On $p$-adic Siegel modular forms of non-real Nebentypus of degree 2

A note on quadratic twisting of epsilon factors for modular forms with arbitrary nebentypus

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…

On p-adic properties of Siegel modular forms

We show that Siegel modular forms of level Γ_0(p^m) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre closely; his proofs however do not generalize to the Siegel case or need some modifications.

A remark on $p$-adic Siegel Eisenstein series

A generalization of Serre's $p$-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a $p$-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is proved.

关于Igusa塔上p进积分的理论

On the p-adic integration over Igusa towers of Siegel modular varieties

本文提出了一种针对Igusa塔的显式p进积分理论，用于研究模Siegel流形的互反律。

A formula for the derivative of the p-adic L-function of the symmetric square of a finite slope modular form

Let f be a modular form of weight k and Nebentypus $ψ$. By generalizing a construction of Dabrowski and Delbourgo, we construct a p-adic L-function interpolating the special values of the L-function $L(s,\mathrm{Sym}^2(f)\otimes ξ)$, where $ξ$ is a Dirichlet character. When s=k-1 and $ξ=ψ^{-1}$, this p-adic L-function vanishes due to the presence of a so-called trivial zero. We give a formula for the derivative at s=k-1 of this p-adic L-function when the form f is Steinberg at p. If the weight of f is even, the conductor is even and squarefree, and the Nebentypus is trivial this formula implie…

Siegel模空间上的不可约和p-adic单值性

Irreducibility and p-adic monodromies on the Siegel moduli spaces

本文将Faltings和Chai（独立于Ekedahl）关于Siegel模空间的普通轨迹的$p$ -adic单值满射性结果推广到任意$p$ -秩层的满射性结果.讨论了具有仿射水平结构的模空间的某些$p$秩层的不可约性和连通性。在具有Iwahori水平结构的Siegel 3-fold上得到了较好的结果。

关于级为 $p$ 的 Siegel–Eisenstein 级数及其 $p$-进性质

On Siegel--Eisenstein series of level $p$ and their $p$-adic properties

本文构造了级为 $p$、带模 $p$ 二次特征的 Siegel–Eisenstein 级数，显式计算其 Fourier 系数，并证明它及 Takemori 构造的非二次特征版本均为 $p$-进 Siegel–Eisenstein 级数。

Siegel模形式的非尖点Hida理论与p进L-函数的平凡零点

Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$-adic $L$-functions

本文通过发展非尖点Siegel模形式的Hida理论，利用Greenberg-Stevens方法构造改进的p进L-函数，研究了P-普通Siegel模形式的标准p进L-函数在半稳定平凡零点处的导数，从而部分证明了Greenberg关于p进L-函数平凡零点的阶数与首项系数的猜想。

Derivative of the standard $p$-adic $L$-function associated with a Siegel form

In this paper we construct a two variables $p$-adic $L$-function for the standard representation associated with a Hida family of parallel weight genus $g$ Siegel forms, using a method previously developed by Böcherer--Schmidt in one variable. When a form of weight $g+1$ is Steinberg at $p$, a trivial zero appears and, using the method of Greenberg--Stevens, we calculate the first derivative of this $p$-adic $L$-function and show that it has the form predicted by a conjecture of Greenberg on trivial zeros.

具有水平的某些Siegel尖点形式由其基本Fourier系数决定

Certain Siegel Cusp Forms with Level are Determined by their Fundamental Fourier Coefficients

本文证明具有特定nebentypus的向量值Siegel尖点形式由与水平互素的基本Fourier系数决定，并在亏格3时加强至四元数极大序情形，同时推广到Jacobi形式。

A p-adic analogue of Siegel's Theorem on sums of squares

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally stud…

Sturm type theorem for Siegel modular forms of degree 2

We attempt to generalize a congruence property of elliptic modular forms proved by Sturm to that of Haupttypus of Siegel modular forms of degree 2 with level. Namely, we give an explicit bound of Fourier coefficients required to determine the congruence of modular forms. We give the analog of Sturm's theorem for Jacobi forms, which is required in the proof. In the case that Nebentypus of Siegel modular forms with prime level, in order to prove the congruence between two modular forms, we show that it suffices to check the congruence of the finitely many Fourier coefficients. Finally, we give t…

