Mahdi Asgari, Ralf Schmidt
We explicitly compute the adjoint L-function of those L-packets of representations of the group GSp(4) over a p-adic field of characteristic zero that contain non-supercuspidal representations. As an application we verify a conjecture of Gross-Prasad and Rallis in this case. The conjecture states that the adjoint L-function has a pole at s=1 if and only if the L-packet contains a generic representation.
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
Representations of the $p$-Adic $GSpin_4$ and $GSpin_6$ and the Adjoint L-Function
本文证明了Gross–Prasad关于p-进GSpin₄和GSpin₆的generic L-包可由伴随L-函数的解析性质确定的猜想，并用一般线性群的局部Langlands L-函数显式表示各L-包的伴随L-函数。
Arthur packets for quasisplit $GSp(2n)$ and $GO(2n)$ over a $p$-adic field
该研究构造了p进域上辛相似群与偶正交相似群的Arthur包，验证了其稳定性及满足扭内视特征关系，推进了p进域上相关群的Arthur包理论研究。
Local Langlands Conjecture for $p$-adic $GSpin_4,$ $GSpin_6,$ and their inner forms
本文针对 p-进 GSpin_4、GSpin_6 及其内形式建立局部朗兰兹猜想，构造 L-包并验证其性质，同时在多种情形下确定了 L-包的精确大小。
We investigate the alternating sum of the l-adic cohomology of the Rapoport-Zink tower for GSp(4) by the Lefschetz trace formula. Under some assumptions on L-packets of GSp(4) and its inner form, we observe that the local Jacquet-Langlands correspondence appears in the cohomology.
We compute the images of motivic cohomology classes for GSp(4) x GL(2) and GSp(4) x GL(2) x GL(2) under the syntomic regulator, and relate them to non-critical values of p-adic L-functions.
In this paper, we study the l-adic cohomology of the Rapoport-Zink tower for GSp(4). We prove that the smooth representation of GSp_4(Q_p) obtained as the i-th compactly supported l-adic cohomology of the Rapoport-Zink tower has no quasi-cuspidal subquotient unless i=2,3,4. Our proof is purely local and does not require global automorphic methods.
p-adic Hodge parameters in the crystalline representations of GSp4
本文研究了GSp4的晶体表示中p进Hodge参数的应用，通过构造显式的局部解析表示来确定表示ρ，并探讨了局部-全局兼容性。
This paper proves the local Langlands conjecture for the non quasi-split inner form Sp(1,1) of Sp(4) over a p-adic field of characteristic 0, by studying the restriction of representations from the non quasi-split inner form GSp(1,1) of GSp(4) to Sp(1,1). The L-packets for Sp(1,1) are constructed based on the earlier work on the local Langlands correspondence for GSp(1,1) by Gan and Tantono. To parameterize them in terms of so-called S-groups, we establish and utilize the local Langlands correspondence for reductive dual groups which participate in the theta correspondence with Sp(1,1) and GSp…
We use the "higher Hida theory" recently introduced by the second author to p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4), contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree 4 (spin) L-function of automorphic representations of GSp(4), and the degree 8 L-function of GSp(4) x GL(2).
We develop a (largely conjectural) theory of p-adic L-functions interpolating square roots of central L-values for automorphic forms on GSp(4) x GL(2) x GL(2), and a relation between these p-adic L-functions and families of Galois cohomology classes interpolating algebraic cycles. Our theory is a generalisation of the theory of "diagonal cycles" developed by Darmon and Rotger for the GL(2) triple product.
$p$-adic rigidity for $\mathrm{GSp}_4$
本文研究了$\mathrm{GSp}_4$上特定非 cuspidal 自动形 Saito-Kurokawa 表示的$p$-adic 刚性，揭示了其在$p$-adic 家族中无法非平凡插值的性质，并识别了$\mathrm{GSp}_4$特征谱上的$p$-adic 变化障碍。
We prove an algebraicity result for certain critical value of adjoint $L$-functions for ${\rm GSp}_4$ over a totally real number field in terms of the Petersson norm of normalized generic cuspidal newforms on ${\rm GSp}_4$. This is a generalization of our previous result arXiv:1902.06429.
