Kay Wingberg
Fontaine and Mazur conjecture that a number field k has no infinite unramified Galois extension such that its Galois group is a p-adic analytic pro-p-group. We consider this conjecture for the maximal unramified p-extension of a CM-field k.
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 is an updated and extended version of ANT-0024. We prove more special cases of the Fontaine-Mazur conjecture regarding p-adic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it.
Towards the Fontaine--Mazur conjecture for GL(2)
该研究结合新型模性结果与数的几何，证明了GL(2)的Fontaine-Mazur猜想的若干新情形，推进了该数论猜想的相关进展。
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…
On Greenberg--Benois $\mathcal{L}$-invariants and Fontaine--Mazur $\mathcal{L}$-invariants
该研究针对Greenberg–Benois L-不变量与Fontaine–Mazur L-不变量的关系问题，通过证明二者的比较定理，对Besser–de Shalit猜想给出了肯定回答。
We prove some cases of the Fontaine-Mazur conjecture for even Galois representations. In particular, we prove, under mild hypotheses, that there are no irreducible two-dimensional ordinary even Galois representations of $\Gal(\Qbar/\Q)$ with distinct Hodge-Tate weights. If $K/\Q$ is an imaginary quadratic field, we also prove (again, under certain hypotheses) that $\Gal(\Qbar/K)$ does not admit irreducible two-dimensional ordinary Galois representations of non-parallel weight. Finally, we prove that any weakly compatible family of two dimensional irreducible Galois representations of $\Gal(\Qb…
We prove, under mild hypotheses, that there are no irreducible two-dimensional_even_ Galois representations of $\Gal(\Qbar/\Q)$ which are de Rham with distinct Hodge--Tate weights. This removes the "ordinary" hypothesis required in previous work of the author. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric (= de Rham) deformations.
In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate $λ$-adic 2-dimensional Galois representation unramified at 3 and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p^2 \}) = \Q $ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of …
Remarks on the Boston Unramified Fontaine-Mazur Conjecture, II
本文研究波士顿对Galois表示的未分枝Fontaine-Mazur猜想的推广，从群论角度将其归约到特定的p-进分析群，并提出虚Golod-Shafarevich性质的解释，同时建立局部-整体原理和素数-伴随原理。
Let p be an odd prime, let S be a finite set of primes q congruent to 1 mod p but not mod p^2 and let G_S be the Galois group of the maximal p-extension of Q un-ramified outside of S. If r is a continuous homomorphism of G_S into GL_2(Z_p) then under certain conditions on the linking numbers of S we show that r=1 if its reduction mod p is 1. We also show that the reduction of r mod p is 1 if r can be put in triangular form mod p^3.
From Fontaine-Mazur conjecture to analytic pro-p groups -- A survey
本文综述了利用Boston提出的纯群论方法研究Fontaine-Mazur猜想的发展历程，回顾了已知成果并介绍了相关未解问题。
We estimate the proportion of function fields satisfying certain conditions which imply a function-field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.
On the Fontaine-Mazur conjecture for $p=3$
本文在 p=3 情形下证明了二维正则 Galois 表示的 Fontaine-Mazur 猜想剩余情形，从而完成所有奇素数正则情形的证明，主要基于 p-进 Langlands 对应、Galois 形变理论及潜在 pro-模性结果的最新进展。
On the Boston's Unramified Fontaine-Mazur Conjecture
本文研究 $p$-进 Galois 表示的非分歧 Fontaine-Mazur 猜想，证明若干基本情形与二维新结果，提出多种攻击策略与判据，并研究非分歧形变环的结构及其有限性。
In this note we prove the non existence of certain irreducible two dimensional representations of the the absolute Galois group. Such results are predicted by Serre's conjecture and we use Fontaine's methods to verify these predictions in a small number of cases. We also use a methods of Faltings to study certain finiteness conjectures for Galois representations. Key words: Serre's conjecture, Fontaine-Mazur conjectures, Galois representations.
Bockstein Spectral Sequences and Applications to the Tame Fontaine Mazur Conjecture
本文利用Bockstein谱序列与李理论工具改进了Labute的方法，验证了无穷多3位集合对应的数域pro-p伽罗瓦群满足一致Fontaine-Mazur性质，还构造了此类无限群并给出高概率检测的数值证据。
Let $F_{\wp}$ be a finite extension of $\mathbb{Q}_p$. By considering partially de Rham families, we establish a Colmez-Greenberg-Stevens formula (on Fontaine-Mazur $\mathcal{L}$-invariants) for (general) $2$-dimensional semi-stable non-crystalline $\mathrm{Gal}(\overline{\mathbb{Q}_p}/F_{\wp})$-representations. As an application, we prove local-global compatibility results for completed cohomology of quaternion Shimura curves, and in particular the equality of Fontaine-Mazur $\mathcal{L}$-invariants and Breuil's $\mathcal{L}$-invariants, in critical case.
