Dimension of the deformation space of ordinary representations in the cyclotomic limit
Ashay A. Burungale, Laurent Clozel, Barry Mazur
本文研究分圆极限下权二常表示形变的无障碍性，证明分圆塔上相应常表示形变环的维数可任意大。
在某些假设下，权为二的常表示形变在分圆极限中是无障碍的。我们证明了分圆塔上的这种常表示形变环可以具有任意大的维数。
The weight two ordinary deformations are unobstructed in the cyclotomic limit under certain assumptions. We show that such an ordinary deformation ring over the cyclotomic tower can have arbitrarily large dimension.
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In this paper, we investigate the $p$-rank of Jacobian of cyclotomic function field $K_m$. Our goal in this paper is to give the necessary and sufficient condition $m$ such that $K_m$ is ordinary.
The deformation theory of ordinary representations of the absolute Galois groups of totally real number fields (over a finite field $k$) has been studied for a long time, starting with the work of Hida, Mazur and Tilouine, and continued by Wiles and others. Hida has studied the behaviour of these deformations when one considers the $p$-cyclotomic tower of extensions of the field. In the limit, one obtains a deformation ring $R_\infty$ classifying the ordinary deformations of the (Galois group of) the $p$-cyclotomic extension. We show that if $R_\infty$ is Noetherian and certain adjoint $μ$-inv…
On the cyclotomic Iwasawa invariants of elliptic curves of rank one
本文针对定义在ℚ上的秩1椭圆曲线，提出显式数值准则，判定其在好素数处的岩泽不变量是否取最小值，涵盖超奇异与常权两种情形。
Laurent Berger attached a p-adic differential equation N_rig(M) with a Frobenius structure to an arbitrary de Rham (phi, Gamma)-module over a Robba ring. In this article, we compare the local epsilon conjecture for the cyclotomic deformation of M with that of N_rig(M). We first define an isomorphism between the fundamental lines of their cyclotomic deformations using the second author's results on the big exponential map. As a main result of the article, we show that this isomorphism enables us to reduce the local epsilon conjecture for the cyclotomic deformation of M to that of N_rig(M).
We establish a duality result proving the `functional equation' of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic $\mathbb{Z}_{p}$ extension of a totally real number field. Further, we use this result to establish a duality or algebraic `functional equation' for the `big' Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as an evidence t…
In this paper, we study the deformations of Kolyvagin systems that are known to exist in a wide variety of cases, by the work of B. Howard, B. Mazur, and K. Rubin for the residual Galois representations, along the cyclotomic Iwasawa algebra. We prove, under certain technical hypotheses, that a cyclotomic deformation of a Kolyvagin system exists. We also briefly discuss how our techniques could be extended to prove that one could deform Kolyvagin systems for other deformations as well. We discuss several applications of this result, particularly relation of these $Λ$-adic Kolyvagin systems to p…
We study potentially crystalline deformation rings for a residual, ordinary Galois representation $\overlineρ: G_{\mathbb{Q}_p}\rightarrow \mathrm{GL}_3(\mathbb{F}_p)$. We consider deformations with Hodge-Tate weights $(0,1,2)$ and inertial type chosen to contain exactly one Fontaine-Laffaille modular weight for $\overlineρ$. We show that, in this setting, the potentially crystalline deformation space is formally smooth over $\mathbb{Z}_p$ and any potentially crystalline lift is ordinary. The proof requires an understanding of the condition imposed by the monodromy operator on Breuil modules w…
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 adapt a technique of Kisin to construct and study crystalline deformation rings of $G_K$ for a finite extension $K/\mathbb{Q}_p$. This is done by considering a moduli space of Breuil--Kisin modules, satisfying an additional Galois condition, over the universal deformation ring. For $K$ unramified over $\mathbb{Q}_p$ and Hodge--Tate weights in $[0,p]$, we study the geometry of this space. As a consequence we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_p$, with Hodge--Tate weights in $[0,p]$, are potentiall…
Let K be an arbitrary number field, and let rho: Gal(Kbar/K) -> GL_2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of rho. When K is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho contains a Zariski dense set of points corresponding to "automorphic" Galois representations. We conjecture that if K is_not_ totally real, then this is never the case, except in three exceptional cases, corresponding to (1) "base change", (2) "CM" forms, and (3) "Even" …
In this paper, we study the Iwasawa theory of a motive whose Hodge-Tate weights are $0$ or $1$ (thence in practice, of a motive associated to an abelian variety) at a non-ordinary prime, over the cyclotomic tower of a number field that is either totally real or CM. In particular, under certain technical assumptions, we construct Sprung-type Coleman maps on the local Iwasawa cohomology groups and use them to define integral $p$-adic $L$-functions and (one unconditionally and other conjecturally) cotorsion Selmer groups. This allows us to reformulate Perrin-Riou's main conjecture in terms of the…
Multi-variable admissible distributions
本文研究多变量权重空间上可接受分布的理论，并应用该理论构造了两变量非普通p-进L函数，以插值Rankin-Selberg L函数的特殊值。
Let R^univ be the universal deformation ring of a residual representation of a local Galois group. Kisin showed that many loci in MaxSpec(R^univ[1/p]) of interest are Zariski closed, and gave a way to study the generic fiber of the corresponding quotient of R^univ. However, his method gives little information about the quotient ring before inverting p. We give a method for studying this quotient in certain cases, and carry it out in the simplest non-trivial case. Precisely, suppose that V_0 is the trivial two dimensional representation and let R be the unique Z_p-flat and reduced quotient of R…
Selmer groups of symmetric powers of ordinary modular Galois representations
本文研究 Hida 族 Galois 表示偶对称幂的分圆 Selmer 群，在 $n\leq4$ 时将其特征幂级数联系于同余理想，并用 $\mathcal{L}$-不变量描述平凡零点首项。
In a recent work of Galatius and Venkatesh, the authors showed the importance of studying simplicial generalizations of Galois deformation functors. They established a precise link between the simplicial universal deformation ring $R$ prorepresenting such a deformation problem (with local conditions) and a derived Hecke algebra. Here we focus on the algebraic part of their study which we complete in two directions. First, we introduce the notion of simplicial pseudo-characters and prove relations between the (derived) deformation functors of simplicial pseudo-characters and that of simplicial …
Variation of Iwasawa Invariants for Ordinary Representations
本文扩展Greenberg的岩泽不变量研究框架，比较普通p进表示的Selmer群在Greenberg邻域内ℤₚ扩张上的岩泽不变量，为精细Selmer群建立类似结果，并为分圆ℤₚ扩张邻域内相关特征理想与p进L函数的联系提供证据。
In this article, we present the first half of our project on the Iwasawa theory of higher rank Galois deformations over deformations rings of arbitrary dimension. We develop a theory of Coleman maps for a very general class of coefficient rings, devise a dimension reduction procedure for locally restricted Euler systems and finally, put these into use in order to prove a divisibility in a 3-variable main conjecture for nearly ordinary families of Rankin-Selberg convolutions.