$p$-adic $L$-functions for ${\rm GSp}(4)\times {\rm GL}(2)$ II
本文为 ${\ m GSp}(4)\ imes{\ m GL}(2)$ 上的尖点 Hida 族构造四变量 $p$-进 $L$-函数，证明完整插值公式，并借助部分插值公式与已有 $p$-进 $L$-函数计算阿基米德 zeta 积分。
We determine test vectors and explicit formulas for all Bessel models for those Iwahori-spherical representations of GSp(4) over a p-adic field that have non-zero vectors fixed under the Siegel congruence subgroup.
$p$-adic $L$-functions for ${\rm GSp}(4)\times{\rm GL}(2)$
本文针对由两类 $p$-常点模形式生成的尖点自守表示，构造了插值 ${\rm GSp}(4)\ imes{\ m GL}(2)$ 八次 $L$-函数临界值的 $p$-进 $L$-函数。
Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$
本文构造了Siegel模形式与模形式扭曲的$p$-adic $L$-函数，利用近过收敛模形式空间和Gan-Gross-Prasad周期，在Coleman族中实现插值。
We study the adjoint Bloch--Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with $\mathrm{GSp}_{2g}$. Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.
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.
Correspondence of discrete series representations of $\mathrm{GSp}_{2n}$ and its inner form
对于 $p>2$ 和 $n\geq 2$，在 $p$-adic 域上的 $\mathrm{GSp}_{2n}$ 及其内形式的离散系列表示之间构造了具有特征方程的对应，推广了 Chan 和 Gan 的工作以及局部 Jacquet-Langlands 对应，并部分揭示了 $\mathrm{GSp}_{2n}$ 及其内形式在 $p$-adic 域上的局部 Langlands 对应。
Wall-crossing and $p$-adic Artin formalism for ${\rm GSp}_4 \times {\rm GL}_2 \times {\rm GL}_2$
本文在 ${\ m GSp}_4 \ imes {\ m GL}_2 \ imes {\ m GL}_2$ 自守形式 $p$-adic 族中发展 $p$-adic Artin 形式主义，基于双重壁穿越原理，提出猜想并给出代数 $p$-adic $L$-函数及内窥镜情形下的证据。
On $p$-adic $L$-functions for $\text{GSp}_4 \times \text{GL}_2$
本文借助高阶 Coleman 理论构造 $\ ext{GSp}_4 \ imes \ ext{GL}_2$ 的新 $p$-进 $L$-函数，通过研究 $H^1$ 类的插值拓展可及权范围，并证明其复 $L$-值插值性质。
We study parabolically induced representations for $GSpin_m(F)$ with $F$ a $p$--adic field of characteristic zero. The Knapp-Stein $R$--groups are described and shown to be elementary two groups. We show the associated cocycle is trivial proving multiplicity one for induced representations. We classify the elliptic tempered spectrum. For $GSpin_{2n+1}(F)$, we describe the Arthur (Endoscopic) $R$--group attached to Langlands parameters, and show these are isomorphic to the corresponding Knapp-Stein $R$--groups.
On Petersson norms of generic cusp forms and special values of adjoint $L$-functions for ${\rm GSp}_4$
本文证明了 ${\ m GSp}_4$ 上一般尖点新形式 Petersson 范数的显式公式，将其表示为伴随 $L$-函数特殊值与局部初等常数的关系。
This article is a companion to several works of the author and others on the arithmetic of automorphic forms for GSp(4), and their associated L-functions and Galois representations. These works require, at various points, an input from smooth representation theory over p-adic local fields: the computation of values of the unique trilinear form on an unramified representation of GSp(4) x GL(2) x GL(2) invariant under the diagonal GL(2) x GL(2), for various different input data. In this note, we carry out these local computations, using a general formula for Shintani functions due to Gejima.
