Irene I. Bouw, Stefan Wewers, Leonardo Zapponi
We prove existence and nonexistence results for certain differential forms in positive characteristic, called {\em good deformation data}. Some of these results are obtained by reduction modulo $p$ of Belyi maps. As an application, we solve the local lifting problem for groups with Sylow $p$-subgroup of order $p$.
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Given a composition of Bely\uı maps $β\circ γ: X \rightarrow Z$, paths between edges of $β$ are extended to form loops, then lifted by $γ$. These liftings are then studied to understand how loops in $Z$ act on edges of $β\circ γ$, demonstrating the group operation in $\mathop{\mathrm{Mon}} β\circ γ\unlhd \mathop{\mathrm{Mon}} γ\wr \mathop{\mathrm{Mon}} β$. Abstracting away the specific Bely\uı map $γ$ and finding the image of $π_{1}(Z)$ in $π_{1}(Y) \wr \mathop{\mathrm{Mon}} β$ instead allows subsequently determining $\mathop{\mathrm{Mon}} β\circ γ$, for any $γ$, using only the monodromy repre…
We study irreducible odd mod $p$ Galois representations $\barρ \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\barρ$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.
Let $X$ and $Y$ be proper birational varieties, say with only rational double points over a perfect field $k$ of positive characteristic. If $X$ lifts to $W_n(k)$, is it true that $Y$ has the same lifting property? This is true for smooth surfaces, but we show by example that this is false for smooth varieties in higher dimension, and for surfaces with canonical singularities. We also answer a stacky analogue of this question: given a canonical surface $X$ with minimal resolution $Y$ and stacky resolution $\mathcal{X}$, we characterize when liftability of $Y$ is equivalent to that of $\mathcal…
We give a different approach to the well-known modularity lifting results of Wiles and Taylor. Instead of Taylor-Wiles systems we use a Galois cohomological discovery of R. Ramakrishna. This paper is 2 years older than math.NT/0210296 where another approach to modularity lifting results was presented.
We prove a modularity lifting theorem for minimally ramified deformations of two-dimensional odd Galois representations, over an arbitrary number field. The main ingredient is a generalization of the Taylor-Wiles method in which we patch complexes rather than modules.
In this paper we study deformations of mod $p$ Galois representations $τ$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $τ_1$ and $τ_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_Σ(F, {\rm Hom}(τ_2, τ_1)) \subset {\rm Ext}^1(τ_2, τ_1)$. We establish that there exists a basis $\mathcal{B}$ of $H^1_Σ(F, {\rm Hom}(τ_2, τ_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among other things that the elements of $\mathcal{…
Lifting $G$-Valued Galois Representations when $\ell \neq p$
本文研究ℓ≠p时任意约化群概形值局部伽罗瓦表示的通用提升空间，构造典范光滑分支推广极小分歧形变条件，引入弱约化群概形解决概形论问题，可用于整体伽罗瓦表示的几何提升。
Forcing ramification in the relative deformation method
本文将Khare-Ramakrishna的分歧强制论证推广到相对形变方法框架，证明可选有限素数集使几何特征0提升在其处分歧，且局部形式光滑性蕴含相关几何Bloch-Kato Selmer群零化。
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。
We prove the weight part of Serre's conjecture in generic situations for forms of $U(3)$ which are compact at infinity and split at places dividing $p$ as conjectured by Herzig. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights $(2,1,0)$ for $K/\mathbb{Q}_p$ unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.