In this paper we show that under certain condition the Fontaine--Mazur $L$-invariant for a Hilbert eigenform coincides with its Teitelbaum type $L$-invariant, and thus prove a conjecture of Chida, Mok and Park.
Under the assumption of the Hodge, Tate and Fontaine-Mazur conjectures we give a criterion for a compatible system of l-adic representations to be isomorphic to the second cohomology of a K3 surface.
In this note we complement a part of a theorem of Fontaine-Mazur. We show that if $(V,ρ)$ is an irreducible finite dimensional representation of the Galois group $Gal({\bar K}/K)$ of a finite extension of $K\Q_p$, of Hodge-Tate type $(0,1)$ then it is potentially semi-stable if and only if it is potentially crystalline. This was proved by Fontaine-Mazur for dimension two and $p\geq 5$ by their classfication theorem.
We prove that there is no geometric $p$-adic representation of the Galois group of $\Q$ which is irreducible, of dimension 2, of conductor 1 and low weight, according to a conjecture of Fontaine and Mazur.
For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine-Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at $p$ arising from Fontaine-Laffaille theory.
In this paper, a conjecture of Mazur, Rubin and Stein concerning certain averages of modular symbols is proved.
Gouvea and Mazur made a precise conjecture about slopes of modular forms. Weaker versions of this conjecture were established by Coleman and Wan. In this note, we exhibit explicit examples contradicting the full conjecture as it currently stands.
Under certain assumptions, we prove an anticyclotomic analogue of the "weak main conjecture" à la Mazur and Tate for modular forms over a large class of cyclic ring class extensions.
In 1995, Ehud de Shalit proved an analogue of a conjecture of Mazur--Tate for the modular Jacobian $J_0(p)$. His main result was valid away from the Eisenstein primes. We complete the work of de Shalit by including the Eisenstein primes, and give some applications such as an elementary combinatorial identity involving discrete logarithms of difference of supersingular $j$-invariants. An important tool is our recent work on the so called "generalized cuspidal $1$-motive".
We provide a new interpretation of the Mazur-Tate Conjecture and then use it to obtain the first (unconditional) theoretical evidence in support of the conjecture for elliptic curves of strictly positive rank.
We give an elementary proof of the Mazur-Tate-Teitelbaum conjecture for elliptic curves by using Kato's element.
Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$ with the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato's Euler system.
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.
Suppose $K$ is unramified over $\mathbb Q _p$ and $Γ_K=\operatorname{Gal}(\bar K/K)$. Let $H$ be a torsion $Γ_K$-equivariant subquotient of crystalline $\mathbb Q _p[Γ_K]$-module with HT weights from $[0,p-2]$. We give a new proof of Fontaine's conjecture about the triviality of action of some ramification subgroups $Γ_K^{(v)}$ on $H$. The earlier author's proof from [1] contains a gap and proves this conjecture only for some subgroups of index $p$ in $Γ_K^{(v)}$.
On the refined `Birch--Swinnerton-Dyer type' conjectures of Mazur and Tate
该研究证明了Mazur和Tate对Birch-Swinnerton-Dyer猜想的精细化猜想的大部分内容，通过等变Tamagawa数猜想的零秩分量方法，还得到了比原预测更精细的结果。
On the anticyclotomic Mazur-Tate conjecture for elliptic curves with supersingular reduction
该短注针对具有超奇异约化的椭圆曲线，研究其Selmer群Fitting理想相关的Mazur-Tate“弱”主猜想的反分圆类似问题。
This paper treats what we call `weak geometric liftings' of Galois representations associated to abelian varieties. This notion can be seen as a generalization of the idea of lifting a Galois representation along an isogeny of algebraic groups. The weaker notion only takes into account an isogeny of the derived groups and disregards the centres of the groups in question. The weakly lifted representations are required to be geometric in the sense of a conjecture of Fontaine and Mazur. The conjecture in question states that any irreducible geometric representation is a twist of a subquotient of …
Localized Gouvêa-Mazur conjecture
本文在剩余 Galois 表示不可约且限制在 Gal(Q̄_p/Q_p) 上可约且非常一般的条件下，证明了局部化 Gouvêa-Mazur 猜想。
On the analyticity of the maximal extension of a number field with prescribed ramification and splitting
本文在偏离Tame Fontaine-Mazur猜想的假设下，完全刻画了可作为数域指定分歧与分裂的极大pro-p扩张Galois群的p-adic解析群。
The Absolute Grothendieck Conjecture is false for Fargues-Fontaine Curves
本文证明绝对格罗滕迪克猜想对法尔格-方丹曲线不成立，给出该猜想在相关对象上的反例。
In this article, we prove a generalization of a theorem (Ogg's conjecture) due to Bary Mazur for arbitrary $N\in \N$ and for {\it number fields}. The main new observation is a modification of a theorem due to Glenn Stevens for the congruence subgroups of the form $\Ga_0(N)$ for any $N \in \N$. This in turn help us to determine the relevant part of the cuspidal subgroups without dependence on Shimura subgroups.
