Non-vanishing of automorphic L-functions of prime power level

Effective log-free zero density estimates for automorphic $L$-functions and the Sato-Tate conjecture

Let $K/\mathbb{Q}$ be a number field. Let $π$ and $π^\prime$ be cuspidal automorphic representations of $\mathrm{GL}_d(\mathbb{A}_K)$ and $\mathrm{GL}_{d^\prime}(\mathbb{A}_K)$, and suppose that either both $d$ and $d'$ are at most 2 or at least one of $π$ and $π^\prime$ is self-dual. When $d=d^\prime=2$, we prove an unconditional and effective log-free zero density estimate for the Rankin-Selberg $L$-function $L(s,π\otimesπ^\prime,K)$. For other choices of $d$ and $d^\prime$, we obtain similar results assuming that either $π$ or $π^\prime$ satisfies the generalized Ramanujan conjecture. With …

A comparison of automorphic and Artin L-series of GL(2)-type agreeing at degree one primes

Let $F/k$ be a cyclic extension of number fields of prime degree. Let $ρ$ be an irreducible $2$-dimensional representation of Artin type of the absolute Galois group of $F$, and $π$ a cuspidal automorphic representation of GL$_2(\mathbb A_F)$, such that the $L$-functions $L(s,ρ_v)$ and $L(s,π_v)$ agree at all (but finitely many of) the places $v$ of degree one over $k$. We prove in this case that we have the global identity $L(s,ρ)=L(s,π)$, with $ρ_v \leftrightarrow π_v$ being given by the local Langlands correspondence at all $v$. In particular, $π$ is tempered and $L(s,ρ)$ is entire.

Hecke eigensystems of automorphic forms (mod $p$) of Hodge type and algebraic modular forms

We show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular forms$\pmod p$ associated to an inner form of $G$.

Second Moments and simultaneous non-vanishing of GL(2) automorphic L-series

We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q, with an error term that has a close to optimal power savings in the exponent. However, one of the contributions to the main term is a special value of a shifted double Dirichlet series. We show that this special value is small on average, and obtain a corresponding estimate for a mean value of the second moment over Q. This mean value is non-zero even when applied to a product of two distinct L-ser…

虚二次域上GL(2)的自守符号与自守L-值

Automorphic symbols and automorphic \emph{L}-values of GL(2) over imaginary quadratic fields

该研究将Kim–Sun的同调策略从经典模曲线推广到算术轨形，证明虚二次域上平行权2 Hecke本征形式经Coates–Wiles特征扭转的L-函数临界值，在正密度素数集合上模素非零的特征占比为正。

Congruence primes for automorphic forms on unitary groups and applications to the arithmetic of Ikeda lifts

In this paper we provide a sufficient condition for a prime to be a congruence prime for an automorphic form $f$ on the unitary group $U(n,n)(A_F)$ for a large class of totally real fields $F$ via a divisibility of a special value of the standard $L$-function associated to $f$. We also study $\ell$-adic properties of the Fourier coefficients of an Ikeda lift $I_ϕ$ (of an elliptic modular form $ϕ$) on $U(n,n)(A_{\mathbf{Q}})$ proving that they are $\ell$-adic integers which do not all vanish modulo $\ell$. Finally we combine these results to show that the condition of $\ell$ being a congruence …

The first non-vanishing quadratic twist of an automorphic L-series

Let pi be an automorphic representation on GL(r, A_Q) for r=1, 2, or 3. Let d be a fundamental discriminant and chi_d the corresponding quadratic Dirichlet character. We consider the question of the least d, relative to the data (level, weight or eigenvalue) of pi, such that the central value of the twisted L-series is nonzero, i.e. L(1/2, pi x chi_d) != 0. For example, let N be the level of pi. Using multiple Dirichlet series, we prove the nonvanishing of a central twisted $L$-value with d << N^{1/2} for GL(1), d << N^1 for GL(2), and d << N^2 for GL(3), the last case assuming that a certain …

特征 $p$ 下向量值 Siegel 自守形式的消失定理

A vanishing theorem for vector-valued Siegel automorphic forms in characteristic $p$

本文证明了特征 p 下向量值 Siegel 自守形式在权处于显式轨迹之外时为零，从而验证了与 Goldring 提出的 Hodge 型 Shimura 簇一般猜想的一个特例。

On the images of the Galois representations attached to generic automorphic representations of GSp(4)

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for almost every prime. Moreover, by using the notion of (n,p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes.

