Chi-Yun Hsu, Zhaorong Jin, Ryotaro Sakamoto
For a non-endoscopic cohomological cuspidal automorphic representation of $\mathrm{GSp}_4 \times \mathrm{GL}_2$, assumed to be $p$-ordinary, we construct an Euler system for the Galois representation associated to it. Both the construction and the verification of tame norm relations are based on Novodvorsky's integral formula for the $L$-function of $\mathrm{GSp}_4 \times \mathrm{GL}_2$.
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Euler systems for $\mathrm{GSp}(4)$ over imaginary quadratic fields
本文研究了GSp(4)上虚二次域的欧拉系统构造，并通过该系统对严格塞勒群进行界限，同时扩展了GSp(4)×GL(2)欧拉系统至动机陈述。
We prove a potentially automorphy theorem for suitable Galois representations $Γ_{F^+} \to \mathrm{GSpin}_{2n+1}(\overline{\mathbb{F}}_p)$ and $Γ_{F^+} \to \mathrm{GSpin}_{2n+1}(\overline{\mathbb{Q}}_p)$, where $Γ_{F^+}$ is the absolute Galois group of a totally real field $F^+$. We also prove results on solvable descent for $\mathrm{GSp}_{2n}(\mathbb{A}_{F^+})$ and use these to put representations $Γ_{F^+} \to \mathrm{GSpin}_{2n+1}(\overline{\mathbb{Q}}_p)$ into compatible systems of $\mathrm{GSpin}_{2n+1}(\overline{\mathbb{Q}}_{\ell})$-valued representations.
Euler systems for GSp(4)
本文利用 GL(2) x GL(2) 的 Eisenstein 类的推前，为与 GSp(4) 上同调尖点自守表示相关的 Galois 表示构造了 Euler 系统。
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 prove a cohomological formula for non-critical residues of degree eight automorphic $L$-functions of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ in the spirit of Beilinson conjecture. We rely on the cohomological interpretation of an automorphic period integral and on the study of Novodvorsky's integral representation of the $L$-functions.
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 find experimental examples of congruences of Hecke eigenvalues between automorphic representations of groups such as $\mathrm{GSp}_2(\mathbb{A})$, $\mathrm{SO}(4,3)(\mathbb{\mathbb{A}})$ and $\mathrm{SO}(5,4)(\mathbb{A})$, where the prime modulus should, for various reasons, appear in the algebraic part of a critical "tensor-product" $L$-value associated to cuspidal automorphic representations of $\mathrm{GL}_2(\mathbb{A})$ and $\mathrm{GSp}_2(\mathbb{A})$. Using special techniques for evaluating $L$-functions with few known coefficients, we compute sufficiently good approximations to detec…
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.
In this paper, we describe an algorithm for computing algebraic modular forms on compact inner forms of $\mathrm{GSp}_4$ over totally real number fields. By analogues of the Jacquet-Langlands correspondence for $\mathrm{GL}_2$, this algorithm in fact computes Hecke eigensystems of Hilbert-Siegel modular forms of genus 2. We give some examples of such eigensystems over $\Q(\sqrt{2})$.
In the present paper, we first construct a pairing on the space of analytic distributions associated with $\mathrm{GSp}_{2g}$. By considering the overconvergent parabolic cohomology groups and following the work of Johansson--Newton, we construct the cuspidal eigenvariety for $\mathrm{GSp}_{2g}$. The pairing on the analytic distributions then induces a pairing on some coherent sheaves of the cuspidal eigenvariety. As an application, we follow the strategy of Bellaïche to study the ramification locus of the cuspidal eigenvariety over the corresponding weight space.
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.
$p$-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。
A universal Euler system for GSp(4)
本文通过多重性一结果确定了GSp(4)自守形式对应的4维旋量伽罗华表示的欧拉系统类依赖于局部测试数据的选择，并展示这些类位于1维空间中，是独立于测试数据的通用类的显式倍数。
Serre weight conjectures for $\mathrm{GSp}_4$
本研究证明了满足明确一般性条件的驯分歧$\u200b\mathrm{GSp}_4\u200b$值伽罗瓦表示的Serre猜想权部分，改进了此前需非明确一般性假设的结果，并推导了相应的模性提升定理。
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。
We find the number $s_k(p,Ω)$ of cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$ with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight $k\ge 3$, and the non-archimedean component at $p$ is an Iwahori-spherical representation of type $Ω$ and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for $s_k(p,Ω)$ generalizes to the vector-valued case and a finite number of ramified places.
We build upon our previous work to study the Newton stratification on the $B_\mathrm{dR}^+$-Grassmannian for $\mathrm{GSp}_{2n}$. Our main result gives an explicit classification of all nonempty Newton strata associated to a Frobenius-conjugacy class satisfying certain conditions on the Newton polygon.
