Finiteness of Selmer groups and deformation rings

Relative deformation theory, relative Selmer groups, and lifting irreducible Galois representations

We study irreducible odd mod $p$ Galois representations $\barρ \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\barρ$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.

Deformations of polarized automorphic Galois representations and adjoint Selmer groups

We prove the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor-Wiles-Kisin patching method together with Kisin's technique of analy…

On the structure of Selmer groups of $p$-ordinary modular forms over $\mathbf{Z}_p$-extensions

We prove analogues of the major algebraic results of Greenberg-Vatsal for Selmer groups of $p$-ordinary newforms over $\mathbf{Z}_p$-extensions which may be neither cyclotomic nor anticyclotomic, under a number of technical hypotheses, including a cotorsion assumption on the Selmer groups. The main complication which arises in our work is the possible presence of finite primes which can split completely in the $\mathbf{Z}_p$-extension being considered, resulting in the local cohomology groups that appear in the definition of the Selmer groups being significantly larger than they are in the cas…

具有大 Selmer p-秩的可约 Galois 表示的形变

Deformations of reducible Galois representations with large Selmer $p$-rank

本文推广 Hamblen–Ramakrishna 形变提升技术，对任意 $p\geq5$ 构造与模形式相关的 Galois 表示，证明其 Bloch-Kato Selmer 群可具有任意大的 $p$-秩。

Bloch--Kato 塞尔默群的2-下降与超椭圆曲线上的有理点 II

2-descent for Bloch--Kato Selmer groups and rational points on hyperelliptic curves II

该研究提出精细化方法证明带单有理魏尔斯特拉斯点的超椭圆曲线的Chabauty--Coleman--Kim集合有限性，给出模2 Bloch--Kato塞尔默群的2进与实进塞尔默条件计算方法，验证了多数LMFDB中秩2亏格2曲线的有限性，还得到超椭圆曲线雅可比簇二次下降的域论描述。

Height one specializations of Selmer groups

We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to four dimensional Galois representations coming from (i) the tensor product of two cuspidal Hida families $F$ and $G$, (ii) its cyclotomic deformation, (iii) the tensor product of a cusp form $f$ and the Hida family $G$, where $f$ is a classical specialization of $F$ with weight $k \geq 2$. We prove control theorems to relate (a) the Selmer group associated to the tensor product of Hida families $F$ and $G$ to the Selmer group associated to its cyclotomic deformation a…

Selmer秩在偶Galois表示族中的密度，Wiles公式与全球互反律

Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

本文通过Galois上同调方法研究偶Galois表示族中Selmer秩的分布，利用Wiles公式和全球互反律证明了Selmer秩增加的素数密度为1/192。

On the vanishing of almost all primary components of the Shafarevich-Tate group of elliptic curves over the rationals

The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory for elliptic curves. The finiteness of the Shafarevich-Tate group of an elliptic curve $E$ over the field of rational numbers is included in the Birch and Swinnerton-Dyer conjectures, and is still an open question. We present an overview of the Shafarevich-Tate and Selmer groups of an elliptic curve in the framework of Galois cohomology. Known results on the finiteness of the Shafarevich-Tate group are mentioned, including results of Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin. We then prove the vanishing…

与超椭圆曲线相关的 Bloch–Kato Selmer 群的精细有效界

Refined effective bounds for Bloch-Kato Selmer groups associated to hyperelliptic curves

本文推广 Dogra 的显式 $2$-下降方法，在无有理 Weierstrass 点及 $2$ 处好通常约化条件下给出 Bloch–Kato Selmer 群维数的有效界，并据此建立深度 $2$ Chabauty–Kim 集有限性的新判定准则。

多重$\u2124_p^d$-扩张上精细Selmer群的伪零性

Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions

该研究建立了伽罗瓦表示及其$p$进族形变相关精细Selmer群的垂直控制定理，为Coates–Sujatha的两个猜想提供新证据，还得到多类对象对应精细Selmer群的伪零性。

Finiteness Theorems for Deformations of Complexes

We consider deformations of bounded complexes of modules for a profinite group G over a field of positive characteristic. We prove a finiteness theorem which provides some sufficient conditions for the versal deformation of such a complex to be represented by a complex of G-modules that is strictly perfect over the associated versal deformation ring.