A characterization of ordinary modular eigenforms with CM
本文证明具有复乘的 $p$-普通模特征形式可由其模 $p^m$ 约化与 $p$-普通 CM 伴随形式模 $p^m$ 的表示同构来刻画。
We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\barσ$ with $\barσ^{\rm ss}=χ^{k-2} \oplus ρ\oplus χ^{k-1}$, where $χ$ is the mod $p$ cyclotomic character and $ρ$ is an absolutely irreducible reduction of the Galois representation $ρ_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f(\mathbf{Q}, ρ_f(1-k)\otimes \mathbf{Q}_p/\mathbf{Z}_p)$ and $H^1_f(\mathbf{Q}, ρ(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\barσ$ t…
Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\…
Ochiai has previously proved that the Beilinson-Kato Euler systems for modular forms interpolate in nearly-ordinary $p$-adic families (Howard has obtained a similar result for Heegner points), based on which he was able to prove a half of the two-variable main conjectures. The principal goal of this article is to generalize Ochiai's work in the level of Kolyvagin systems so as to prove that Kolyvagin systems associated to Beilinson-Kato elements interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus) and use what we call universal Kolyvagin systems to attemp…
Towards a finite-slope universal Rankin-Selberg p-adic L-function
本文提出在通用变形家族中研究有限斜率兰利-塞尔伯格 $p$-adic $L$-函数的框架，并探讨了残余表示与变形家族之间的关系。
We study the behavior of Iwasawa invariants among ordinary deformations of a fixed residual Galois representation taking values in a reductive algebraic group G. In particular, under the assumption that these Selmer groups are cotorsion modules over the Iwasawa algebra, we show that the vanishing of the mu invariant is independent of the deformation, while the lambda invariant depends only on the ramification. This generalizes work of Greenberg-Vatsal and Emerton-Pollack and the author in the case G = GL(2).
On a parity result for the symmetric square of modular forms with congruent residual representations
本文证明了具有同余剩余Galois表示的两个模形式相关联的对称平方表示在分圆扩张上Selmer群λ-不变量的奇偶性结果，并处理了普通与非普通两种情况。
P-adic Rankin-Selberg L-functions in universal deformation families and functional equations
本研究构造了与普通（Hida）模形式族和任意泛形变模形式族乘积相关的p-adic Rankin-Selberg L函数，证明其插值经典临界值并推导函数方程，拓展了普通特征簇的研究维度。
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.
The purpose of this paper is to build on results in {\it{higher codimension Iwasawa theory}}. The setting of our results involves Galois representations arising as cyclotomic twist deformations associated to (i) the tensor product of two cuspidal Hida families $F$ and $G$, and (ii) the tensor product of three cuspidal Hida families $F$, $G$ and $H$. On the analytic side, we consider (i) a pair of $3$-variable Rankin-Selberg $p$-adic $L$-functions constructed by Hida and (ii) a balanced $4$-variable $p$-adic $L$-function (due to Hsieh and Yamana) and an unbalanced $4$-variable $p$-adic $L$-func…
P-adic L-functions in universal deformation families
本文构造了 GL(2) 通用形变空间上的 p-adic L-函数例子，并提出猜想认为其自然参数空间并非通常的特征簇，而是依赖于抛物子群选择的更大空间（即“大抛物特征簇”）。
Deformations and Lifts of Calabi-Yau Varieties in Characteristic $p$
本文用导出代数几何研究特征 $p$ 中 Calabi-Yau 簇的形变，证明 Bogomolov-Tian-Todorov 定理的混合特征类比，并给出普通 Calabi-Yau 簇到特征 $0$ 的典范提升，推广 Serre-Tate 定理。
Consider a formal (mixed-characteristic) deformation functor D of a representation of a finite group G as automorphisms of a power series ring k[[t]] over a perfect field k of positive characteristic. Assume that the action of G is weakly ramified, i.e., the second ramification group is trivial. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. We prove that the only such D that are not pro-representable occur if k has characteristic two and G is o…
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