Chandrashekhar Khare
In this note we relate the property of a semisimple l-adic Galois representation being "F-split" for a number field F to its having abelian image. By F-split we mean that the characteristic polynomials of Frobenii all split in F.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This paper surveys the methods that have been used to attack the conjecture, still open, that an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.
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.
We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.
An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…
This paper justifies an assertion in (Elder, Proc AMS 137 (2009), no 4, 1193--1203) that Galois scaffolds make the questions of Galois module structure tractable. Let $k$ be a perfect field of characteristic $p$ and let $K=k((T))$. For the class of characteristic $p$ elementary abelian $p$-extensions $L/K$ with Galois scaffolds described in mentioned paper, we give a necessary and sufficient condition for the valuation ring $\mathfrak{O}_L$ to be free over its associated order $\mathcal{A}_{L/K}$ in $K[\Gal(L/K)]$. Interestingly, this condition agrees with the condition found by Y. Miyata, con…
We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.
We give an effective version of a result reported by Serre asserting that the images of the Galois representations attached to an abelian surface with $\End(A)= \mathbb{Z}$ are as large as possible for almost every prime. Our algorithm depends on the truth of Serre's conjecture for two dimensional odd irreducible Galois representations. Assuming this conjecture we determine the finite set of primes with exceptional image.
This paper explores some first-order properties of commuting-liftable pairs in pro-$\ell$ abelian-by-central Galois groups of fields. The main focus of the paper is to prove that minimized inertia and decomposition groups of many valuations are first-order definable using a predicate for the collection of commuting-liftable pairs.
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
In this paper we use the Galois module structure for the classical parameterizing spaces for elementary p-abelian extensions of a field K to give necessary and sufficient conditions for the solvability of any embedding problem which is an extension of Z/p^nZ with elementary p-abelian kernel. This allows us to count the total number of solutions to a given embedding problem when the appropriate modules are finite, and leads to some nontrivial automatic realization and realization multiplicity results for Galois groups.
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.
We determine the Galois module structure of the parameterizing space of elementary $p$-abelian extensions of a field $K$ when $\text{Gal}(K/F)$ is any finite $p$-group, under the assumption that the maximal pro-$p$ quotient of the absolute Galois group of $F$ is a free, finitely generated pro-$p$ group, and that $F$ contains a primitive $p$th root of unity if $\text{char}(F) \neq p$.
Galois invariants of finite abelian descent and Brauer sets
该文研究全局域上代数簇的有限阿贝尔下降、Brauer-Manin障碍对应的阿德尔点子集的伽罗瓦不变量还原基域障碍集的条件，并将常曲线的相关结果推广到函数域上的等平凡曲线。
Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $ρ_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $ρ_A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a nonempty open subvariety $U\subseteq \mathbb{P}^n_K$. We show that away from a set of density $0$, the image of $ρ_A$ will be very large; more precisely, it will have uniformly bounded in…
We define a variant of normal basis, called a {\em Galois scaffolding}, that allows for an easy determination of valuation, and has implications for Galois module structure. We identify fully ramified, elementary abelian extensions of local function fields of characteristic $p$, called {\em one-dimensional}, that, in a particular sense, are as simple as cyclic degree $p$ extensions, and prove the statement in the title above.
We propose a conjectural characterization of when the dynamical Galois group associated to a polynomial is abelian, and we prove our conjecture in several cases, including the stable quadratic case over ${\mathbb Q}$. In the postcritically infinite case, the proof uses algebraic techniques, including a result concerning ramification in towers of cyclic $p$-extensions. In the postcritically finite case, the proof uses the theory of heights together with results of Amoroso-Zannier and Amoroso-Dvornicich, as well as properties of the Arakelov-Zhang pairing.
Let $A$ be an absolutely simple abelian variety without (potential) complex multiplication, defined over the number field $K$. Suppose that either $\dim A=2$ or $A$ is of $\operatorname{GL}_2$-type: we give an explicit bound $\ell_0(A,K)$ such that, for every prime $\ell>\ell_0(A,K)$, the image of the absolute Galois group of $K$ in $\operatorname{Aut}(T_\ell(A))$ is as large as it is allowed to be by endomorphisms and polarizations.