In the first part of the book, we classify the automorphic representations of {\rm GSp}(2) which are invariant under tensor product with a given quadratic idèle class character, via the lifting of automorphic representations of twisted endoscopic groups. In the second part of the book, we classify the admissible representations of {\rm GSp}(2, k), k a p-adic field, which are invariant under tensor product with a given quadratic character of k^\times. This classification is given in terms of twisted character identities among the admissible representations of {\rm GSp}(2, k) and those of its tw…
The present paper studies Hecke rings derived by the automorphism groups of certain algebras $L_p$ over the ring of $p$-adic integers. Our previous work considered the case where $L_p$ is the Heisenberg Lie algebra (of dimension 3) over the ring of $p$-adic integers. Although this Hecke ring is noncommutative, we showed that a formal power series with coefficients in this Hecke ring satisfies an identity similar to the rationality of the Hecke series for $GL_2$ due to E.~Hecke. In the present paper, we establish an analogous result in the case of the Heisenberg Lie algebra of dimension 5 over …
Spherical varieties and p-adic families of cohomology classes
本文证明了与对称空间相关的上同调类p-进族的扭转相容性，表明单一族可插值不同权重的经典上同调类，推广了Euler系统与p-进L-函数的先前结果，并给出了GSp(4)及相关群的Euler系统新应用。
Let F be a p-adic field and let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over F. We show here that if n is odd then all the genuine irreducible representations of G(n) are induced from a normal subgroup of finite index closely related to G`(n). Thus, we reduce, in this case, the theory of genuine admissible representations of G(n) to the better understood corresponding theory of G`(n). For odd n we also prove the uniqueness of certain Whittaker functionals along with Rodier type of Heredit…
Let $p$ be a prime, and let $Γ=\Sp_g(\Z)$ be the Siegel modular group of genus $g$. We study $p$-adic families of zeta functions and Siegel modular forms. $L$-functions of Siegel modular forms are described in terms of motivic $L$-functions attached to $\Sp_g$, and their analytic properties are given. Critical values for the spinor $L$-functions and $p$-adic constructions are discussed. Rankin's lemma of higher genus is established. A general conjecture on a lifting from $ GSp_{2m} \times GSp_{2m}$ to $GSp_{4m}$ (of genus $g=4m$) is formulated. Constructions of $p$-adic families of Siegel modu…
We prove the Bloch--Kato conjecture for certain critical values of degree 8 $L$-functions associated to cusp forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$. We also construct a $p$-adic Eichler--Shimura isomorphism in Hida families for $\mathrm{GSp}_4$, relating $H^2$ of automorphic vector bundles with $\mathbf{D}_{\mathrm{cris}}$ of a subquotient of étale cohomology.
A theta operator for the group $\mathrm{GSp}_4$
本文利用修订的 Howe 方法，在 Siegel 三维簇上构造了 $\mathrm{GSp}_4$ 的 $p$-进模形式层上的 theta 算子。
Algebraicity of $L$-values for $\text{GSp}_4 \times \text{GL}_2$ and $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$
本文证明 $\text{GSp}_4\ imes\ ext{GL}_2$ 临界 $L$ 值及相关 Gan–Gross–Prasad 周期的代数性，以新方法重证 Morimoto 结果，并研究阿基米德与 $p$-进 zeta 积分。
We construct global cohomology classes in the middle degree cohomology of the Shimura variety of the symplectic group $GSp_6$ compatible when one varies the level at $p$. These classes are expected constituents of an Euler system for the Galois representations appearing in these cohomology groups. As an application, we show how these classes provide elements in the Iwasawa cohomology of these representations and, by applying Perrin-Riou's machinery, $p$-adic L-functions associated to them.
Siegel $\mathfrak{p}^2$ Vectors for Representations of $GSp(4)$
本文研究 $GSp(4)$ 的不可约表示中 Siegel 同余子群 $\mathfrak{p}^2$ 级的不变向量空间维数及 Atkin-Lehner 对合的符号。
Let pi be a cuspidal, automorphic representation of GSp(4) attached to a Siegel modular form of degree 2. We refine the method of Furusawa to obtain an integral representation for the degree-8 L-function L(s,pi x tau), where tau runs through certain cuspidal, automorphic representation of GL(2). Our calculations include the case of square-free level for the p-adic components of tau, and a wide class of archimedean types including Maass forms. As an application we obtain a special value result for L(s,pi x tau).