关于Iwasawa理论中精细Selmer群的$μ$等于零猜想

On the $μ$ equals zero conjecture for the fine Selmer group in Iwasawa theory

本文研究了与某些Galois表示相关的精细Selmer群的Iwasawa理论，证明了在Galois形变环满足特定自然性质时$μ$-不变量消失，并概述了$μ=0$猜想对多种Galois表示成立的条件。

On the complete faithfulness of the $p$-free quotient modules of dual Selmer groups

In this paper, we consider the question of the complete faithfulness of the $p$-free quotient module of the dual Selmer groups of elliptic curves defined over a noncommutative $p$-adic Lie extension. Our question will refine previous questions on the complete faithfulness of dual Selmer groups. We also consider the question of the triviality of the central torsion submodules of these Iwasawa modules and we see that this latter question is intimately related to the former. We will also formulate and study analogous questions for the dual Selmer groups of Hida deformations. We then give positive…

Finiteness of unramified deformation rings

We prove that the universal unramified deformation ring $R^{\mathrm{unr}}$ of a continuous Galois representation $\overlineρ: G_{F^{+}} \rightarrow \mathrm{GL}_n(k)$ (for a totally real field $F^{+}$ and finite field $k$) is finite over $\mathcal{O} = W(k)$ in many cases. We also prove (under similar hypotheses) that the universal deformation ring $R^{\mathrm{univ}}$ is finite over the local deformation ring $R^{\mathrm{loc}}$.

Massey积与精细Selmer群的Iwasawa理论

Massey products and the Iwasawa theory of fine Selmer groups

本文用Massey积重新表述Coates-Sujatha猜想，证明μ-不变量的消失可传递至邻域扩张，并建立形变环无阻碍性与精细Selmer群μ-不变量消失的等价关系。

关于GL(N)的上同调与伴随Selmer群

On the cohomology of GL(N) and adjoint Selmer groups

本文在局部-整体相容等条件下将Galatius-Venkatesh定理推广到CM域上GL(N)，构造从导出形变环基本群的可除群到扭对偶伴随动机Selmer群的Hecke等变单射，并推广到高阶同伦群与Hida族。

Kato欧拉系统在模形式中的精细应用

Refined applications of Kato's Euler systems for modular forms

本文探讨了Kato欧拉系统在高权模形式中的精细应用，证明了Mazur-Tate猜想并给出岩泽主猜想的新解释，同时涉及对BSD猜想的应用。

Integral Points on Elliptic Curves and Modularity

In this paper we prove the finiteness of the set of S-integral points of a punctured rational elliptic curve without complex multiplication using the Chabauty-Kim method. This extends previous results of Kim in the complex multiplication case. The key input of our approach is the use of modularity techniques to prove the vanishing of certain Selmer groups involved in the Chabauty-Kim method.

伪特征变形与马祖尔的有限性条件

Deformations of pseudocharacters and Mazur's finiteness condition

本文研究了满足马祖尔有限性条件的群的伪特征变形环的noetherian性质，并探讨了相关模空间的表示理论。

Iwasawa invariants of Galois deformations

We study the behavior of Iwasawa invariants among ordinary deformations of a fixed residual Galois representation taking values in a reductive algebraic group G. In particular, under the assumption that these Selmer groups are cotorsion modules over the Iwasawa algebra, we show that the vanishing of the mu invariant is independent of the deformation, while the lambda invariant depends only on the ramification. This generalizes work of Greenberg-Vatsal and Emerton-Pollack and the author in the case G = GL(2).

关于一族违反哈塞原理的椭圆曲线与亏格1曲线的Selmer群与秩

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

本研究针对一族j不变量为零的椭圆曲线及其3-同源曲线，结合整二元三次型等价类，在Tate-Shafarevich群有限假设下推导秩的奇偶性与上下界，并构造出显式违反哈塞原理的亏格1曲线类。

Euler systems for Rankin--Selberg convolutions of modular forms

We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.