Given a rational variety $V$ defined over $K$, we consider a principally polarized abelian variety $A$ of dimension $g$ defined over $V$. For each prime l we then consider the galois representation on the $l$-torsion of $A_t$, where $t$ is a $K$-rational point of $V$. The largest possible image is $GSp_{2g}(\mathbb{F}_l)$ and in the cases $g=1$ and 2, we are able to get surjectivity for all $l$ and almost all $t$. In the case $g=1$ this recovers a theorem originally proven by William Duke.
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.
Let $K$ be a number field, $f\in K[x]$ and $α\in K$. A recent conjecture of Andrews and Petsche predicts that the dynamical Galois group of the pair $(f,α)$ is abelian if and only if the pair $(f,α)$ is $K^{\text{ab}}$-conjugated to $(g,β)$, where $g$ is a power or a Chebyshev map and $β$ is $ζ$ or $ζ+ζ^{-1}$, respectively, and $ζ$ is a root of unity. We review three completely different approaches that allow to prove several cases of the conjecture.
For any finite group G and integer i, let $\mathcal{H}^i(G)$ be the set of all the isomorphism classes of the Galois cohomology groups $\hat{H}^i(K/k,E_K)$, where K/k runs over all the unramified G-extension of number fields and E_K denotes the global unit group of K. We will determine $\mathcal{H}^i(G)$ for i=0,1,2, and 4 in the case where G is a finite p-group.
Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we also describe the full Galois group of $K(A_{\text{tors}})/K$. This makes explicit previous results of Serre and Ribet, and strengthens a theorem of Banaszak, Gajda and Krasoń. Our bounds are especially sharp in case the CM type of $A$ is nondegenerate.
Non-abelian arboreal Galois groups associated to PCF rational maps
本文证明了对于PCF有理映射和非预周期基点，数域的树状Galois扩张绝非阿贝尔的，为多项式确立了Andrews-Petsche猜想的新已知情形，并指出反例若存在必稀疏。
This is a revision of a McMaster University preprint, with extension. In this paper we prove that over local or global fields of characteristic 0, the Corestriction Principle holds for kernel and image of all maps which are connecting maps in group cohomology and the groups of $R$-equivalences. Some related questions over arbitrary fields of characteristic 0 are also discussed. AMS Mathematics Subject Classification (1991): Primary 11E72, Secondary 18G50, 20G10
Let $A$ be a principally polarized abelian variety of dimension $g$ over a number field $K$. Assume that the image of the adelic Galois representation of $A$ is an open subgroup of $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. Then there exists a positive integer $m$ so that the Galois image of $A$ is the full preimage of its reduction modulo $m$. The least $m$ with this property, denoted $m_A$, is called the image conductor (also called the level) of $A$. Jones recently established an upper bound for $m_A$, in terms of standard invariants of $A$, in the case that $A$ is an elliptic curve witho…
On a Galois property of fields generated by the torsion of an abelian variety
本文证明数域上阿贝尔簇挠点生成域的伽罗瓦子扩张（伽罗瓦群指数有限）必含于某有限扩张的阿贝尔扩张中，并由此推出这些域具有 Northcott 性质。
In a recent paper, Moshe Jarden proposed a conjecture, later named the Kuykian conjecture, which states that if A is an abelian variety defined over a Hilbertian field K, then every intermediate field of K(A_{tor})/K is Hilbertian. We prove that the conjecture holds for Galois extensions of K in K(A_{tor}).
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.
In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of non-isomorphic $K$ having isomorphic $A_K$ were obtained on the basis of a classification by Kubota of idele class character groups in terms of their infinite families of Ulm invariants, and did not yield a description of $A_K$. In this paper, we provide a direct `computation' of the profinite group $A_K$ for imaginary quadratic $K$, and use it to obtain many differe…
We prove new results concerning the additive Galois module structure of certain wildly ramified finite non-abelian extensions of Q. In particular, when K/Q is a Galois extension with Galois group G isomorphic to A4, S4 or A5, we give necessary and sufficient conditions for the ring of integers of K to be free over its associated order in the rational group algebra Q[G].