关于p进Siegel-Eisenstein级数II：如何避免p的正则性条件

On $p$-adic Siegel--Eisenstein series II: How to avoid the regularity condition for $p$

本文针对p进Siegel-Eisenstein级数与经典模形式的一致性问题，移除了p为正则素数的条件，证明当次数n≤2k+1时两者仍一致。

从模$p^m$奇异形式理论的视角看$p$-进Siegel Eisenstein级数

On $p$-adic Siegel Eisenstein series from a point of view of the theory of mod $p^m$ singular forms

该研究利用模$p$幂奇异形式理论，证明一般次数的两类$p$-进Siegel Eisenstein级数是水平$p$的亏格theta级数的线性组合，并由此推导出已有相关结果。

关于Zeta函数和Siegel模形式族

On Zeta Functions and Families of Siegel Modular Forms

设$p$是素数，$Γ=\Sp_g(\Z)$是亏格为$g$的Siegel模群.我们研究$p$ -adic族的zeta函数和Siegel模形式.用附在$\Sp_g$上的动机$L$ -函数来描述Siegel模形式的$L$ -函数，并给出它们的解析性质.讨论了旋量$L$ -函数和$p$ -adic构造的临界值.建立了高等亏格的Rankin引理。给出了从$ GSp_{2m} \times GSp_{2m}$提升到$GSp_{4m}$（属$g=4m$）的一般猜想.利用Ikeda-Miyawaki构造给出了Siegel模形式的$p$ -adic族的构造.

Spherical functions on $U(n,n)/(U(n) \times U(n))$ and hermitian Siegel series

We study spherical functions on the space isomorphic to $U(2n)/(U(n)\times U(n))$ over a $p$-adic field; those functional equations with respect to the action of the Weyl group, the location of possible poles and zeros, explicit formulas, and spherical Fourier transforms. Then, as an application, we give a functional equation of $p$-adic local hermitian Siegel series.

Heuristics in direction of a p-adic Brauer--Siegel theorem

Let p be a fixed prime number. Let K be a totally real number field of discriminant D\_K and let T\_K be the torsion group of the Galois group of the maximal abelian p-ramified pro-p-extension of K (under Leopoldt's conjecture). We conjecture the existence of a constant C\_p>0 such that log(\#T\_K) $\le$ C\_p log(\sqrt(D\_K)) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer--Siegel Theorem, wearing here on the valuation of the residue at s=1 (essentially equal to \#T\_K) of the p-adic …

Galois level and congruence ideal for $p$-adic families of finite slope Siegel modular forms

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…

P-Adic families of Siegel modular cuspforms

Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.

Siegel-Hilbert尖点本征系统的多变量p进族及其伽罗瓦表示

Several variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations

设F是全实域，G=GSp（4）_{/F}.本文证明了在一个弱假设下，给定一个Hecke本征系λ对G中的一个固定抛物P是（p，P）-平凡的，存在一个包含λ的多元p-adic族Hecke本征系（它们都是（p，P）-近似平凡的）的下划线{λ}.假设λ对于正则系数系统是上同调的。如果F=Q，则变量的个数为3。此外，在这种情况下，我们构造了与下划线{lambda}相关联的伽罗瓦表示的三变量p-adic族rho_{underline{lambda}}。最后，根据几何假设（这将是满意的，如果一个证明，在家庭的伽罗瓦表示来自格罗滕迪克动机），我们表明，ρ_{下划线{lambda}}是几乎普通的双抛物线的P.这段文字是我们的第一个预印本（在“预出版德L 'universite巴黎北部”）的更新版本，并将出现在“科学年鉴德L' E N S”。

Cusp forms as p-adic limits

Ahlgren and Samart relate three cusp forms with complex multiplication to certain weakly holomorphic modular forms using $p$-adic bounds related to their Fourier coefficients. In these three examples, their result strengthens a theorem of Guerzhoy, Kent, and Ono which pairs certain CM forms with weakly holomorphic modular forms via $p$-adic limits. Ahlgren and Samart use only the theory of modular forms and Hecke operators, whereas Guerzhoy, Kent, and Ono use the theory of harmonic Maass forms. Here we extend Ahlgren and Samart's work to all cases where the cusp form space is one-dimensional a…