We prove the expected algebraicity property for the critical values of character twists of the standard $L$-function associated to vector-valued holomorphic Siegel cusp forms of archimedean type $(k_1, k_2, \ldots, k_n)$, where $k_n \geq n+1$ and all $k_i$ are of the same parity. For the proof, we use an explicit integral representation to reduce to arithmetic properties of differential operators on vector-valued nearly holomorphic Siegel cusp forms. We establish these properties via a representation-theoretic approach.
We prove a conjectural formula relating the Bessel period of certain automorphic forms on $\mathrm{GSp}_4$ to a central $L$-value. This formula is proposed by Liu \cite{liu} as the refined Gan-Gross-Prasad conjecture for the groups $(\SO(5), \SO(2))$. The conjecture has been previously proved for certain automorphic forms on $\mathrm{GSp_4}$ from lifts. In this paper, we extend the formula to Siegel modular forms of $\Sp_4(\bZ)$.
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。
The explicit Local Langlands Correspondence for $\mathrm{GSp}_4$, $\mathrm{Sp}_4$ and stability (with an application to Modularity Lifting)
本研究给出 $\mathrm{GSp}_4$ 与 $\mathrm{Sp}_4$ 显式局部朗兰兹对应的纯局部证明，通过 $L$-包稳定性等性质完成唯一刻画，并将其应用于模性提升问题。
We give a two variable Rankin-Selberg integral inspired by consideration of Garrett's pullback formula. For a globally generic cusp form on $\mathrm{GL}_2\times \mathrm{GSp}_4$, the integral represents the product of the $\mathrm{Std}\times \mathrm{Spin}$ and $\mathbf{1} \times \mathrm{Std}$ $L$-functions. We prove a result concerning an Archimedean principal series representation in order to verify a case of Jiang's first-term identity relating certain non-Siegel Eisenstein series on symplectic groups. Using it, we obtain a new proof of a known result concerning possible poles of these $L$-fu…
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.
Emerton--Gee stacks, Serre weights, and Breuil--Mézard conjectures for $\mathrm{GSp}_4$
本研究构造了秩4辛射影平展 $(φ,Γ)$-模的模栈并研究其几何性质，证明了局部模型的单分支性，进而在一般性条件下建立了 $\rm{GSp}_4$ 情形下的Breuil–Mézard猜想、Serre猜想权部分及模性提升结果。
We give two global integrals that unfold to a non-unique model and represent the partial Spin $L$-function on $\mathrm{GSp}_6$. We deduce that for a wide class of cuspidal automorphic representations $π,$ the partial Spin $L$-function is holomorphic except for a possible simple pole at $s=1$, and that the presence of such a pole indicates that $π$ is an exceptional theta lift from $\mathrm{G}_2$. These results utilize and extend previous work of Gan and Gurevich, who introduced one of the global integrals and proved these facts for a special subclass of these $π$ upon which the aforementioned …
Algebraicity of Spin $L$-functions for $\mathrm{GSp}_6$
本研究证明了$\u200b\u200b\u200bmathrm{GSp}_6$上全纯西格尔特征形式对应的尖点自守表示的Spin $L$-函数在对称中心右侧临界点的代数性，通过关联艾森斯坦级数性质的策略，突破了群无已知模问题、关联模型非唯一的难点。
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。
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。
On the semi-infinite Deligne--Lusztig varieties for $\mathrm{GSp}$
本文证明了GSp的半无限Deligne–Lusztig变种与无限水平仿射Deligne–Lusztig变种同构，并重新诠释了X_h的表示构造方法。
We give a Rankin-Selberg integral representation for the Spin (degree eight) $L$-function on $\mathrm{PGSp}_6$. The integral applies to the cuspidal automorphic representations associated to Siegel modular forms. If $π$ corresponds to a level one Siegel modular form $f$ of even weight, and if $f$ has a non-vanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$-function $Λ(π,Spin,s)$ of $π$.