The Euler characteristic of $\ell$-adic local systems on $\mathcal{A}_n$
研究$\ell$-adic局部系统在$\mathcal{A}_n$上的欧拉特征，证明了与自守表示相关的$\mathrm{GSpin}$-valued伽罗瓦表示的存在性，并给出了紧支撑欧拉特征的显式公式。
Furusawa has given an integral representation for the degree 8 L-function of GSp(4) x GL(2) and has carried out the unramified calculation. The local p-adic zeta integrals were calculated in our earlier work under the assumption that the GSp(4) representation πis unramified and the GL(2) representation τhas conductor p. In the present work we generalize to the case where the GL(2) representation has arbitrarily high conductor. The result is that the zeta integral represents the local Euler factor L(s,π\times τ) in all cases. As a global application we obtain a special value result for a GSp(4)…
Let $K$ be an imaginary quadratic field, with associated quadratic character $α$. We construct an analytic $p$-adic $L$-function interpolating the twisted adjoint $L$-values $L(1, \mathrm{ad}(f) \otimes α)$ as $f$ varies in a Hida family; these special values are non-critical in the sense of Deligne. Our approach is based on Greenberg--Stevens' idea of $Λ$-adic modular symbols, which considers cohomology with values in a space of $p$-adic measures.
We prove surjectivity criteria for $p$-adic representations and we apply them to abelian varieties over number fields. In particular, we provide examples of Jacobians over $\dbQ$ of dimension $d\in\{1,2,3\}$ whose 2-adic representations have as images $GSp_{2d}(\dbZ_2)$.
Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a quadratic twist the isogeny class of $E$. To do this, we prove a result on the determination of isobaric representations of $\text{GL}(3, \mathbb{A}_\mathbb{Q})$ by certain $L$-values of $p$-power twists.
Factorization of Algebraic $p$-adic $L$-functions Attached to Adjoint Representations of Coleman Families: Non-critical Case
该研究针对模形式Coleman族伴随表示的代数$p$-进$L$-函数，借助多类数论工具证明含Selmer复形行列式的分解公式，建立$p$-进Artin形式体系并推广了前人结果至$p$-非平凡情形。
On $p$-adic adjoint $L$-functions for Bianchi cuspforms: the $p$-split case
本文通过 Bianchi 三维簇过收敛上同调上的 Hecke 等变配对，在 $p$ 分裂情形构造 Bianchi 尖点形式的 $p$-进伴随 $L$-函数，刻画特征簇分歧轨迹并证明其在特定点非零。
$p$-adic monodromy and mod $p$ unlikely intersections, I
本文提出特征p下的Mumford-Tate和André-Oort猜想的类比，并通过Ax-Schanuel类型的猜想证明了这两个猜想，同时解决了Chai的Tate线性猜想。
Simple supercuspidal representations of $\mathrm{GSp}_4$ and test vectors
本文研究GSp_4上的简单超cuspidal表示，证明其导数指数为5，并推导了与这些表示相关的矩阵系数公式及关键局部积分公式，进而得到关于自守表示的显式全局周期公式。
Hirzebruch-Zagier cycles in $p$-adic families and adjoint $L$-values
本文研究全实域二次扩张上的Hirzebruch-Zagier循环的$p$-进族构造，并利用基变换理论给出Hida族的多元$p$-进伴随$L$-函数的几何构造。
Let G be a connected simple adjoint p-adic group not isomorphic to a projective linear group PGL(m,D) of a division algebra D, or an adjoint ramified unitary group of a split hermitian form in 3 variables. We prove that G admits an irreducible admissible supercuspidal (i.e. supersingular) representation over any field of characteristic p.