Iwasawa Theory for Artin Representations, I

This article is the first of a pair of articles dealing with the Iwasawa theory of modular forms of weight 1 and, more generally, of Artin representations satisfying certain conditions. The main results in this part analyze the structure of certain Selmer groups for the Artin representation. In particular, it is shown that the Selmer groups are co-torsion as $Λ$-modules. For each Selmer group, we consider a generator of the characteristic ideal of its Pontrjagin dual. We call that the algebraic p-adic L-function. We also construct an analytic $p$-adic L-function via deformation to higher weigh…

孪生素数椭圆曲线 $y^2=x(x-2)(x-p)$ 的解析秩与代数秩

On the algebraic and analytic ranks of the twin-prime elliptic curve $y^2=x(x-2)(x-p)$

研究孪生素数椭圆曲线E_p的解析秩与代数秩，利用2-Selmer计算和Cassels-Tate配对，在低余秩BSD假设下证明二者相等、Tate-Shafarevich群有限并给出精确BSD公式。

关于平坦偶变形环

On flat even deformation rings

研究显示在非平凡双Selmer群存在时，某些全球偶变形环为有限且在Z_p上平坦。通过全球类域论技术，计算了偶表示的显式例子，并观察到若Leopoldt猜想对相关数域成立，则全球偶变形环在最小水平上平坦。

大Galois表示在多重Z_p扩展上的代数函数方程

Algebraic functional equation for big Galois representations over multiple $\mathbb{Z}_p$-extensions

本文提出了一种方法，用于在多重Z_p扩展上建立大Galois表示的代数函数方程，涵盖Iwasawa理论多种场景，并扩展了已知结果，回答了Greenberg的问题。

非交换阿贝尔簇的全局函数域上的伊瓦萨理论

Non-commutative Iwasawa theory of abelian varieties over global function fields

本文研究了全局函数域上阿贝尔簇的非交换伊瓦萨理论，证明了相关猜想并扩展了塞奇定理。

相对形变方法中的分歧强制

Forcing ramification in the relative deformation method

本文将Khare-Ramakrishna的分歧强制论证推广到相对形变方法框架，证明可选有限素数集使几何特征0提升在其处分歧，且局部形式光滑性蕴含相关几何Bloch-Kato Selmer群零化。

模点在埃西恩情况下的扎里斯基密度

Modular points and dimensions of Eisenstein deformation spaces

本文通过局部-全局兼容性结果和非普通有限性结果，研究了残余可约情况下模点的扎里斯基密度，并探讨了其在海克环等维性和R=\mathbb{T}定理中的应用。

关于全局函数域上光滑层的格林伯格μ = 0猜想

Greenberg's $μ=0$ conjecture for lisse sheaves over global function fields

本文在正特征下证明了Greenberg的μ=0猜想的类比，针对全局函数域上的lisheaves，展示了Selmer群的结构及相关变形环的性质。

Deformations of Saito-Kurokawa type and the Paramodular Conjecture (with an appendix by Cris Poor, Jerry Shurman, and David S. Yuen)

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\barσ$ with $\barσ^{\rm ss}=χ^{k-2} \oplus ρ\oplus χ^{k-1}$, where $χ$ is the mod $p$ cyclotomic character and $ρ$ is an absolutely irreducible reduction of the Galois representation $ρ_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f(\mathbf{Q}, ρ_f(1-k)\otimes \mathbf{Q}_p/\mathbf{Z}_p)$ and $H^1_f(\mathbf{Q}, ρ(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\barσ$ t…

Galois 变形环与剩余可约情形下的模性

Galois deformation rings and modularity in the residually reducible case

本文证明了数域的剩余可约 p-adic Galois 表示可由模形式产生，通过研究 Fontaine-Laffaille 条件下的万有变形环，并利用伪特征和 Selmer 群界，证明了环为离散赋值环，且应用于 4 维表示证明了模性。

关于具有三个Jordan-Hölder因子的剩余Galois表示的形变环与模性

On deformation rings of residual Galois representations with three Jordan-Holder factors and modularity

研究具有三个互不同构绝对不可约Jordan-Hölder因子的模p非半单Galois表示的Fontaine-Laffaille自对偶形变，在Selmer群阶的条件下证明泛形变环为离散赋值环，并在Hecke代数信息充分时证明R=T定理，应用于阿贝尔曲面及某些6维表示。

