Chandrashekhar Khare
In this note we produce examples of converging sequences of Galois representations, and study some of their properties. Some of the results here are used in the preprint math.NT/0210296.
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We prove new automorphy lifting theorems for residually reducible Galois representations of unitary type in which the residual representation is permitted to have an arbitrary number of irreducible constituents.
This paper is a continuation of the author's previous work, where we studied the variation of the number of isomorphic classes of $G_{\mathbb{Q}}$-stable lattices in p-adic families of residually reducible ordinary Galois representations. In this paper, we study the number of isomorphic classes of $G_{\mathbb{Q}}$-stable free lattices in a residually reducible Hida deformation and the variation of their related two-variable algebraic $p$-adic $L$-functions. This result gives an answer of a question asked by Ochiai.
We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called `Taylor--Wiles hypothesis'. We apply this to the problem of the modularity of elliptic curves over totally real fields
We prove modularity of some two dimensional, 2-adic Galois representations over totally real fields that are nearly ordinary and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2.
We prove automorphy lifting theorems for 2-dimensional Galois representations of absolute Galois groups of totally real fields when the residual representation is of "exceptional" type. This exceptional case is when we are in characteristic 5, the residual representation has projective image PGL_2(F_5), and the fixed field of its kernel contains a primitive 5th root of unity. In this case the Taylor-Wiles patching method does not suffice and we prove our theorem by combining patching with the technique of automorphy by p-adic approximation.
On the $μ$-invariants of residually reducible Galois representations
研究剩余可约伽罗瓦表示塞尔梅群的岩泽$μ$-不变量，定义细化该不变量的岩泽不变量，证明存在模伽罗瓦表示提升可约模$p$伽罗瓦表示且其塞尔梅群$μ$-不变量≠指定整数，细化剩余可约情形下的塞尔猜想。
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 show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the +/--Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this we relate the oddness of the associated polarized Galois representations (in the sense of the Bellaïche-Chenevier sign being +1) to the parity condition for criticality. Under an assumption similar to Vandiver's conjecture this also provides evidence for the Fontaine-Mazur conjecture for polarized Galois representations of any even dimension.
Let $E$ be an elliptic curve over an imaginary quadratic field $K$, and $p$ be an odd prime such that the residual representation $E[p]$ is reducible. The $μ$-invariant of the fine Selmer group of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$ is studied. We do not impose the Heegner hypothesis on $E$, thus allowing certain primes of bad reduction to decompose infinitely in the anticyclotomic $\mathbb{Z}_p$-extension. It is shown that the fine $μ$-invariant vanishes if certain explicit conditions are satisfied. Further, a partial converse is proven.
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。
We give a direct approach to recover some of the results of Wiles and Tayor on modularity of certain 2-dimensional p-adic representations of the absolute Galois group of Q.
The $\ell$-adic Galois polylogarithm is an arithmetic function on an absolute Galois group with values in $\ell$-adic numbers, which arises from Galois actions on $\ell$-adic étale paths on ${\mathbb P}^1 \backslash \{0,1,\infty\}$. In the present paper, we discuss a relationship between $\ell$-adic Galois polylogarithms and triple $\ell$-th power residue symbols in some special cases studied by a work of Hirano-Morishita. We show that a functional equation of $\ell$-adic Galois polylogarithms by Nakamura-Wojtkowiak implies a reciprocity law of triple $\ell$-th power residue symbols.
Let $p$ be a prime number. There are properties called ``overconvergence'' and ``$F$-analyticity'' for $p$-adic Galois representations of a $p$-adic field $F$. By Berger's work, it is known that $F$-analyticity is stricter than overconvergence. In this article, we show that, in many cases, an overconvergent Galois representation is $F$-analytic up to a twist by a character. This result emphasizes the necessity of the theory of $(φ,Γ)$-modules over the multivariable Robba ring, by which we expect to study all $p$-adic Galois representations.
We prove that the density of ramified primes in semisimple p-adic representations of Galois groups of number fields is 0. Ravi Ramakrishna has produced examples of such representations that are infinitely ramified.
We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory for curves over a $p$-adic field.
Comparing Galois representations in the residually reducible case
本文提出一种算法，用于判定剩余像为p-群的伽罗瓦表示的半单化是否同构，并应用于证明特定不可约表示由五个小范数素数处的Frobenius特征多项式决定，且自动验证了Grenié的2-adic例子。
We are studing Galois actions on fundamental groups. Using towers of coverings we construct measures on the products of finite copies of Z_p. Using these measures we can calculate coefficients of Galois representations. In the simplest case of measures on Z_p, we get Kubota_Leopoldt p-adic L-functions and p-adic Hurwitz zeta functions.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
In this paper we show that two dimensional (mod p) Galois representations satisfying mild hypotheses can be lifted to p-adic Galois representations ramified at infinitely many primes such that the characteristic polynomials of Frobenius at a density one set of unramified primes are defined over the rational integers. In particular we show that often one can "density one compatibly" lift mod p and mod q Galois representations.
