Jeffrey D. Achter, Joshua Holden
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.
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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.
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.
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。
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 survey we explain the main ingredients and results of the analogue of Fontaine-Theory in equal positive characteristic which was recently developed by Genestier-Lafforgue and the author.
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。
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.
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.
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.
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。
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.
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.
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.
An integral analogue of Fontaine's crystalline functor
研究在完美域\(k\)上维特向量\(W\)上的光滑形式概型\(\mathfrak{X}\)，构造函子\(\mathbb{D}_{\mathrm{crys}}\)，证明其在特定条件下的性质及诱导范畴等价关系，还用它加强\(p -\)可除群的棱柱形迪厄多内理论研究。
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 $T$.
The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.
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…
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.
Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the p-adic L-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, Teitelbaum in dimension 1 and the …
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…
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".
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。
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.
An analogue of Kida's formula for Mazur-Tate elements
研究\(\mathbb{Q}\)上椭圆曲线马祖尔 - 泰特元素的岩泽不变量，证明其\(\mu\) - 不变量消失在扩张中保持，\(\lambda\) - 不变量有过渡公式，统一了相关解析岩泽不变量的木田公式，还得到特定\(p\) - 进\(L\) - 函数的木田型公式。
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…
A characteristic p analogue of the André--Pink--Zannier conjecture
该研究针对特征p下的André--Pink--Zannier猜想，证明了Hodge型Shimura簇中具大单值群的常值函数域点情形及相应Hecke等分布代数类似物，通过Hecke对应分析与Weyl特殊点充足性证明完成论证。
The moduli space of (1,3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third étale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group Γ_1(6). By Tate's conjecture, this should imply that Y, …
Prismatic cohomology and de Rham-Witt forms
本文为任意棱镜构造Fontaine映射的类似物，在完美情形下定义de Rham-Witt形式到棱镜上同调的典范映射并证明其为同构，进而给出$p$-完备多项式代数棱镜上同调的显式描述。
In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \par This is a …
Iwasawa Theory for K3 Surfaces over Finite Fields
本文为有限域上K3曲面建立岩泽理论，证明Mazur控制定理类比、Brauer群的岩泽类数公式及主猜想。
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。
On the arithmetic of special values of $L$-functions for certain abelian varieties with a rational isogeny
本研究针对具有有理同源的特定阿贝尔簇，证明其偶二次扭的$p$-艾森斯坦部分的模$p$版本BSD猜想，是Mazur关于奇二次扭相关结果的偶二次扭类比。
Schur $σ$-groups of type $(3,3)$ for $p=3$
研究p=3时类型(3,3)的Schur σ-群，证明10种情况下的群结构，并通过计算验证启发式方法的有效性。
The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals …
Courbes et fibrés vectoriels en théorie de Hodge $z$-adique globale
本文研究函数域上Fargues-Fontaine曲线的全局性质，通过模空间重新表述全球Langlands猜想，并在可交换群情况下验证了该猜想。
The purpose of this paper is to formulate and study a common refinement of a version of Stark's conjecture and its $p$-adic analogue, in terms of Fontaine's $p$-adic period ring and $p$-adic Hodge theory. We construct period-ring-valued functions under a generalization of Yoshida's conjecture on the transcendental parts of CM-periods. Then we conjecture a reciprocity law on their special values concerning the absolute Frobenius action. We show that our conjecture implies a part of Stark's conjecture when the base field is an arbitrary real field and the splitting place is its real place. It al…
In this paper, we deduce the vanishing of Selmer groups for the Rankin-Selberg convolution of a cusp form with a theta series of higher weight from the nonvanishing of the associated $L$-value, thus establishing the rank 0 case of the Bloch-Kato conjecture in these cases. Our methods are based on the connection between Heegner cycles and $p$-adic $L$-functions, building upon recent work of Bertolini, Darmon and Prasanna, and on an extension of Kolyvagin's method of Euler systems to the anticyclotomic setting. In the course of the proof, we also obtain a higher weight analogue of Mazur's conjec…
In 1923 Schur considered the following problem. Let f(X) be a polynomial with integer coefficients that induces a bijection on the residue fields Z/pZ for infinitely many primes p. His conjecture, that such polynomials are compositions of linear and Dickson polynomials, was proved by M. Fried in 1970. Here we investigate the analogous question for rational functions, also we allow the base field to be any number field. As a result, there are many more rational functions for which the analogous property holds. Some infinite series come from rational isogenies of elliptic curves and deformations…
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…
Effective surjectivity of Galois representations of products of elliptic curves over function fields
研究函数域上非等平凡、非同构椭圆曲线乘积的伽罗瓦表示的有效满射性，借助Griffon - Pazuki的界和Frey - Mazur猜想函数域类似物得出等距度界，运用相关技术证明了有效满射性结果。
An introduction to $p$-adic $L$-functions
本文以三种等价方式介绍p进L-函数与Iwasawa理论，并陈述Iwasawa主猜想及Vandiver素数情形下的证明。
Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p-adic L-functions for the symmetric powers of f, thus verifying general conjectures of Dabrowski and Panchishkin in this special case. We also construct their "mixed" plus and minus counterparts and prove an analogue of Pollack's decomposition of the admissible p-adic L-functions into mixed plus and minus p-adic L-functions. On the arithmetic side, we define corresponding mixed plus and minus Selmer groups. We unite the arithmetic with the analytic by firs…
Let $F$ be a global function field of characteristic $p>0$, $\mathcal F/F$ a Galois extension with $Gal(\tilde F/F)\simeq \mathbb{Z}_p^{\mathbb N}$ and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_l$ ($l$ any prime) as $L$ varies through the subextensions of $\mathcal F$ via appropriate versions of Mazur's Control Theorem. In the case $l=p$ we let $\mathcal F=\bigcup \mathcal F_d$ where $\mathcal F_d/F$ is a $\mathbb{Z}_p^d$-extension. With a mild hypothesis on $Sel_E(F)_p$ (essentially a consequence of the Birch and Swinnerton-Dyer conjecture) we pr…
We study the prime number race for elliptic curves over the function field of a proper, smooth and geometrically connected curve over a finite field. This constitutes a function field analogue of prior work by Mazur, Sarnak and the second author. In this geometric setting we can prove unconditional results whose counterparts in the number field case are conditional on a Riemann Hypothesis and a linear independence hypothesis on the zeros of the implied L-functions. Notably we show that in certain natural families of elliptic curves, the bias generically dissipates as the conductor grows. This …
Tamagawa number conjecture for CM modular forms and Rankin--Selberg convolutions
本文针对定义在数域上、具有虚二次域复乘法的椭圆曲线，证明了解析秩1时Birch–Swinnerton-Dyer公式的p部分（p>3且在该虚二次域中分裂），方法还可推广至CM阿贝尔簇、CM模形式及相关p逆定理的类比。
On Boston's Unramified Conjecture for $\GL_2$ and McLeman's $(3,3)$-Conjecture
本文证明 Boston 非分歧猜想在二维奇情形及 McLeman $(3,3)$-猜想的充分性，即虚二次域 $p$-类秩为二且 Galois 群具 Zassenhaus 型 $(3,3)$ 时，$p$-类域塔有限。
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