On Ihara's lemma for Hilbert Modular Varieties

Let ρbe a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρhas large image and admits a low weight crystalline modular deformation we show that any low weight crystalline deformation of ρunramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara type lemma for the local component at ρof the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at 1 of the adjoint L-function …

CM域上自守伽罗瓦表示的刚性

Rigidity of Automorphic Galois Representations over CM Fields

本文证明了CM域上自守伽罗瓦表示的伴随Bloch-Kato Selmer群消没，确立了其无de Rham形变的刚性，并借鉴多位学者的方法验证了该表示本身是de Rham的。

Irreducibility of limits of Galois representations of Saito-Kurokawa type

We prove (under certain assumptions) the irreducibility of the limit $σ_2$ of a sequence of irreducible essentially self-dual Galois representations $σ_k: G_{\mathbf{Q}} \to \mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$ approaches 2 in a $p$-adic sense) which mod $p$ reduce (after semi-simplifying) to $1 \oplus ρ\oplus χ$ with $ρ$ irreducible, two-dimensional of determinant $χ$, where $χ$ is the mod $p$ cyclotomic character. More precisely, we assume that $σ_k$ are crystalline (with a particular choice of weights) and Siegel-ordinary at $p$. Such representations arise in the study of $p$-adi…

On certain L-functions for deformations of knot group representations

We study the twisted knot module for the universal deformation of an ${\rm SL}_2$-representation of a knot group, and introduce an associated $L$-function, which may be seen as an analogue of the algebraic $p$-adic $L$-function associated to the Selmer module for the universal deformation of a Galois representation. We then investigate two problems proposed by Mazur: Firstly we show the torsion property of the twisted knot module over the universal deformation ring under certain conditions. Secondly we verify the simplicity of the zeroes of the $L$-function by some concrete examples for 2-brid…

On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields

In this paper we study deformations of mod $p$ Galois representations $τ$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $τ_1$ and $τ_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_Σ(F, {\rm Hom}(τ_2, τ_1)) \subset {\rm Ext}^1(τ_2, τ_1)$. We establish that there exists a basis $\mathcal{B}$ of $H^1_Σ(F, {\rm Hom}(τ_2, τ_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\mathcal{…

On deformation rings of residually reducible Galois representations and R=T theorems

We study the crystalline universal deformation ring R (and its ideal of reducibility I) of a mod p Galois representation rho_0 of dimension n whose semisimplification is the direct sum of two absolutely irreducible mutually non-isomorphic constituents rho_1 and rho_2. Under some assumptions on Selmer groups associated with rho_1 and rho_2 we show that R/I is cyclic and often finite. Using ideas and results of (but somewhat different assumptions from) Bellaiche and Chenevier we prove that I is principal for essentially self-dual representations and deduce statements about the structure of R. Us…

Arithmetic representations of fundamental groups II: finiteness

Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero. For example, we show that if $X$ is a normal, conne…

由超椭圆曲线乘积所支配的曲面上的零循环

Zero-cycles on surfaces dominated by products of hyperelliptic curves

研究由\(y^2 = f_1(x_1)f_2(x_2)\)给出的曲面上一度零循环，在相关泰特 - 沙法列维奇群有限条件下，通过结合纤维化方法等证明局部到整体结果。

On the infinite fern of Galois representations of unitary type

Let E be a CM number field, F its maximal totally real subfield, c the generator of Gal(E/F), p an odd prime totally split in E, and S a finite set of places of E containing the places above p. Let r : G_{E,S} --> GL_3(F_p^bar) be a modular, absolutely irreducible, Galois representation of type U(3), i.e. such that r^* = r^c, and let X(r) be the rigid analytic generic fiber of its universal G_{E,S}-deformation of type U(3). We show that each irreducible component of the Zariski-closure of the modular points in X(r) has dimension at least 6[F:Q]. We study an analogue of the infinite fern of Gou…

高维情形下的无限蕨类结构

The infinite fern in higher dimensions

本研究针对p处分裂的酉群自守模p伽罗瓦表示，结合局部形变环切空间控制与特征簇几何论证，证明无限蕨类在其极化形变空间的非空不可约分支并集中扎里斯基稠密，推广了低维情形的相关结果。