Dirichlet twists of $\operatorname{GL}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series ass…

Cohomology vanishing for Ekedahl--Oort strata on Hilbert--Blumenthal Shimura varieties

We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl--Oort strata of the special fiber of Hilbert--Blumenthal Shimura varieties. This work vastly extends to all Ekedahl--Oort strata previous results of Diamond--Kassaei and Goldring--Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\leq 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring--Koskivirta on the set of possible weights of automorphic forms.

Weighted average values of automorphic $L$-functions

Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)} L\left(\frac{1}{2}+it,f\right)^2λ_f(p^j),\qquad j=1,2, $$ where $λ_f(p^j)$ is the $p^j$-th normalized Fourier coefficient of $f$ and $L(s,f)$ is the $L$-function associated to $f$.

New error term for the fourth moment of automorphic L functions

We improve the error term in the asymptotic formula for the twisted fourth moment of automorphic L functions of prime level and weight two proved by Kowalski, Michel and Vanderkam. As a consequence, we obtain a new subconvexity bound in the level aspect and improve the lower bound on proportion of simultaneous non-vanishing.

可解自守形式的参数

Parameters of solvable automorphic forms

本文基于塔特与塞雷1974年的通信，研究了素数幂层级的马索波形在复偶表示下的分类，揭示了某些素数下存在不同类型的马索自守形式。

k-hypermonogenic automorphic forms

In this paper we deal with monogenic and $k$-hypermonogenic automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. Monogenic automorphic forms are exactly the 0-hypermonogenic automorphic forms. In the first part we establish an explicit relation between $k$-hypermonogenic automorphic forms and Maaß wave forms. In particular, we show how one can construct from any arbitrary non-vanishing monogenic automorphic form a Clifford algebra valued Maaß wave form. In the second part of the paper we compute the Fourier expansion of the $k$-hypermonogenic Eisenstein series which provide u…

正特征Siegel簇上自守向量丛的凝聚上同调的消失结果

Vanishing results for the coherent cohomology of automorphic vector bundles over the Siegel variety in positive characteristic

本文证明了正特征Siegel簇上自守向量丛的凝聚上同调在特定权条件下的消失结果，并提供了Sage算法以显式计算。

p-adic L-functions of automorphic forms and exceptional zeros

Let F be a number field, p a prime number. We construct the (multi-variable) p-adic L-function of an automorphic representation of $GL_2(A_F)$ (with certain conditions at places above p and $\infty$), which interpolates the complex L-function at the central critical point. We use this construction to prove that the p-adic L-function of a modular elliptic curve E over F has vanishing order greater or equal to the number of primes above p at which E has split multiplicative reduction, as predicted by the exceptional zero conjecture. This is a generalization of analogous results by Spieß (arXiv:1…

Functoriality of automorphic L-invariants and applications

We study the behaviour of automorphic L-Invariants associated to cuspidal representations of GL(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard non-vanishing hypothesis on automorphic L-functions and some technical restrictions on the automorphic representation and the base field we get a simple proof of the equality of automorphic and arithmetic L-invariants. This together with Spiess' results on p-adic L-functions yields a new proof of the exceptional zero conjecture for modular elliptic curves - at …

$\operatorname{SL}_2(\mathbb{R})$ 自守核的加权平均，第一部分：非振荡函数

Weighted averages of $\operatorname{SL}_2(\mathbb{R})$ automorphic kernel, Part I: non-oscillatory functions

本文证明了一个计算由 $\operatorname{SL}_2(\mathbb{Z})$ 同余子群参数化的和的加权平均的定理，通过将谱方法直接应用于自守核以更好地保留对称性，并应用于二次多项式获得了关于最大素因子和根等分布的新结果。

On higher congruences between automorphic forms

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π_0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π_0 and other automorphic forms. We apply this result to several situations including the congruences described by Mazur's Eisenstein ideal.

Griffiths-Schmid conditions for automorphic forms via characteristic $p$

We establish vanishing results for spaces of automorphic forms in characteristic $0$ and characteristic $p$. We prove that for Hodge-type Shimura varieties, the weight of any nonzero automorphic form in characteristic $0$ satisfies the Griffiths-Schmid conditions, by purely algebraic, characteristic $p$ methods. We state a conjecture for general Hodge-type Shimura varieties regarding the vanishing of the space of automorphic forms in characteristic $p$ in terms of the weight. We verify this conjecture for unitary PEL Shimura varieties of signature $(n-1,1)$ at a split prime.

Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\…

Periodic twists of $GL_3$-automorphic forms

We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.

Analytic ranks of automorphic L-functions and Landau-Siegel zeros

We relate the study of Landau-Siegel zeros to the ranks of Jacobians $J_0(q)$ of modular curves for large primes $q$. By a conjecture of Brumer-Murty, the rank should be equal to half of the dimension. Equivalently, almost all newforms of weight two and level $q$ have analytic rank $\leq 1$. We show that either Landau-Siegel zeros do not exist, or that almost all such newforms have analytic rank $\leq 2$. In particular, almost all odd newforms have analytic rank equal to one. Additionally, for a sparse set of primes $q$ we show the rank of $J_0(q)$ is asymptotically equal to the rank predicted…

Automorphic forms for triangle groups: Integrality properties

We classify all primes appearing in the denominators of the Hauptmodul $J$ and modular forms for non-arithmetic triangle groups with a cusp. These primes have a congruence condition in terms of the order of the generators of the group. As a corollary we show that for the Hecke group of type $(2,m,\infty)$, the prime $p$ does not appear in the denominator of $J$ if and only if $p\equiv \pm 1\pmod m$.

CM 域上一般线性群的正则代数自守伽罗瓦表示的伴随布洛赫 - 加藤塞尔默群

Adjoint Bloch--Kato Selmer groups of regular algebraic automorphic Galois representations

该研究证明了 CM 域上一般线性群正则代数自守表示相关伽罗瓦表示的伴随布洛赫 - 加藤塞尔默群消失，关键在于仅对 p 进伽罗瓦表示而非其模 p 剩余表示施加条件。

Toroidal automorphic forms, Waldspurger periods and double Dirichlet series

The space of toroidal automorphic forms was introduced by Zagier in the 1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier coefficients along all embedded non-split tori. The interest in this space stems (amongst others) from the fact that an Eisenstein series of weight s is toroidal for a given torus precisely if s is a non-trivial zero of the zeta function of the quadratic field corresponding to the torus. In this paper, we study the structure of the space of toroidal automorphic forms for an arbitrary number field F. We prove that it decomposes into a space span…

Special values of adjoint L-functions and congruences for automorphic forms on GL(n) over a number field

We prove an integrality result for the value at s=1 of the adjoint L-function associated to a cohomological cuspidal automorphic representation on GL(n) over any number field. We then show that primes (outside an exceptional set) dividing this special value give rise to congruences between automorphic forms. We also prove a non-vanishing property at infinity for the relevant Rankin-Selberg L-functions on GL(n) x GL(n).

深度方面的自动形式L⁴范数

$L^4$-norms of automorphic forms in the depth aspect

该研究证明了在深度方面，自动形式的L⁴范数的最优界，通过分析具有有界谱参数和平凡中心字符的新形式，得出了关于p和ε的渐进行为。

Level-raising for automorphic forms on $GL_n$

Let $E$ be a CM number field and $F$ its maximal real subfield. We prove a level-raising result for regular algebraic conjugate self-dual automorphic representations of $GL_n(\mathbb{A}_E)$. This generalizes previously known results of Thorne by removing certain hypotheses occurring in that work. In particular, the level-raising prime $p$ is allowed to be unramified as opposed to inert in $F$, the field $E/F$ is not assumed to be everywhere unramified, and the field $E$ is allowed to be a general CM field.

Deligne's conjecture for automorphic motives over CM-fields

The present paper is devoted to the relations between Deligne's conjecture on critical values of motivic $L$-functions and the multiplicative relations between periods of arithmetically normalized automorphic forms on unitary groups. In the first place, we combine the Ichino--Ikeda--Neal-Harris (IINH) formula -- which is now a theorem -- with an analysis of cup products of coherent cohomological automorphic forms on Shimura varieties to establish relations between certain automorphic periods and critical values of Rankin-Selberg and Asai $L$-functions of ${\rm GL}(n)\times{\rm GL}(m)$ over CM …

Twists of automorphic L-functions at the central point

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.