We construct various explicit Herr complexes that compute the Galois cohomology of a $p$-adic representation of the absolute Galois group of a complete discrete valuation field of characteristic $0$ with a perfect residue field of characteristic $p$, using the associated $(φ,τ)$-modules (defined by Xavier Caruso), instead of $(φ,Γ)$-modules. We also give an application to $p$-divisible groups.
The fact that the double zeta values at n and m can be written as a sum of products of two zeta values and of zeta value at m+n, whenever n+m is odd is due to Euler. We shall show a weak version of this result for the Galois l-adic realization.
Let $K$ be a $p$-adic field and $E$ an elliptic curve over $K$ with potential good reduction. For some large Galois extensions $L$ of $K$ containing all $p$-power roots of unity, we show the vanishing of certain Galois cohomology groups of $L$ with values in the $p$-adic representation associated with $E$. This generalizes some results of Coates, Sujatha and Wintenberger.
We describe the generic blocks in the category of smooth locally admissible mod $2$ representations of $\mathrm{GL}_2(\mathbb{Q}_2)$. As an application we obtain new cases of Breuil--Mézard and Fontaine--Mazur conjectures for $2$-dimensional $2$-adic Galois representations.
Let $K$ be a complete, discretely valued field with finite residue field and $G_K$ its absolute Galois group. The subject of this note is the study of the set of positive integers $d$ for which there exists an absolutely irreducible $\ell$-adic representation of $G_K$ of dimension $d$ with rational traces on inertia. Our main result is that non-Sophie Germain primes are not in this set when the residue characteristic of $K$ is $> 3$. The result stated in the title is a special case.
We present a family of algorithms for computing the Galois group of a polynomial defined over a $p$-adic field. Apart from the "naive" algorithm, these are the first general algorithms for this task. As an application, we compute the Galois groups of all totally ramified extensions of $\mathbb{Q}_2$ of degrees 18, 20 and 22, tables of which are available online.
For a $p$-adic local field $K$ with perfect residue field, L. Herr constructed a complex which computes the Galois cohomology of a $p$-torsion representation of the absolute Galois group of $K$ by using the theory of $(Φ,Γ)$-modules. We shall generalize his work to the imperfect residue field (the residue field has a finite $p$-basis) case.
The Residually Indistinguishable Case of Ribet's Method for GL2
本文提出了GL2情形下Ribet引理的一般版本，去除了相关特征在剩余域上必须可区分的假设，从而扩展了构造p-adic伽罗瓦表示非平凡扩张的方法。
The Galois action on the pro-$\ell$ étale fundamental groupoid of the projective line minus three points with rational base points gives rise to a non-commutative formal power series in two variables with $\ell$-adic coefficients, called the $\ell$-adic Galois associator. In the present paper, we focus on how Landen's functional equation of trilogarithms and its $\ell$-adic Galois analog can be derived algebraically from the $S_3$-symmetry of the projective line minus three points. Twofold proofs of the functional equation will be presented, one is based on Zagier's tensor criterion devised in…
Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.
Let V be a p-adic representation of the absolute Galois group G of Q_p that becomes crystalline over a finite tame extension, and assume p odd. We provide necessary and sufficient conditions for V to be isomorphic to the Tate module V_p(A) of an abelian variety A defined over Q_p. These conditions are stated on the filtered (ϕ,G)-module attached to V.
Drinfeld-modules and $A$-motives are the function field analogous of elliptic curves and abelian varieties. For the latter one can construct the $l$-adic Galois representation and can ask if its image is open. For Drinfeld-modules this question was answered by Pink and his coauthors. We clarify the rank one case for $A$-motives and show that the image of Galois is open if and only if the virtual dimension is prime to the characteristic of the ground field.
Hopf-Galois module structure of degree p extensions of p-adic fields
本文研究了p-adic域p次扩张的Hopf-Galois模结构，完整刻画了整数环在其关联阶上为自由模的条件。
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 $p\nmid N$.
We prove new automorphy lifting theorems for essentially conjugate self-dual Galois representations into $GL_n$. Existing theorems require that the residual representation have 'big' image, in a certain technical sense. Our theorems are based on a strengthening of the Taylor-Wiles method which allows one to weaken this hypothesis.
We consider continuous representations of the Galois group G of a number field K taking values in the completion C of an algebraic closure A of the field of l-adic numbers. We give a construction of irreducible representations of G in GL(2,C) which cannot be conjugated into GL(2,A) but for which there exist density-one sets of primes with Frobenius traces in A. We also show that any such representation must be ramified at infinitely many primes of K.
For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mat…
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
Multivariable $p$-adic Hodge theory for products of Galois groups
本文利用Colmez-Sen-Tate下降构造Galois群乘积表示的多变量过收敛$(φ,Γ)$-模，并定义相应周期环以统一推广并恢复多项已有多变量$p$-进Hodge理论结果。
We show that an infinite family of odd complex 2-dimensional Galois representations ramified at 5 having nonsolvable projective image are modular, thereby verifying Artin's conjecture for a new case of examples. Such a family contains the original example studied by Buhler. In the process, we prove that an infinite family of residually modular Galois representations are modular by studying $Λ$-adic Hecke algebras.