On a conjecture of Mazur predicting the growth of Mordell--Weil ranks in $\mathbb{Z}_p$-extensions
本文研究Mazur猜想，通过显式计算条件验证椭圆曲线在$\mathbb{Z}_p$扩张中Mordell-Weil秩的增长。
Let $N$ and $p$ be prime numbers $\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur's Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \mathbf{Z}_p)$ for the complex conjugation. We give a conjectural explicit description of the group $I\cdot H_+/I^2\cdot H_+$ in terms of the second $K$-group of the cyclotomic field $\mathbf{Q}(ζ_N)$. We prove that this conjecture follows from a conjecture of Sharifi about some Eisenstein ideal of level $Γ_1(N)$. Following the work of Fukaya--Kato, we prove partial results on Sharifi's conjecture. This allows us to prove partial res…
B. Mazur has considered the question of density in the Euclidean topology of the set of ${\Bbb Q}$-rational points on a variety $X$ defined over ${\Bbb Q}$, in particular for Abelian varieties. In this paper we consider the question of closures of the image of finitely generated subgroups of $T({\Bbb Q})$ in $Γ\backslash T({\Bbb R})$ where $T$ is a torus defined over ${\Bbb Q}$, $Γ$ an arithmetic subgroup such that $Γ\backslash T({\Bbb R})$ is compact. Assuming Schanuel's conjecture, we prove that the closures correspond to sub {\it algebraic} tori of $T$.
On the Mazur--Tate refined conjecture for the anticyclotomic tower at inert primes
本文针对$p\geq5$为惰性素数且椭圆曲线在该处超奇异约化的情形，利用Burungale--Büyükboduk--Lei的结果，证明了虚二次域反圆塔上的弱Mazur--Tate精细猜想。
On Fontaine's conjecture for torsion crystalline local systems
本文基于棱镜上同调理论，对光滑连通p-进形式概形上的挠晶体局部系证明了Fontaine猜想的类比，并由此证明具有固定Hodge-Tate权区间的晶体局部系在万有变形环中构成闭子概形。
Mazur's Growth Number Conjecture in the Rank One Case
针对秩一情形，在素数$p\geq5$、椭圆曲线好普通约化、虚二次域满足条件等假设下，通过Heegner点的$p$-进高度非零这一条件，证明了Mazur关于$\u2124_p^2$-扩张中Selmer余秩增长的猜想成立。
Let $E/\mathbb{Q}$ be an elliptic curve which has split multiplicative reduction at a prime $p$ and whose analytic rank $r_{an}(E)$ equals one. The main goal of this article is to relate the second order derivative of the Mazur-Tate-Teitelbaum $p$-adic $L$-function $L_p(E,s)$ of $E$ to Nekovář's height pairing evaluated on natural elements arising from the Beilinson-Kato elements. Along the way, we extend a Rubin-style formula of Nekovář (or in an alternative wording, correct another Rubin-style formula of his) to apply in the presence of exceptional zeros. Our height formula allows us, among …
On the Breuil-Schneider conjecture II: Potentially crystalline non-generic case
研究布勒伊 - 施奈德猜想中潜在晶态非一般情形，通过在一般方丹 - 拉法耶假设下研究自守分量非光滑点情形，证明该猜想，此假设依极值模塞尔权重等表述，让相关算子可逆。
Let $k/\mathbb F_p$ denote a finite field. For any split connected reductive group $G/W(k)$ and certain CM number fields $F$, we deform certain Galois representations $\overlineρ:Gal(\overline F/F) \to G(k)$ to continuous families $X_{\overlineρ}$ of Galois representations $Gal(\overline F/F) \to G(\overline{\mathbb Q_p})$ lifting $\overlineρ$ such that the space of points of $X_{\overlineρ}$ which are geometric (in the sense of the Fontaine-Mazur conjecture) with parallel Hodge-Tate weights has positive codimension in $X_{\overlineρ}$. Thus the set of points in $X_{\overlineρ}$ which could (c…
We study the Fitting ideals over the finite layers of the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ of Selmer groups attached to the Rankin--Selberg convolution of two modular forms $f$ and $g$. Inspired by the Theta elements for modular forms defined by Mazur and Tate in ``Refined conjectures of the Birch and Swinnerton-Dyer type'', we define new Theta elements for Rankin--Selberg convolutions of $f$ and $g$ using Loeffler--Zerbes' geometric $p$-adic $L$-functions attached to $f$ and $g$. Under certain technical hypotheses, we generalize a recent work of Kim--Kurihara on elliptic cu…