Irreducibility of automorphic Galois representations of GL(n), n at most 5

Let pi be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GL_n(A_F), where F is a totally real field and n is at most 5. We show that for all primes l, the l-adic Galois representations associated to pi are irreducible, and for all but finitely many primes l, the mod l Galois representations associated to pi are also irreducible. We also show that the Lie algebras of the Zariski closures of the l-adic representations are independent of l.

Linnik-type problems for automorphic L-functions; Problemes de type Linnik pour les fonctions L de formes automorphes

The thesis gave a fine study on the distribution of the coefficients of automorphic L-functions for GL(m) with m>1. In particular we have treated two types of problems: change of signs of these coefficients (when they are real) and their decompensation on the prime numbers. Chapter 2 is devoted to the results known concerning the first problem in the case of classical modular forms and to explain our motivation of research. In Chapter 3, we treated the case of primitive Maass forms and obtained an estimate of the subconvexity-type for the least integer n such that the n-th coefficient is negat…

大偶数和奇数正交族自守L函数的一级密度

One-level densities of large even and odd orthogonal families of automorphic L-functions

本文在广义黎曼猜想条件下，研究了大偶数和奇数正交族自守L函数的一级密度，扩展了傅里叶变换测试函数的支持范围，并获得了这些L函数及其在中心点导数的最强非消失结果。

Oscillations of coefficients of Dirichlet series attached to automorphic forms

For $m\ge 2$, let $π$ be an irreducible cuspidal automorphic representation of $GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $a_π(n)$ be the $n^{th}$ coefficient of the $L$-function attached to $π$. Goldfeld and Sengupta have recently obtained a bound for $\sum_{n\le x} a_π(n)$ as $x \rightarrow \infty$. For $m\ge 3$ and $π$ not a symmetric power of a $GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $π$, their bound is better than all previous bounds. In this paper, we further improve the bound of Golfeld and Se…

On the rationality and holomorphy of Langlands-Shahidi L-functions over function fields

We prove three main results: all Langlands-Shahidi automorphic $L$-functions over function fields are rational; after twists by highly ramified characters our automorphic $L$-functions become polynomials; and, if $π$ is a globally generic cuspidal automorphic representation of a split classical group or a unitary group ${\bf G}_n$ and $τ$ a cuspidal (unitary) automorphic representation of a general linear group, then $L(s,π\times τ)$ is holomorphic for $\Re(s) > 1$ and has at most a simple pole at $s=1$. We also prove the holomorphy and non-vanishing of automorphic exterior square, symmetric s…

Hecke-Maass cusp forms on PGL_n of large levels with non-vanishing L-values

We introduce a new trace formula of Kuznetsov type involving the central standard L-values and the Whittaker periods of cuspidal automorphic representations of PGL_n(Q) which are spherical at the archimedean place. As an application, we show a simultaneous non-vanishing of standard L-values at n-1 points on the critical strip for infinitely many Hecke-Maass cuspidal newforms.

On the Rigid Cohomology of Certain Shimura Varieties

We construct the compatible system of $l$-adic representations associated to a regular algebraic cuspidal automorphic representation of $GL_n$ over a CM (or totally real) field and check local-global compatibility for the $l$-adic representation away from $l$ and finite number of rational primes above which the CM field or the automorphic representation ramify. The main innovation is that we impose no self-duality hypothesis on the automorphic representation.

函数域上尖点自守表示的循环基变换

Cyclic base change of cuspidal automorphic representations over function fields

本文利用模性提升定理与Smith理论，证明了函数域上尖点自守表示沿循环扩张的基变换，并推广至局部域不可约容许表示及环面超尖点表示的情形。

Existence of Hilbert cusp forms with non-vanishing $L$-values

We give a derivative version of the relative trace formula on PGL(2) studied in our previous work, and obtain a formula of an average of central values (derivatives) of automorphic $L$-functions for Hilbert cusp forms. As an application, we prove existence of Hilbert cusp forms with non-vanishing central values (derivatives) such that the absolute degrees of their Hecke fields are sufficiently large.