Models of CM elliptic curves with a prescribed $\ell$-adic Galois image
该研究针对素数ℓ与类数1或2的虚二次序，刻画了对应复乘椭圆曲线在最小定义域上的所有ℓ-进伽罗瓦表示，并确定了具有指定ℓ-进伽罗瓦表示的椭圆曲线同构类。
We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with $p$-adic CM forms.
In this paper we present a description of the Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and $SL_2(\mathbb{F}_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over $\mathbb{Q}_2$ is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.
Bloch-Kato groups over perfectoid fields and Galois theory of $p$-adic periods
该研究将完美oid域上de Rham伽罗瓦表示的Bloch-Kato群结构与p进周期环B_dR⁺的伽罗瓦理论关联，解答了Coates和Greenberg的问题并推广相关结果，方法依赖Fargues-Fontaine曲线上向量丛的分类。
On Constructing Extensions of Residually Isomorphic Characters
本文阐述了一项合作工作，在剩余不可区分情形下证明了适用于$\mathrm{GL}_2$的Ribet引理版本，通过构造取值于“大”$\mathbf{T}$-模的Galois上同调类来构造特征的扩张。
This paper investigates Galois branched covers of the open $p$-adic disc and their reductions to characteristic $p$. Using the field of norms functor of Fontaine and Wintenberger, we show that the special fiber of a Galois cover is determined by arithmetic and geometric properties of the generic fiber and its characteristic zero specializations. As applications, we derive a criterion for good reduction in the abelian case, and give an arithmetic reformulation of the local Oort Conjecture concerning the liftability of cyclic covers of germs of curves.
We complete the picture of local and global arithmetic duality theorems for short complexes of finite Galois modules and tori over $p$-adic function fields. In view of the duality theorems, we deduce a $12$-term Poitou--Tate exact sequence which relates global Galois cohomology groups to restricted topological products of local Galois cohomology groups.
We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over Q when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a Taylor-Wiles patching argument for the completed homology in this case. As a key input, we generalize the work of Skinner-Wiles in the ordinary case. In addition, we also treat the residually irreducible case at the end of the paper. Combining with people's earlier work, we can prove the Fontaine-Mazur conjecture completely in the regular case when p is at le…
We estimate several probability distributions arising from the study of random, monic polynomials of degree $n$ with coefficients in the integers of a general $p$-adic field $K_{\mathfrak{p}}$ having residue field with $q= p^f$ elements. We estimate the distribution of the degrees of irreducible factors of the polynomials, with tight error bounds valid when $q> n^2+n$. We also estimate the distribution of Galois groups of such polynomials, showing that for fixed $n$, almost all Galois groups are cyclic in the limit $q \to \infty$. In particular, we show that the Galois groups are cyclic with p…
Given two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's u…
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…
Duality-reflection formulas of multiple polylogarithms and their $\ell$-adic Galois analogues
本文利用射影直线去掉三点后的 S₃-对称性所诱导的关联子代数关系，证明了复数与 ℓ-进多重多对数在指标对偶和变量反射两方面的一致公式。
On the Landen formula for multiple polylogarithms and its $\ell$-adic Galois analogue
本文通过代数和几何方法证明了复数多重polylogarithms的Landen公式，并将其推广到ℓ-adic伽罗瓦设定，引入了特定于该设定的低阶项。
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 …
We give a classification of all possible $2$-adic images of Galois representations associated to elliptic curves over $\mathbb{Q}$. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.
On the surjectivity of $(T)$-adic Galois Representations of Drinfeld $A$-Modules of Rank 2 and 3: Density results
本文研究了秩为2和3的Drinfeld A-模的(T)-adic伽罗瓦表示的满射性条件，并计算了此类模的密度。
Using the overconvergent cohomology modules introduced by Ash and Stevens, we construct eigenvarieties associated with reductive groups and establish some basic geometric properties of these spaces, building on work of Ash-Stevens, Urban, and others. We also formulate a precise modularity conjecture linking trianguline Galois representations with overconvergent cohomology classes. In the course of giving evidence for this conjecture, we establish several new instances of \emph{p-}adic Langlands functoriality. Our main technical innovations are a family of universal coefficients spectral sequen…
On possibilities of 3-adic Galois images associated to isogeny-torsion graphs
本文研究非CM椭圆曲线在素数3时的同源-挠图与3-adic伽罗瓦表示之间的关系，并给出与图顶点相关的伽罗瓦像的完整分类。
Complete classification of $7$-adic Galois images for non-CM elliptic curves over $\Q$
本文解决了Furio和Lombardo的猜想1.6，从而完成了非CM椭圆曲线在有理数域上的7-adic Galois像的完全分类，证明基于作者2026年6月至9月的AI训练，且为独立完成。
Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the ell-adic etale cohomology modules of X. We prove that this family of representations varying over ell is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ell become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting f…
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…
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