Contents: Rational functions with given monodromy on generic curves (I. Bouw & S. Wewers); Can deformation rings of group representations not be local complete intersections? (T. Chinburg); Lifting an automorphism group to finite characteristic (G. Cornelissen); Flat connections and representations of the fundamental group in characteristic p>0 (C. Gasbarri) Questions on p-torsion of hyperelliptic curves (D. Glass & R. Pries); Automorphisms of curves and stable reduction (C. Lehr & M. Matignon); p-groups as automorphism groups of curves in characteristic p; Lifting Galois covers of smooth curv…
A minimal modularity lifting theorem for Siegel modular forms
该研究在亏格为二的西格尔模形式下，证明极小模性提升定理，剩余表示源于稳定吉田提升。基于\(R = \mathbb{T}\)定理，建立通用极小普通伽罗瓦变形环自由性及希达族唯一性。
Building on lifting results of Ramakrishna, Khare and Ramakrishna proved a purely Galois-theoretic level-raising theorem for two-dimensional odd representations of the Galois group of Q. In this paper, we generalize these techniques from type A1 to general (semi-)simple groups. We then strengthen our previous results on constructing geometric Galois representations with exceptional monodromy groups, achieving such constructions for almost all l, rather than a density-one set, and achieving greater flexibility in the Hodge numbers of the lifts; the latter improvement requires the new level-rais…
Let $K$ be a finite extension of $\mathbb{Q}_p$, and choose a uniformizer $π\in K$, and put $K_\infty:=K(\sqrt[p^\infty]π)$. We introduce a new technique using restriction to $\Gal(\ol K/K_\infty)$ to study flat deformation rings. We show the existence of deformation rings for $\Gal(\ol K/K_\infty)$-representations ``of height $\leqslant h$'' for any positive integer $h$, and we use them to give a variant of Kisin's proof of connected component analysis of a certain flat deformation rings, which was used to prove Kisin's modularity lifting theorem for potentially Barsotti-Tate representations.…
We show that if a differential equations $\mathscr{F}$ over a quasi-smooth Berkovich curve $X$ has a certain compatibility condition with respect to an automorphism $σ$ of $X$, and if the automorphism is sufficiently close to the identity, then $\mathscr{F}$ acquires a semi-linear action of $σ$ (i.e. lifting that on $X$). This generalizes the previous works of Yves André, Lucia Di Vizio, and the author about $p$-adic $q$-difference equations. We also obtain an application to Morita's $p$-adic Gamma function, and to related values of $p$-adic $L$-functions.
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 prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the automorphic forms in question contribute to a single degree of cohomology. In practice, this imposes several restrictions -- one must be in a Shimura variety setting and the automorphic forms must be of regular weight at infinity. In this paper, we essentially show how to remove these restrictions. Our most general result is a modularity lifting theorem which, on…
We prove a companion forms theorem for mod l Hilbert modular forms. This work generalises results of Gross and Coleman--Voloch for modular forms over Q, and gives a new proof of their results in many cases. The methods used are completely different to previous work in this area, and rely on modularity lifting theorems and the general theory of deformations of Galois representations.
We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lo…
We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\GL_2(\A_F)$, $F$ a totally real field, which are not of CM type. The argument is based on the potential automorphy techniques developed by Taylor et. al., but makes use of automorphy lifting theorems over ramified fields, together with a 'topological' argument with local deformation rings. In particular, we give a new proof of the conjecture for modular forms, which does not make use of poten…
If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if $\sD_X$ is the sheaf of differential operators on $X$ [EGAIV], it is well known that giving an action of $\sD_X$ on an $\sO_X$-module $\sE$ is equivalent to giving an infinite sequence of $\sO_X$-modules descending $\sE$ via the iterates of the Frobenius endomorphism of $X$. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic formal schemes such that $p$ belongs to an…
The commutative algebra of congruence ideals and applications to number theory
本文基于Wiles的工作，将数值准则推广到更高余数维度，引入同余模的概念，并探讨其在变形理论和Hecke代数中的应用，以及同余理想在模ularity lifting和因子化公式中的作用。
Emerton--Gee stacks, Serre weights, and Breuil--Mézard conjectures for $\mathrm{GSp}_4$
本研究构造了秩4辛射影平展 $(φ,Γ)$-模的模栈并研究其几何性质，证明了局部模型的单分支性，进而在一般性条件下建立了 $\rm{GSp}_4$ 情形下的Breuil–Mézard猜想、Serre猜想权部分及模性提升结果。
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