Analytic constructions of p-adic L-functions and Eisenstein series

The Fourier coefficients of the Siegel-Eisenstein series are p-adically continued for all primes p, as meromorphic functions, using the reciprocal of a product of L-functions. A construction of p-adic meromorphic families of such series is given in relation to the geometry of homogeneous spaces. Applications are given to p-adic L-functions, to Siegel's Mass Formula, to p-adic analytic families of automorphic representations. Based on author's talk for the Conference Automorphic Forms and Related Geometry, Assessing the Legacy of I.I. Piatetski-Shapiro

具有p-adic整系数的Siegel簇的上同调及其应用

Cohomology of Siegel Varieties with p-adic integral coefficients and Applications

在与Siegel模形式相关的伽罗瓦表示存在的假设下（它只对亏格至多为2已知），我们证明了Siegel簇的p-adic整系数的上同调，当局部化在Hecke代数的非Eisenstein极大理想上时，是无挠的，只要素数p相对于系数系统的权是大的.证明使用p-adic Hodge理论，对偶BGG复模p，以计算mod p上同调的Hodge-Tate权重。我们将这个结果应用于辛群的Hida p-adic族的构造，并应用于这些群的泰勒-怀尔斯系统的构造的第一步。

Mock modular Eisenstein series with Nebentypus

By the theory of Eisenstein series, generating functions of various divisor functions arise as modular forms. It is natural to ask whether further divisor functions arise systematically in the theory of mock modular forms. We establish, using the method of Zagier and Zwegers on holomorphic projection, that this is indeed the case for certain (twisted) "small divisors" summatory functions $σ_ψ^{\mathrm{sm}}(n)$. More precisely, in terms of the weight 2 quasimodular Eisenstein series $E_2(τ)$ and a generic Shimura theta function $θ_ψ(τ)$, we show that there is a constant $α_ψ$ for which $$ \math…

$p$-adic $L$-functions on metaplectic groups

With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of half-integral weight is lopsided; the analytic theory is strong whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture -- the $p$-adic $L$-function obtained by interpolating the complex $L$-function at special values. This is achieved through the Rankin-Selberg method and the explicit Fourier expansion of non-holomorphic Siegel Eisenstein series. The construction of the $p$-stabilisation in this …

Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form

Let p be an odd prime and F a totally real number field. Let f be a Hilbert cuspidal eigenform of parallel weight 2, trivial Nebentypus and ordinary at p. It is possible to construct a p-adic L-function which interpolates the complex L-function associated to the symmetric square representation of f. This p-adic L-function vanishes at s=1 even if the complex L-function does not. Assuming p inert and f Steinberg at p, we give a formula for the p-adic derivative at s=1 of this p-adic L-function, generalizing unpublished work of Greenberg and Tilouine. Under some hypotheses on the conductor of f w…

Linearity on ordinary Siegel moduli schemes and joint unlikely almost intersections

The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of André--Oort and André--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.

Integrality of nearly (holomorphic) Siegel modular forms

In order to considering the integrality of nearly holomorphic (vector-valued) Siegel modular forms, we introduce nearly Siegel modular forms and study their integrality. We show that the integrality of nearly Siegel modular forms in terms of their Fourier expansion implies the integrality of their CM values. Furthermore, we show that there exists a one-to-one correspondence between integral nearly Siegel modular forms and integral nearly holomorphic ones. By these results, the integrality of CM values holds for nearly holomorphic Siegel modular forms and for nearly overconvergent p-adic Siegel…

A $p$-adic Hermitian Maass lift

For $K$ an imaginary quadratic field with discriminant $-D_K$ and associated quadratic Galois character $χ_K$, Kojima, Gritsenko and Krieg studied a Hermitian Maass lift of elliptic modular cusp forms of level $D_K$ and nebentypus $χ_K$ via Hermitian Jacobi forms to Hermitian modular forms of level one for the unitary group $U(2,2)$ split over $K$. We generalize this (under certain conditions on $K$ and $p$) to the case of $p$-oldforms of level $pD_K$ and character $χ_K$. To do this, we define an appropriate Hermitian Maass space for general level and prove that it is isomorphic to the space o…

On $p$-adic quaternionic Eisenstein series

We show that certain $p$-adic Eisenstein series for quaternionic modular groups of degree 2 become "real" modular forms of level $p$ in almost all cases. To prove this, we introduce a $U(p)$ type operator. We also show that there exists a $p$-adic Eisenstein series of the above type that has transcendental coefficients. Former examples of $p$-adic Eisenstein series for Siegel and Hermitian modular groups are both rational (i.e., algebraic).