Let $K$ be a number field with ring of integers $\mathcal{O}_K$ and $G$ a finite group of odd order. If $K_h$ is a weakly ramified $G$-Galois $K$-algebra, then its square root $A_h$ of the inverse different is a locally free $\mathcal{O}_{K}G$-module and hence determines a class in the locally free class group $\mbox{Cl}(\mathcal{O}_KG)$ of $\mathcal{O}_KG$. We show that for $G$ abelian and under suitable assumptions, the set of all such classes is a subgroup of $\mbox{Cl}(\mathcal{O}_KG)$.
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。
The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat curve. We give examples of torsion points whose associated Galois structure is trivial, as well as points of infinite order whose associated Galois structure is non-trivial.
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.
Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting \ell-torsion representation ρ_{A,\ell} from G_F to GSp(A[\ell](\bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(\mathbb{F}_\ell) as the Galois group of a finite tame extension of number fields F'/F such that F is unramified above \ell.
Let L/K be an extension of number fields where L/\Q is abelian. We define such an extension to be Leopoldt if the ring of integers O_L of L is free over the associated order A_L/K. Furthermore we define an abelian number field K to be Leopoldt if every finite extension L/K with L/Q abelian is Leopoldt in the sense above. Previous results of Leopoldt, Chan & Lim, Bley, and Byott & Lettl culminate in the proof that the n-th cyclotomic field Q^(n) is Leopoldt for every n. In this paper, we generalize this result by giving more examples of Leopoldt extensions and fields, along with explicit genera…
Irreducibility of Newton strata in Picard modular surfaces and split local Galois representations
本文通过计算分歧素数处Picard模曲面特殊纤维中非寻常Newton层闭包的单值群与不可约性，证明Picard模形式的伴随形式存在性等价于其关联局部Galois表示的分歧性质。
Let $L/K$ be a finite Galois extension whose Galois group $G$ is non-abelian and characteristically simple. Using tools from graph theory, we shall give a closed formula for the number of Hopf-Galois structures on $L/K$ with associated group isomorphic to $G$.
This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with several explicit examples.
On the local-global principle for twists of abelian varieties and Galois representations
本文研究了阿贝尔簇扭的局部-全局原理，定义并研究了相关的上同调集，并证明了其有限性和平凡条件。
We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove arithmetical analogues of results shown by Moonen and Zarhin in the context of complex abelian varieties (of dimension at most 5).
The main result of the paper is a reciprocity law which proves that compatible systems of semisimple, abelian mod $p$ representations (of arbitrary dimension) of absolute Galois groups of number fields, arise from Hecke characters. In the last section of the paper analogs for Galois groups of function fields of these results are explored, and a question is raised whose answer will require developments in transcendence theory in characteristic $p$.
We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.
Abelian dynamical Galois groups over global function fields
本文在全局函数域上证明了Andrews--Petsche猜想的一个类比：当多项式次数小于域特征时，阿贝尔动力伽罗瓦群恰为等平凡的，并给出了完整分类猜想。
Ramification groups of some finite Galois extensions of maximal nilpotency class over local fields of positive characteristic
本文将 Brylinski 关于等特征局部域上阿贝尔伽罗瓦扩张分歧群的结果，推广到阶不超过 $p^{p+1}$ 且具有最大幂零类的非阿贝尔伽罗瓦扩张情形。
Let $n$ denote either a positive integer or $\infty$, let $\ell$ be a fixed prime and let $K$ be a field of characteristic different from $\ell$. In the presence of sufficiently many roots of unity in $K$, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal $\ell^n$-abelian Galois group of $K$ using the maximal $\ell^N$-abelian-by-central Galois group of $K$, whenever $N$ is sufficiently large relative to $n$.
In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This in particular implies Grothendieck's conjecture on the perfectness of his pairing between the Neron component groups of an abelian variety and its dual. The point is that our formulation is well-suited with Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.