Mazur's growth number conjecture and congruences
本文研究虚二次域上 Mazur 关于 Selmer 秩增长的猜想在 Galois 表示 $p$-同余下的保持性，证明了对特定三元组及模形式 Greenberg Selmer 群的类似结果，并得出非反圆扩张中 Mordell-Weil 秩的有界性。
Mazur's main conjecture at Eisenstein primes
本文证明了椭圆曲线在满足特定同源条件时的Mazur主猜想，利用反cyclotomic Iwasawa理论和Beilinson--Flach类的Euler系统同余论证。
We apply the methods of Fukaya, Kato and Sharifi to refine Mazur's study of the Eisenstein ideal. Given prime numbers $N$ and $p\geq 5$ such that $p\mid φ(N)$, we study the quotient of the cohomology group of modular curve $X_{0}(N)$ by the square of the Eisenstein ideal. We study two invariants $b,c$ attached to this quotient and compute $c$. We propose a conjecture about the invariant $b$ which relates the structure of the ray class group of conductor $N$ to the modular symbols of $X_{0}(N)$. Assuming this conjecture, We compute the invariant $b$.
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.
We show that Mazur's conjecture on the real topology of rational points on varieties implies that there is no diophantine model of the rational integers in the rational numbers. We also prove that there is a diophantine model of the polynomial ring over a finite field in the ring of rational functions over that finite field. Both proofs depend upon Matijasevich's theorem.
Using Calegari's result on the Fontaine-Mazur conjecture, we study the irreducibility of pure, regular, rank 3 weakly compatible systems of self-dual l-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over Q with geometric genus bigger than 1.
- Let K be a totally imaginary number field. Denote by G ur K (2) the Galois group of the maximal unramified pro-2 extension of K. By comparing cup-products in {é}tale cohomology of SpecO K and cohomology of uniform pro-2 groups, we obtain situations where G ur K (2) has no non-trivial uniform analytic quotient, proving some new special cases of the unramified Fontaine-Mazur conjecture. For example, in the family of imaginary quadratic fields K for which the 2-rank of the class group is equal to 5, we obtain that for at least 33.12% of such K, the group G ur K (2) has no non-trivial uniform an…
The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their $p$-torsion Galois representations is one-to-one over function fields of small genus complex curves for sufficiently large $p$ relative to the genus.
We give the first examples of infinite sets of primes S such that Hilbert's Tenth Problem over Z[S^{-1}] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E' over Z[S^{-1}] such that the topological closure of E'(Z[S^{-1}]) in E'(R) has infinitely many connected components.
Ogg's conjectures over function fields
本文综述了数域中由Mazur证明的Ogg猜想的函数域类比命题、研究现状与证明方法，指出其基于原有思想但存在技术差异，并提出了新的推广方向。
We prove a portion of a conjecture of B. Conrad, F. Diamond, and R. Taylor, yielding some new cases of the Fontaine-Mazur conjectures, specifically, the modularity of certain potentially Barsotti-Tate Galois representations. The proof follows the template of Wiles, Taylor-Wiles, and Breuil-Conrad-Diamond-Taylor, and relies on a detailed study of the descent, across tamely ramified extensions, of finite flat group schemes over the integers of a local field. This makes crucial use of the filtered ϕ_1-modules of C. Breuil.
Strengthenings of Mazur's Conjecture for Higher Heegner Points
本文通过伽罗瓦轨道与海根轨道的相互作用，定量加强了马祖尔猜想关于更高海根点非 torsion 性的结论，避免了对水平结构的严格限制。
Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations
本文证明了Mazur猜想的一个特殊情形，通过研究Kummer曲面上的奇异有理曲线并推广至更一般的超椭圆曲面，扩展了前人结果并给出了无限制条件下的证明。
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。