Images de représentations galoisiennes associées à certaines formes modulaires de Siegel de genre $2$

We study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes. These representations are symplectic of dimension $4$. Following a method of Dieulefait, we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is "full" for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases.

广义Logistic $p$ -adic动力系统的混沌行为

On the chaotic behavior of a generalized logistic $p$-adic dynamical system

本文描述吸引区域$p$ -adic动力系统$G(x)=(ax)^2(x+1)$ .此外，我们还描述了系统的Siegel圆盘，因为系统的轨道结构与$p$ -adic Siegel圆盘的几何形状有关。

Siegel 模形式权重 $(3,3)$ 的 Kato 显式互反律

Kato explicit reciprocity law for Siegel modular forms of weight $(3, 3)$

本文将 Kato 显式互反律从模曲线推广到两条模曲线的乘积，并利用嵌入 Siegel 三维簇，得到权重 $(3,3)$ 的 Siegel 模形式所附 $p$-进表示的唯一次临界扭转的显式互反律。

三次$p$ -adic动力系统的混沌

On chaos of a cubic $p$-adic dynamical system

本文描述了$p$ -adic动力系统$f(x)=x^3+ax^2$吸引区域.此外，我们还描述了系统的Siegel圆盘，因为系统的轨道结构与$p$ -adic Siegel圆盘的几何形状有关。

Power series expansions of modular forms and $p$-adic interpolation of the square roots of Rankin-Selberg special values

Let $f$ be a newform of even weight $2κ$ for $D^\times$, where $D$ is a possibly split indefinite quaternion algebra over $\mathbb{Q}$. Let $K$ be a quadratic imaginary field splitting $D$ and $p$ an odd prime split in $K$. We extend our theory of $p$-adic measures attached to the power series expansions of $f$ around the Galois orbit of the CM point corresponding to an embedding $K\hookrightarrow D$ to forms with any nebentypus and to $p$ dividing the level of $f$. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic $p$-level str…

A p-adic Eisenstein measure for vector-weight automorphic forms

We construct a p-adic Eisenstein measure with values in the space of vector-weight p-adic automorphic forms on certain unitary groups. This measure allows us to p-adically interpolate special values of certain vector-weight C-infinity automorphic forms, including Eisenstein series, as their weights vary. We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz's p-adic families of Eisenstein series for Hilbert modular forms.

关于有理$P$ -Adic动态系统

On Rational $P$-Adic Dyanamical Systems

本文研究了有理$p$ -adic动力系统在复$p$ -adic域$\C_p$中的轨线行为.研究了这类动力系统的Siegel盘和吸引子。我们证明了Siegel圆盘对于动力系统的不同不动点既可以重合也可以不相交。此外，我们还找到了系统吸引子的盆。证明了这类动力系统在单位球面上关于Haar测度不是遍历的。

Higher Hida theory and p-adic L-functions for GSp(4)

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).

On Siegel eigenvarieties at Saito-Kurokawa points

We study the geometry of the $p$-adic Siegel eigenvariety $\mathcal{E}$ of paramodular tame level at certain Saito-Kurokawa points having a critical slope. For $k \geq 2$ let $f$ be a cuspidal new eigenform of $\mathrm{S}_{2k-2}(Γ_0(N))$ ordinary at a prime $p\nmid N$ with sign $ε_f=-1$ and write $α$ for the $p$-adic unit root of the Hecke polynomial of $f$ at $p$. Let $π_α$ be the semi-ordinary $p$-stabilization of the Saito-Kurokawa lift of the cusp form $f$ to $\mathrm{GSp}(4)$ of weight $(k,k)$ and paramodular tame level. Under the assumption that the dimension of the Selmer group $H^1_{f,…

On the images of Galois representations attached to low weight Siegel modular forms

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 $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.

关于一个多项式$p$ -adic动力系统

On one polynomial $p$-adic dynamical system

本文描述了复$p$-adic域上的$p$ -adic动力系统$f(x)=x^{2n+1}+ax^{n+1}$吸引区域和Siegel圆.