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)…
Rankin-Selberg integrals for $\mathrm{GSpin}$ groups with application to the global Gan-Gross-Prasad conjecture
本文利用 Bessel 模型为 $\mathrm{GSpin}$ 群构造了张量积部分 $L$-函数乘积的 Rankin-Selberg 积分，并以此证明了拟分裂 $\mathrm{GSpin}$ 群一般表示的全局 Gan-Gross-Prasad 猜想的一个方向。
On the Bloch-Kato conjecture for GSp(4)
研究通过欧拉系统证明 reciprocity 法律，得出Iwasawa主猜想和Bloch-Kato猜想在解析秩0时的包含关系。
Mod $p$ local-global compatibility for $\mathrm{GSp}_4(\mathbb{Q}_p)$ in the ordinary case
本文在偶次全实域与常情形、一般性及 Taylor–Wiles 条件下，证明模 $p$ 的 $\mathrm{GSp}_4$ 自守形式 Hecke 同型部分上的局部群作用可决定相应 Galois 表示在 $p$ 位的限制。
A two variable Rankin-Selberg integral for $\mathrm{GU}(2,2)$ and the degree 5 $L$-function of $\mathrm{GSp}_4$
本文提出了一种双变量 Rankin-Selberg 积分，用于表示 exterior square $L$-functions 的乘积，并作为 residue 得到了被二次特征扭曲的 $\mathrm{GSp}_4$ 5 次 $L$-函数的积分表示。
Horizontal norm compatibility of cohomology classes for $\mathrm{GSp}_{6}$
本文针对 $\mathrm{GSp}_{6}$ 的八次旋量 $L$-因子，建立了 Siegel 模六维流形上同调类的水平范数关系，并通过绕过重数一假设失效的新方法，为后续构造八维 Galois 表示的 Euler 系奠定基础。
On the Bloch-Kato conjecture for some four-dimensional symplectic Galois representations
本文证明了对GSp_4自守表示对应的四维辛Galois表示，当中心L值非零时Selmer群消失，并利用提升同余构造上同调类，同时处理秩一情形，推进了Bloch-Kato猜想。
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。
Overconvergent Eichler-Shimura morphisms for $\mathrm{GSp}_4$
本文为亏格二的超收敛 Siegel 模形式族构造显式 Eichler-Shimura 态射，实现 Faltings-Chai 分解的 p-进插值，关键工具为 Boxer-Pilloni 高阶 Coleman 理论与带支集 pro-Kummer 平展上同调。
Tempered currents and Deligne cohomology of Shimura varieties, with an application to $\mathrm{GSp}_6$
本文利用缓增电流给出任意Shimura簇Deligne-Beilinson上同调的新描述，并应用于计算Siegel六维簇motivic上同调类的Beilinson高阶调节子像，将其与Spin $L$-函数非临界特殊值联系，验证Beilinson猜想。
A comparison problem for abelian surfaces and descent for symplectic orbital integrals
本文通过计算GSp_4中特定轨道积分，研究有限域上阿贝尔曲面同源类中椭圆曲线乘积的分布，并给出子群上球函数轨道积分的显式结果。
We study the L-functions associated to Siegel modular forms (equivalently, automorphic representations of ${\rm GSp}(4,\mathbb{A}_{\mathbb{Q}})$) both theoretically and numerically. For the L-functions of degrees 10, 14, and 16 we perform representation theoretic calculations to cast the Langlands L-function in classical terms. We develop a precise notion of what it means to test a conjectured functional equation for an L-function, and we apply this to the degree 10 adjoint L-function associated to a Siegel modular form.
Let $p$ be an odd prime, $N$ a square-free odd positive integer prime to $p$, $π$ a $p$-ordinary cohomological irreducible cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ of principal level $N$ and Iwahori level at $p$. Using a $p$-integral version of Rallis inner product formula and modularity theorems for $\mathrm{GSp}_{4/\mathbb{Q}}$ and $\mathrm{U}_{4/\mathbb{Q}}$, we establish an identity between the $p$-part of the critical value at $1$ of the degree $5$ $L$-function of $π$ twisted by the non-trivial quadratic Dirichlet character $ξ$ associated to the exten…
We construct classes in the middle degree plus one motivic cohomology of the Siegel Shimura variety of almost any dimension. We compute their image by Beilinson's higher regulator in terms of Rankin-Selberg type automorphic integrals. Our construction generalises the one for $\mathrm{GSp}(4)$ and for $\mathrm{GSp}(6)$. For Siegel varieties associated to small genus symplectic groups, we also show how these integrals unfold.
Potential Automorphy of K3 Surfaces with Large Picard Rank
本文通过证明高 Picard 等级 K3 表面的潜在模性，结合 GSp₄ 型阿贝尔簇的兼容系统理论，拓展了模性定理的应用范围。
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。
On $p$-refined Friedberg-Jacquet integrals and the classical symplectic locus in the $\mathrm{GL}_{2n}$ eigenvariety
本文提出关于 $P$-parahoric $p$-细化下 Friedberg-Jacquet 积分非零性与函子转移等价性的猜想，通过证明 $\mathrm{GL}_{2n}$ 特征簇中经典辛族维数的上界，获得了支持该猜想的若干结果。
Computing the mod-3 Galois image of a principally polarized abelian surface over the rationals
针对有理数域上主极化阿贝尔曲面的mod-3伽罗瓦表示像的计算问题，提出以GSp(4,F₃)子群共轭类分布为关键要素的算法，用多种技术解决ℓ=3时Gassmann等价子群的区分难题，且无需已知自同态信息。
In this article, the cohomology of the arithmetic group $\mathrm{Sp}_4(\mathbb{Z})$ with coefficients in any finite dimensional highest weight representation $\mathcal{M}_λ$ have been studied. Euler characteristic with coefficients in $\mathcal{M}_λ$ have been carried out in detail. Combining the results obtained on Euler characteristic and the work of Harder on Eisenstein cohomology, the description of the cuspidal cohomology has been achieved. At the end, we employ our study to compute the dimensions for the cohomology spaces $H^{\bullet}(\mathrm{Sp}_4(\mathbb{Z}), \mathcal{M}_λ)$.
Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields
本文研究全实域上亏格2超椭圆曲线的模2 Galois表示，在其像为$S_5$且在$\mathrm{GSp}_4(\mathbb{F}_2)\cong S_6$下传递时，证明它对应平行权2的Hilbert–Siegel Hecke特征尖点形式。
Global liftings between inner forms of GSp(4)
本文研究数域上约化群到其拟分裂内形式的自守提升，证明许多情形下非平凡上同调弱整体提升的存在性，并针对 GSp(4) 内形式解决局部提升的精细细节，证实了 Ibukiyama-Kitayama 关于平方自由水平参数模新形式的猜想。
Let $K$ be a imaginary quadratic field where the prime $p$ splits. Our goal in this article is to prove results towards the Iwasawa main conjectures for $p$-nearly-ordinary families associated to $\mathrm{GL}_2\times \mathrm{Res}_{K/\mathbb{Q}}\mathrm{GL}_1$ with a minimal set of assumptions. The main technical input is an improvement on the locally restricted Euler system machinery that allows the treatment of residually reducible cases, which we apply with the Beilinson--Flach Euler system.
Let $A$ be a semistable principally polarized abelian variety of dimension $d$ defined over the rationals. Let $\ell$ be a prime and let $\barρ_{A,\ell} : G_{\mathbb{Q}} \rightarrow \mathrm{GSp}_{2d}(\mathbb{F}_\ell)$ be the representation giving the action of $G_{\mathrm{Q}} :=\mathrm{Gal}(\bar{\mathrm{Q}}/\mathrm{Q})$ on the $\ell$-torsion group $A[\ell]$. We show that if $\ell \ge \max(5,d+2)$, and if image of $\barρ_{A,\ell}$ contains a transvection then $\barρ_{A,\ell}$ is either reducible or surjective. With the help of this we study surjectivity of $\barρ_{A,\ell}$ for semistable princi…
Mod-$p$ Galois representations not arising from abelian varieties
该研究针对伽罗瓦表示是否源自阿贝尔簇的问题，证明了当维数$g\ge2$且排除三类特殊$(g,p)$值时，存在具有分圆相似特征的$\mathrm{GSp}(2g,\mathbb{F}_p)$值伽罗瓦表示无法对应$\mathbb{Q}$上$g$维阿贝尔簇的$p$挠表示。
Ginzburg's conjecture on the unramified computation of Eulerian integrals
该研究证明了D. Ginzburg关于特定欧拉全局积分非分歧计算的猜想，还研究了含退化艾森斯坦级数的类似积分并计算其非分歧局部因子。
Anti-cyclotomic Euler system of diagonal cycles
本文为 GL_n×GL_{n+1} 上特定 RACSDC 自守表示的伽罗瓦表示构造分裂反圆欧拉系，并据此推进了 Beilinson–Bloch–Kato 猜想在秩 1 情形的部分结果。
Let $π$ be a cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbf{A_Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If $π$ is not CAP or endoscopic, then we show that its associated $\ell$-adic Galois representations are irreducible and crystalline for $100\%$ of primes $\ell$. If, moreover, $π$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for $100\%$ of primes $\ell$, the image of its mod $\ell$ Galois representation contains $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.
Inspired by a construction of Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\mathrm{GSp}_4$ for the product of the standard and spin $L$-functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGL}_4$ representing the product of the $L$-functions attached to the three fundamental representations of the Langlands $L$-group $\mathrm{SL}_4(\mathbf{C})$. The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\mathrm{PGU}(2…
On Rankin-Selberg integral structures and Euler systems for $\mathrm{GL}_2\times \mathrm{GL}_2$
本文研究Rankin-Selberg周期与区分问题与球面Whittaker表示积分结构的相互作用，证明Loeffler猜想，表明构造模形式乘积的motivic Rankin-Selberg Euler系统时，局部Euler因子在p处的tame norm关系中被积分可除。
Euler Characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$, and their cohomological consequences
该研究计算了SL₄(ℤ)与GL₄(ℤ)的欧拉示性数，推导了其拟多项式形式、上同调的消失结果与下界，结合Horozov的工作得到对称幂相关的恒等式与猜想。
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