A $p$-adic adjoint $L$-function and the ramification locus of the Hilbert modular eigenvariety
本文研究了全实域上 Hilbert 模特征簇的分歧轨迹与伴随 $L$-值 $p$-进性质的关系，通过解析扭曲 Poincaré 配对和 $L$-理想理论建立联系，推广了 $F=\mathbb{Q}$ 的情形。
Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $τ$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G τ$ and we show that $\mathrm{Ind…
We prove several surjectivity criteria for $p$-adic representations. In particular, we classify all adjoint and simply connected group schemes $G$ over the Witt ring $W(k)$ of a finite field $k$ such that the epimorphism $G(W_2(k))\twoheadrightarrow G(k)$ has a section.
Let F be a p-adic field of odd residual characteristic. Let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and the symplectic group attached to the 2n dimensional symplectic space over F. Let T be a genuine, possibly reducible, unramified principal series representation of G(n). In these notes we give an explicit formulas for a spanning set for the space of Spherical Whittaker functions attached to T. For odd n, and generically for even n, this spanning set is a basis. The signicant property of this set is that each of its elements is unchanged under the action…
Let F be a totally real field and G=GSp(4)_{/F}. In this paper, we show under a weak assumption that, given a Hecke eigensystem lambda which is (p,P)-ordinary for a fixed parabolic P in G, there exists a several variable p-adic family underline{lambda} of Hecke eigensystems (all of them (p,P)-nearly ordinary) which contains lambda. The assumption is that lambda is cohomological for a regular coefficient system. If F=Q, the number of variables is three. Moreover, in this case, we construct the three variable p-adic family rho_{underline{lambda}} of Galois representations associated to underline…
In this paper, we prove Deligne's conjecture for symmetric sixth $L$-functions of Hilbert modular forms. We extend the result of Morimoto based on a different approach. We define automorphic periods associated to globally generic $C$-algebraic cuspidal automorphic representations of ${\rm GSp}_4$ over totally real number fields whose archimedean components are (limits of) discrete series representations. We show that the algebraicity of critical $L$-values for ${\rm GSp}_4 \times {\rm GL}_2$ can be expressed in terms of these periods. In the case of Kim-Ramakrishnan-Shahidi lifts of ${\rm GL}_…
We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to $p$-adic $L$-functions under a conjectural formula for the Fitting ideals of some equivariant congruence modules for abelian base change.
The Hiraga-Ichino-Ikeda Conjecture for Principal Series of Split p-adic Groups
本文利用局部 Langlands 对应证明了分裂 p-adic 群主级数中不可约离散序列表示的 HII 猜想，验证了形式次数与伴随 γ-因子之间的预测公式，并先从连通中心情形推广到一般情形。
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $ρ$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $ρ$. This generalizes previous work of the author and James Newton.
We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…
An anticyclotomic Euler system for adjoint modular Galois representations
本文为基变换到虚二次域的椭圆模形式伴随表示构造了反分圆 Euler 系统，并将其与酉群的 $p$-adic $L$-函数联系起来，从而推导出秩零 Bloch-Kato 猜想的新情形及 Iwasawa 主猜想的整除性。
A canonical generator for congruence ideals of Hida families
本文通过Ohta配对构造标准伴随$p$-adic $L$-函数，生成Hida族的同余理想，并将其与原始伴随Selmer群的特征函数联系起来。
Compatibility of $F$-isocrystals on adjoint Shimura varieties
研究伴随志村簇上典范\(\ell -\)进局部系和典范\(F -\)等晶体的相容性，方法依赖典范\(p -\)进局部系晶体性、Margulis超刚性及相关构造，扩展了前两位作者过去的结果。
Cycles for Rankin-Selberg $L$-functions, I: automorphic periods
该研究建立自守周期显式公式，为推广Bertolini-Darmon-Prasanna公式，证明适配p进插值的GU(1,1)周期公式，引入p亏格Hecke算子以关联自守周期与p进Abel-Jacobi映射像，服务于p进Rankin-Selberg L-函数特殊值研究。
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。