Let $L/K$ be a finite Galois, totally ramified $p$-extension of complete local fields with perfect residue fields of characteristic $p>0$. In this paper, we give conditions, valid for any Galois $p$-group $G={Gal}(L/K)$ (abelian or not) and for $K$ of either possible characteristic (0 or $p$), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper. But since our conditions can be difficult to check, we specialize to elementary abelian extensions …
Rank growth of abelian varieties over certain finite Galois extensions
该研究证明了满足特定条件的有限伽罗瓦扩张可迫使阿贝尔簇实现秩增长，结合有限群表示论得到秩增长显式下界，还将结果应用于雅可比簇并探讨其与无限秩猜想的联系。
In this paper we obtain realizations of the 4-dimensional general symplectic group over a prime field of characteristic $\ell>3$ as the Galois group of a tamely ramified Galois extension of $\mathbb{Q}$. The strategy is to consider the Galois representation $ρ_{\ell}$ attached to the Tate module at $\ell$ of a suitable abelian surface. We need to choose the abelian varieties carefully in order to ensure that the image of $ρ_{\ell}$ is large and simultaneously maintain a control on the ramification of the corresponding Galois extension. We obtain an explicit family of curves of genus 2 such tha…
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。
This paper continues the study of certain two-dimensional Galois representations attached to modular forms (mod p) via a construction due to Deligne. In particular, we prove a criterion for determining when the representation is split when restricted to a decomposition group at p.While this problem was already well understood for "nonexceptional" modular forms, here we specifically address the "exceptional" case. Although the problem is not solved in all generality, it is reasonable to hope that this proof will in fact provide the framework of a proof in the more general case.
Let $K$ be a field whose characteristic is prime to a fixed integer $n$ with $μ_n \subset K$, and choose $ω\in μ_n$ a primitive $n$th root of unity. Denote the absolute Galois group of $K$ by $\operatorname{Gal}(K)$, and the mod-$n$ central-descending series of $\operatorname{Gal}(K)$ by $\operatorname{Gal}(K)^{(i)}$. Recall that Kummer theory, together with our choice of $ω$, provides a functorial isomorphism between $\operatorname{Gal}(K)/\operatorname{Gal}(K)^{(2)}$ and $\operatorname{Hom}(K^\times,\mathbb{Z}/n)$. Analogously to Kummer theory, in this note we use the Merkurjev-Suslin theore…
Let K be a field and \ell be a prime such that char K \neq \ell. In the presence of sufficiently many roots of unity in K, we show how to recover some of the inertia/decomposition structure of valuations inside the maximal (\Z/\ell)-abelian resp. pro-\ell-abelian Galois group of K using its (Z/\ell)-central resp. pro-\ell-central extensions.
In this paper the new techniques and results concerning the structure theory of modules over non-commutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions K of number fields k "up to pseudo-isomorphism". In particular, a close relationship is revealed between the Selmer group of abelian varieties, the Galois group of the maximal abelian unramified p-extension of K as well as the Galois group of the maximal abelian outside S unramified p-extension where S is a finite set of certain places of k. Moreover, we determine the Galois module structure …
Descent properties for an abelian variety with extended Galois representation
本文研究了具有扩展伽罗华表示的阿贝尔簇的下降性质，证明了在特定条件下，类型I的阿贝尔簇的逆向成立。
For an abelian number field of odd degree, we study the structure of its 2-Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow class groups in families where the degree and Galois group are fixed.
Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.
Let $G$ be a Frobenius group with an abelian Frobenius kernel $F$ and let $k$ be a finite extension of $\mathbb{Q}$. We obtain an upper bound for the number of degree $|F|$ algebraic extensions $K/k$ with Galois group $G$ with the norm of the discriminant $\mathcal{N}_{k/\mathbb{Q}}(d_{K/k})$ bounded above by $X$. We extend this method for any group $G$ that has an abelian normal subgroup. If $G$ has an abelian normal subgroup, then we obtain upper bounds for the number of degree $|G|$ extensions $N/k$ with Galois group $G$ with bounded norm of the discriminant. Malle made a conjecture about w…
Equidistant liftings of elementary abelian Galois covers of curves
本文研究初等阿贝尔群作用的局部提升问题，通过对数微分形式给出等距提升的必要条件，并证明$\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$作用的新提升障碍。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。