Joel Bellaiche, Gaetan Chenevier
This is the final version of a book about p-adic families of Galois representations, Selmer groups, eigenvarieties and Arthur's conjectures
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Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F. We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.
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.
The first part of the paper is a survey of recent results about the cohomology of $(ϕ,Γ)$-modules and its applications to the theory of Selmer complexes. In the second part we formulate a version of the Main Conjecture for $p$-adic representations which are semistable at $p$ and discuss some related topics.
The main conjecture of Iwasawa theory is a conjecture on the relation between a Selmer group and a conjectural $p$-adic $L$-function. This conjectural $p$-adic $L$-function is expected to satisfy a conjectural functional equation in a certain sense. In view of the main conjecture and this conjectural functional equation, one would expect to have certain algebraic relationship between the Selmer group attached to a Galois representation and the Selmer group attached to the Tate twist of the dual of the Galois representation. It is precisely a component of this algebraic relationship that this p…
In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so is the other. Our results also compare the $π$-primary submodules of the two dual fine Selmer groups. We then apply our results to compare the structure of Galois group of the maximal abelian unramified pro-$p$ extension of an admissible $p$-adic Lie extension and the structure of the dual fine Selmer group over the said admissible $p$-adic Lie extension.
A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions at good ordinary primes $p$. We extend Greenberg's result to more general $p$-adic Galois representations, including a large subclass of those attached to $p$-ordinary modular forms of level $Γ_0(N)$ with $p\nmid N$.
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。
Let $A$ be an abelian variety defined over a number field $F$ with supersingular reduction at all primes of $F$ above $p$. We establish an equivalence between the weak Leopoldt conjecture and the expected value of the corank of the classical Selmer group of $A$ over a $p$-adic Lie extension (not neccesasily containing the cyclotomic $\Zp$-extension). As an application, we obtain the exactness of the defining sequence of the Selmer group. In the event that the $p$-adic Lie extension is one-dimensional, we show that the dual Selmer group has no nontrivial finite submodules. Finally, we show that…
Let p be a prime number and M a quadratic number field, M not equal to Q(\sqrt{p}) if p is congruent to 1 modulo 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D_{2p} and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least p^d.
Adjoint Selmer groups of automorphic Galois representations of unitary type
证明了酉类型自守Galois表示的伴随Selmer群消失，并为非CM Hilbert模形式和全实域椭圆曲线提供确切结果。。
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。
Structure of fine Selmer groups in abelian p-adic Lie extensions
本文研究abelian p-adic Lie扩张中椭圆曲线的fine Selmer群，给出其平凡性与伪零性的新结果，并阐明其与广义Greenberg猜想的关系。
On Selmer complexes, Stark systems and derived $p$-adic heights
论文发展了Nekovář的塞勒复形理论，证明在温和假设下，塞勒复形与Poitou-Tate复形准同构，并应用此结果构建了控制塞勒群的Heegner点斯塔克系统，同时证明Bertolini-Darmon导出p-进高度配对与Nekovář的相同。
We show how non-vanishing of p-adic L functions controls the dimensions of Selmer varieties associated to the complement of the origin in an elliptic curve with CM. As a corollary, one obtains a π_1-proof of the theorem of Siegel for such curves.
We compare the Pontryagin duals of fine Selmer groups of two congruent $p$-adic Galois representations over admissible pro-$p$, $p$-adic Lie extensions $K_\infty$ of number fields $K$. We prove that in several natural settings the $π$-primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $μ$-invariants. In the special case of a $\mathbb{Z}_p$-extension $K_\infty/K$, we also compare the Iwasawa $λ$-invariants of the fine Selmer groups, even in situations …
Let $E$ be an elliptic curve over $\mathbb{Q}$ with good supersingular reduction at a prime $p\geq 3$ and $a_p=0$. We generalise the definition of Kobayashi's plus/minus Selmer groups over $\mathbb{Q}(μ_{p^\infty})$ to $p$-adic Lie extensions $K_\infty$ of $\mathbb{Q}$ containing $\mathbb{Q}(μ_{p^\infty})$, using the theory of $(ϕ,Γ)$-modules and Berger's comparison isomorphisms. We show that these Selmer groups can be equally described using the "jumping conditions" of Kobayashi via the theory of overconvergent power series. Moreover, we show that such an approach gives the usual Selmer group…
Let $E$ be an elliptic curve defined over a number field $F$. In this paper, we study the structure of the $p^\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division points of an abelian variety $A$ defined over $F$. The main focus of the paper is the calculation of the $\Gal(F_\infty/F)$-Euler characteristic of the $p^\infty$-Selmer group of $E$. The final section illustrates the main theory with the example of an elliptic curve of conductor 294.