Galois representations associated with a non-selfdual automorphic representation of GL(3)

In 1994, van Geemen and Top constructed a non-selfdual motive of rank three over $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdual automorphic representation of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$ of level $Γ_0(128)$. They experimentally confirmed the coincidence of the local $L$-factors at finitely many primes using computer. In this paper, we shall prove the coincidence of the local $L$-factors at every prime. To show this, we use the recent results of Harris-Lan-Taylor-Thorne and Scholze on the construction of Galois representations, and Grenié's results to compare thr…

关于 Weil 表示的向量值自守形式

On vector valued automorphic forms for the Weil representation

本文发展了与 Weil 表示相关的向量值自守形式理论，并确定了每个素数 p 下相应的球面 Hecke 代数结构。

Reflective modular forms on lattices of prime level

One of the main open problems in the theory of automorphic products is to classify reflective modular forms. In [Sch06] Scheithauer classified strongly reflective modular forms of singular weight on lattices of prime level. In this paper we classify symmetric reflective modular forms on lattices of prime level. This yields a full classification of lattices of prime level which have reflective modular forms. We also present some applications.

Geometric level raising for p-adic automorphic forms

We present a level raising result for families of p-adic automorphic forms for a definite quaternion algebra D over the rational numbers. The main theorem is an analogue of a theorem for classical automorphic forms due to Diamond and Taylor. We show that certain families of forms old at a prime l intersect with families of l-new forms (at a non-classical point). One of the ingredients in the proof of Diamond and Taylor's theorem (which also played a role in earlier work of Taylor) is the definition of a suitable pairing on the space of automorphic forms. In our situation one cannot define such…

Endoscopic congruences modulo adjoint $L$-values for $\mathrm{GSp}(4)$

We establish the existence of congruences between a fixed endoscopic cuspidal automorphic representation $Π$ of $\mathrm{GSp}(4)$ of square-free conductor and stable cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at $1$ of the adjoint $L$-function of $Π$ normalized by a suitable period.

On the Non-vanishing of the Central Value of Certain L-functions: Unitary Groups

Let $π$ be an irreducible cuspidal automorphic representation of a quasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field $F$. Under the assumption that $π$ has a generic global Arthur parameter, we establish the non-vanishing of the central value of $L$-functions, $L(\frac{1}{2},π\timesχ)$, with a certain automorphic character $χ$ of ${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general ${\mathfrak n}\geq 5$ by assuming a conjecture on certain refined properties of global Arthur packets. In consequence, we obtain some simultaneous non-vanishing results…

A log-free zero-density estimate and small gaps in coefficients of $L$-functions

Let $L(s, π\timesπ^\prime)$ be the Rankin--Selberg $L$-function attached to automorphic representations $π$ and $π^\prime$. Let $\tildeπ$ and $\tildeπ^\prime$ denote the contragredient representations associated to $π$ and $π^\prime$. Under the assumption of certain upper bounds for coefficients of the logarithmic derivatives of $L(s, π\times\tildeπ)$ and $L(s, π^\prime\times\tildeπ^\prime)$, we prove a log-free zero-density estimate for $L(s, π\timesπ^\prime)$ which generalises a result due to Fogels in the context of Dirichlet $L$-functions. We then employ this log-free estimate in studying …

Automorphic forms and cohomology theories on Shimura curves of small discriminant

We compute the homotopy groups of spectra associated by a theorem of Lurie to the Shimura curves of discriminants 6, 10, and 14, beginning with a computation of integral rings of automorphic forms on these curves. As an application, we find that a generalized truncated Brown-Peterson spectrum BP<2> is an E_\infty ring spectrum at the prime 3.

对模形式对称立方转移的两个性质研究

Two properties of symmetric cube transfers of modular forms

本文研究对称立方转移的自守表示的导数和局部自守表示类型，通过ε因子的变化来分类可能的转移类型，重点分析p=3的情况。

Bessel periods and the non-vanishing of Yoshida lifts modulo a prime

We give an explicit construction of vector-valued Yoshida lifts and derive a formula of the Bessel periods of Yoshida lifts, by which we prove the non-vanishing modulo a prime of Yoshida lifts attached to a pair of elliptic modular newforms. As a consequence, we obtain a new proof of the non-vanishing of Yoshida lifts.

某些模 $L$-函数的素数扭曲的非零定理

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。

Non-vanishing of quadratic twists of modular $L$-functions of prime-related moduli

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.

Automorphic representations with prescribed ramification for unitary groups

Let F be a totally real number field, n a prime integer, and G a unitary group of rank n defined over F that is compact at every infinite place. We prove an asymptotic formula for the number of cuspidal automorphic representations of G whose factors at finitely many places are prescribed up to inertia. The results and the methods used are a generalization to this setting of those used by Weinstein for GL_2.