用于 ${\rm GSp}(4)\ imes{\ m GL}(2)$ 的 $p$-进 $L$-函数

$p$-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。

近高阶Coleman理论与$\mathrm{GSp}(4) \times \mathrm{GL}(2)$和$\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$的$p$-adic $L$-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。

半整权弱全纯模形式的$p$ -adic极限

$p$-adic Limit of Weakly Holomorphic Modular Forms of Half Integral Weight

Serre得到了$p=2,3,5,7$在$SL_2(\mathbb{Z})$上的模形式的积分傅里叶系数的p-adic极限.本文将Serre的结果推广到$Γ_{0}(4N)$上的半整权弱全纯模形式，并推广到$N=1,2,4$上。利用半整权的模形式的傅里叶系数之间的线性关系给出了一个证明。作为应用，我们得到了Borcherds指数的同余，Eisentein级数的商的同余以及$L$ -函数在某一点的值的同余.利用Ikeda提升，得到了Maass空间上Siegel模形式的傅里叶系数的同余关系.

p进L函数与带复乘的椭圆曲线的Selmer簇

p-adic L-functions and Selmer varieties associated to elliptic curves with complex multiplication

我们展示了如何非零的p-adic L函数控制的尺寸Selmer品种与补充的原点在一个椭圆曲线CM。作为推论，我们得到了Siegel定理在这类曲线上的π_1-证明.

Hida族中的解析秩一阶传播

Analytic rank one propagation in Hida families

该研究在算术代数几何相关猜想成立的假设下，证明了非 CM 新形式对应的 Hida 族中解析秩一阶的传播性质，为 Greenberg 的极小性猜想提供了证据。

球簇与上同调类的非寻常族

Spherical varieties and non-ordinary families of cohomology classes

该研究构建了与约化群球对相关的局部对称空间上同调的非寻常p进族类，给出平展情形下到伽罗瓦上同调的映射，方法可用于新的p进欧拉系和p进L函数构造，并以非寻常西格尔模形式的范数相容类为例推广了已有欧拉系的p部分。

算术模形式的p进Banach模与Coleman族的三重积

p-adic Banach modules of arithmetical modular forms and triple products of Coleman's families

对于一个素数$p\ge 5$，我们考虑三个经典的尖特征形$f_j(z)$的重量$k_1, k_2, k_3$，导体$N_1, N_2, N_3$，和neumypus字符$ψ_j \bmod N_j$。根据H.Hida和R.Coleman，可以将每个$f_j$包括到权重$k_j$的尖点本征形式$f_{j,k_j}$的{$p$-adic解析族} $k_j \mapsto \{f_{j,k_j}\}$中，使得$f_{j,k_j}=f_j$和所有它们的傅立叶系数$a_n(f_{j, k_j})$由某些$p$-adic解析函数$k_j{}\mapsto a_{n, j}(k_j{})$给出。本文的目的是描述一个四变量$p$ -adic $L$ -函数，它附加到三个固定斜率的尖点本征形的三个Coleman族的Garrett三重积$k_j \mapsto \{f_{j,k_j}\}$ $σ_j=v_p(α_{p, j}^{(1)}(k_j{}))\ge 0$上，其中$α_{p,j}^{(1)} = \al_{p,j}^{(1)}(k_j{})$是Atkin算子$U=U_p$作用于$U(\sum_{n\ge 0}^\infty a_{n}q^n) = \sum_{n \ge 0}^\infty a_{np} q^n$的Fourier展开的本征值（它依赖于$k_j{}$）.我们考虑$p$ -adic…

关于具有双曲二重的模形式Hida家族的局部特征化

On local characterizations of Hida families of Siegel modular forms

本文通过局部特征化方法，研究了具有双曲二重的模形式Hida家族，并证明了相关Selmer群的伪零性及R=TT定理。

群 $\mathrm{GSp}_4$ 上的 theta 算子

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。

$\ell$-adic局部系统在$\mathcal{A}_n$上的欧拉特征

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。

带有Bianchi模形式的有符号p-adic L-函数

Signed $p$-adic $L$-functions of Bianchi modular forms

本文研究了Bianchi模形式的有符号p-adic L-函数，通过p-稳定化形式和局部解析分布，将双变量幂级数分解为有界系数的线性组合。

Siegel $\mathfrak{p}^2$ 向量用于 $GSp(4)$ 的表示

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。

关于 $\mathrm{GSp}_4$ 的超收敛 Eichler-Shimura 态射

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。