Samit Dasgupta has proved a formula factoring a certain restriction of a 3-variable Rankin-Selberg $p$-adic $L$-function as a product of a 2-variable $p$-adic $L$-function related to the adjoint representation of a Hida family and a Kubota-Leopoldt $p$-adic $L$-function. We prove a result involving Selmer groups that along with Dasgupta's result is consistent with the main conjectures associated to the 4-dimensional representation (to which the 3-variable $p$-adic $L$-function is associated), the $3$-dimensional representation (to which the $2$-variable $p$-adic $L$-function is associated) and…
We study the growth of the Galois invariants of the $p$-Selmer group of an elliptic curve in a degree $p$ Galois extension. We show that this growth is determined by certain local cohomology groups and determine necessary and sufficient conditions for these groups to be trivial. Under certain hypotheses this allows us to give necessary and sufficient conditions for there to be growth in the full $p$-Selmer group in a degree $p$ Galois extension.
Computing Selmer groups associated to mod p Galois representations
本文提出了一种计算与模 p Galois 表示相关的 Selmer 组的方法，并通过 Magma 实现，用于比较其秩与上同调类的周期，支持模 p Bloch-Kato 猜想的松散版本。
In this article, we study the pseudo-isomorphism class of the dual fine Selmer group $X$ attached to a $p$-adic Galois deformation whose deformation ring $Λ$ is isomorphic to the ring of formal power series. By using the "Kolyvagin system" arising from a given Euler system $c$, we shall construct ideals $C_i(c)$ of $Λ$, and prove that the ideals $C_i(c)$ approximate the higher Fitting ideals of $X$ under suitable hypothesis. In particular, we shall prove that the ideals $C_i(c)$ arising from the Euler system of Beilinson-Kato elements determine the pseudo-isomorphism classes of the dual fine S…
Let $F$ be a global function field of characteristic $p>0$ and $A/F$ an abelian variety. Let $K/F$ be an $ł$-adic Lie extension ($ł\neq p$) unramified outside a finite set of primes $S$ and such that $\Gal(K/F)$ has no elements of order $ł$. We shall prove that, under certain conditions, $Sel_A(K)_ł^\vee$ has no nontrivial pseudo-null submodule.
The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves
本文证明了在特定数域F（有理数域或多重二次数域）、素数p（2、3、5）及p幂次伽罗瓦扩张K/F下，按高度排序的椭圆曲线E/Q的E/K p-Selmer群平均维数有界，还给出Mordell-Weil群表示论不变量的界。
In this paper, we study the graph of homothety classes of stable free lattices in a two-dimensional representation over a local UFD. This generalizes a classical result of the case where the base ring is a discrete valuation ring due to Serre. As applications, we consider the case when the representation comes from a residually reducible Hida family and we study the control theorem of Selmer groups. These results enable us to know the precise statement of the main conjecture in residually reducible case as we will remark in section 4 and section 5.
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…
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。
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.
Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd prime where $E$ has good supersingular reduction. Let $F_1$ denote the characteristic power series of the Pontryagin dual of the fine Selmer group of $E$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ and let $F_2$ denote the greatest common divisor of Pollack's plus and minus $p$-adic $L$-functions or Sprung's sharp and flat $p$-adic $L$-functions attached to $E$, depending on whether $a_p(E)=0$ or $a_p(E)\ne0$. We study a link between the divisors of $F_1$ and $F_2$ in the Iwasawa algebra. This gives new insights into probl…
Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups
本文利用 Abel 簇稀疏二次扭族中 2-Selmer 秩的分布结果，统一研究 Mordell–Weil 群 Galois 模结构及 Hasse 原理失效等问题。
Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even…
Selmer stability for elliptic curves in Galois $\ell$-extensions
本文研究了椭圆曲线在Galois ℓ-扩张中的Selmer稳定性，证明了在特定条件下存在无限多个扩张使Selmer群消失，并给出了渐近下界。
Selmer stability in families of congruent Galois representations
本文研究同余模形式族中Selmer群的变化，探讨在固定素数p≥5下，通过Greenberg局部条件定义的Selmer群稳定性。证明在某些条件下，满足p-秩稳定性的模形式数量至少增长为X(logX)^{α-1}。
In this paper, we compare the structure of Selmer groups of certain classes of Galois representations over an admissible $p$-adic Lie extension. Namely, we show that the $π$-primary submodules of the Pontryagin dual of the Selmer groups of two Galois representations have the same elementary representations when the two Galois representations in question are either Tate dual to each other or are congruent to each other. In the first situation, our result gives a partial answer to the question of Greenberg on whether the Pontryagin dual of the Selmer groups of two Galois representations that are…
In this paper, we will study the pseudo-nullity of the fine Selmer group and its related question. Namely, we investigate certain situations, where one can deduce the pseudo-nullity of the dual fine Selmer group of a general Galois module over a admissible $p$-adic Lie extension $F_{\infty}$ from the knowledge that pseudo-nullity of the Galois group of the maximal abelian unramified pro-$p$ extension of $F_{\infty}$ at which every prime of $F_{\infty}$ above $p$ splits completely. In particular, this gives us a way to construct examples of the pseudo-nullity of the dual fine Selmer group of a …
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。
In this paper, we compare the Akashi series of the Pontryagin dual of the Selmer groups of two Galois representations over a strongly admissible p-adic Lie extension. Namely, we show that whenever the two Galois representations in question are congruent to each other, the Akashi series of one is a unit if and only if the Akashi series of the other is also a unit. We will also obtain similar results for the Euler characteristics of the Selmer groups and the characteristic elements attached to the Selmer groups.
In this paper, we study the Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We will show that under appropriate congruence condition, if the dual Selmer group of one satisfies the $\M_H(G)$-property, so will the other. In the event that the $\M_H(G)$-property holds, and assuming certain further hypothesis on the decomposition of primes in the $p$-adic Lie extension, we compare the ranks of the $π$-free quotient of the two dual Selmer groups. We then apply our results to compare the characteristic elements attached to the Selmer groups. We also s…
The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the Q_pG-representation naturally associated to the p-infinity Selmer group of A/F is self-dual. The second part describes a method for obtaining information about parities of Selmer ranks from the local Tamagawa numbers of A in intermediate extensions of F/K.
We consider limits of p-adic Galois representations, study different notions of convergence for such representations, and prove Cebotarev-type density theorems for them.
Let $E$ be an elliptic curve---defined over a number field $K$---without complex multiplication and with good ordinary reduction at all the primes above a rational prime $p \geq 5$. We construct a pairing on the dual $p^\infty$-Selmer group of $E$ over any strongly admissible $p$-adic Lie extension $K_\infty/K$ under the assumption that it is a torsion module over the Iwasawa algebra of the Galois group $G=\operatorname{Gal}(K_\infty/K)$. Under some mild additional hypotheses this gives an algebraic functional equation of the conjectured $p$-adic L-function. As an application we construct comp…
In this paper we study when two congruent $l$-adic Galois representations have congruent Selmer groups. We obtain results for representations from cyclotomic characters, Hecke characters and adjoints of modular forms.
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…
Congruent elliptic curves over some $p$-adic Lie extensions
本文研究奇素数p下p进李扩张中p-同余椭圆曲线的Iwasawa不变量变化，同时分析经典Selmer群与精细Selmer群。
We take some initial steps towards illuminating the (hypothetical) $p$-adic local Langlands functoriality principle relating Galois representations of a $p$-adic field $L$ and admissible unitary Banach space representations of $G(L)$ when $G$ is a split reductive group over $L$. The outline of our work is derived from Breuil's remarkable insights into the nature of the correspondence between 2-dimensional crystalline Galois representations of the Galois group of $\Qdss_p$ and Banach space representations of $GL_{2}(\Qdss_p)$.
On the Iwasawa invariants of BDP Selmer groups and BDP P-adic L-fucntions
本文将分圆情形下权2赫克本征尖点形式的p准素塞尔默群与p进L函数的岩泽不变量关系，推广到反分圆情形，研究虚二次域下满足广义Heegner假设的两个形式的BDP塞尔默群与BDP p进L函数的岩泽不变量关联。
Fine Selmer groups of modular forms
研究模形式的细Selmer群与理想类群的Iwasawa不变量，比较其秩和μ不变量。
We discuss $p$-adic unipotent Albanese maps for curves of positive genus, extending the theory of $p$-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of `Birch and Swinnerton-Dyer type' to non-linear theorems of Faltings-Siegel type.
