两方面同时改进，证明了当 M^{1/5}<P<M^{2/5} 时的混合次凸界。
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.
Relative Trace Formula and Twisted $L$-functions: the Burgess Bound
本文发展了正则化相对迹公式，证明了扭曲 $L$-函数 $L(1/2,π\timesχ)$ 的精细混合次凸性界，并特别得到了关于解析前导子的 Burgess 次凸性界。
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。
Determination of $\textrm{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions
本文利用 GL(3) 二次扭 L 函数中心值一阶矩的渐近公式，证明所有奇无平方因子 d 上的中心值成比例即可确定两个 Hecke–Maass 尖点形式相同或互为对偶。
In this paper, we consider the (partial) symmetric square $L$-function $L^S(s,π,Sym^2\otimesχ)$ of an irreducible cuspidal automorphic representation $π$ of $\GL_r(\A)$ twisted by a Hecke character $χ$. In particular, we will show that the $L$-function $L^S(s,π,Sym^2\otimesχ)$ is holomorphic except at $s=0$ and $s=1$, and moreover the possible poles could occur only when $χ^rω^2=1$, where $ω$ is the central character of $π$. Our method of proof is essentially a (nontrivial) modification of the one by Bump and Ginzburg in which they considered the case $χ=1$.
Let $f$ be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let $χ$ be a primitive character of conductor $M$. For the twisted $L$-function $L(s,f\otimes χ)$ we establish the hybrid subconvex bound $$ L(1/2+it,f\otimesχ)\ll (M(3+|t|))^{1/2-1/18+\varepsilon}, $$ for $t\in \mathbb R$. The implied constant depends only on the form $f$ and $\varepsilon$.
The second moment of twisted modular $L$-functions and Dirichlet $L$-functions at the central point
在塞尔伯格特征值猜想假设下，研究人员证明了特定扭曲GL(2) L-函数与中心处Dirichlet L-函数二阶矩的带幂节省误差项的渐近公式，明确了求和集的约束条件。
Recurrence and congruences for the smallest parts function
本文基于Hecke迹研究最小部分函数的递归公式，并推导出同余表达式，同时证明了该函数的非同余性质。
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.
This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch $L$-functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group $H^{J}$, called here the sub-Jacobi group, which is a semi-direct product of $GL(1, {\mathbb R})$ with the Heisenberg group $H({\mathbb R})$. The Heisenberg group action on L^2-functions on the Heisenberg nilmanifold $H({\mathbb Z}) \backslash H({\mathbb R})$ decomposes as $\bigoplus_{N \in {\mathbb Z}} H_N$, where each space $H_N~ (N \neq 0)$ consists of $|N|$ copies o…
We study the theory of finite-order p-adic functions and distributions on ray class groups of number fields, and apply this to the construction of (possibly unbounded) p-adic L-functions for automorphic forms on GL(2) which may be non-ordinary at the primes above p. As a consequence, we obtain a "plus-minus" decomposition of the p-adic L-functions of automorphic forms for GL(2) over an imaginary quadratic field with p split and Hecke eigenvalues 0 at the primes above p, confirming a conjecture of B.D. Kim.
The theme of this work is the study of the Nekovář-Selmer group H^1_f(K,T) attached to a twisted Hida family T of Galois representations and a quadratic number field K. The results that we obtain have the following shape: if a twisted L-function of a suitable modular form in the Hida family has order of vanishing r at most 1 at the central critical point then the rank of H^1_f(K,T) as a module over a certain local Hida-Hecke algebra is equal to r. Under the above assumption, we also show that infinitely many twisted L-functions of modular forms in the Hida family have the same order of vanishi…
Automorphic functions for square-zero extensions of curves over finite fields
该研究针对有限域上曲线的平方零扩张，推广了PGL₂的自守函数相关结果，证明了PGL₃下某猜想的新案例，还确定了对应球面尖与赫克有限函数的最优支撑界。
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