Kolyvagin systems of rank 0 and the structure of the Selmer group of elliptic curves over abelian extensions
通过放宽特定素数的秩0 Selmer结构，构造辅助Kolyvagin系统，描述Selmer群的Fitting理想，并用模符号刻画椭圆曲线在有限阿贝尔扩张上的Galois结构。
Let $p$ be a prime number, and let $K$ be a $p$-adic local field. We study a class of semistable $p$-adic Galois representations of $K$, which we call {\it triangulordinary} because it includes the ordinary ones yet allows non-étale behavior in the associated $(ϕ,Γ_K)$-modules over the Robba ring. Our main result provides a description of the Bloch--Kato local condition of such representations. We also propose a program, using variational techniques, that would give a definition of the Selmer group along the eigencurve of Coleman--Mazur, including notably its nonordinary locus.
Functional equations of algebraic Rankin-Selberg $p$-adic $L$-functions
本文为 Selmer 复形建立代数函数方程，从而得到代数 $p$-进 $L$ 函数的函数方程，并证明其完美性，应用于 Iwasawa 理论中 Rankin-Selberg 乘积的研究。
In this article, we set up a strategy to prove one divisibility towards the main Iwasawa conjecture for the Selmer groups attached to the twisted adjoint modular Galois representations associated to Hida families. This conjecture asserts the equality of the p-adic L-function interpoling the critical values of the symmetric square of the modular forms in these families and the characteristic ideal of the associated Selmer group. The idea is to introduce a third characteristic ideal containing informations on the congruences between cuspidal Siegel modular forms of genus 2 and the Klingen type E…
For an elliptic curve over the rational number field and a prime number $p$, we study the structure of the classical Selmer group of $p$-power torsion points. In our previous paper \cite{Ku6}, assuming the main conjecture and the non-degeneracy of the $p$-adic height pairing, we proved that the structure of the Selmer group with respect to $p$-power torsion points is determined by some analytic elements $\tildeδ_{m}$ defined from modular symbols. In this paper, we do not assume the main conjecture nor the non-degeneracy of the $p$-adic height pairing, and study the structure of Selmer groups, …
On the Pseudonullity of Fine Selmer groups over function fields
本文研究椭圆曲线在数域及特征 $p$ 函数域的 $p$-adic Lie 扩张上精细 Selmer 群的伪零性，验证了猜想 B 的新情形。
Let $A$ be an abelian variety defined over a global function field $F$ of positive characteristic $p$ and let $K/F$ be a $p$-adic Lie extension with Galois group $G$. We provide a formula for the Euler characteristic $χ(G,Sel_A(K)_p)$ of the $p$-part of the Selmer group of $A$ over $K$. In the special case $G=\mathbb{Z}_p^d$ and $A$ a constant ordinary variety, using Akashi series, we show how the Euler characteristic of the dual of $Sel_A(K)_p$ is related to special values of a $p$-adic $\mathcal{L}$-function.
We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were proven to exist by the seond named author.
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 develop a theory of `non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of $p$-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate-Shafarevich and Selmer groups of abelian varieties.
We consider distinction of representations in the context of $p$-adic Galois symmetric spaces. We provide new sufficient conditions for distinction of parabolically induced representations in terms of similar conditions on the inducing data and deduce a characterization for distinction of representations parabolically induced from cuspidal. We explicate the results further for classical groups and give several applications, in particular, concerning the preservation of distinction via Langlands functoriality. We relate our results with a conjecture of Dipendra Prasad.
Let $p$ be an odd prime, and let $K/K_0$ be a quadratic extension of number fields. Denote by $K_\pm$ the maximal $\mathbb{Z}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_+$ abelian over $K_0$ and $K_-$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_\pm$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_\pm$. Our method or proof generalizes a method of Mazur--Rubin, building upon results of N…
Using the theory of $(ϕ,Γ)$-modules and the formalism of Selmer complexes we construct the p-adic height for p-adic representations with coefficients in an affinoid algebra over $Q_p$.
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