Payman L. Kassaei
We prove an analogue of the main result of Buzzard and Taylor (Annals of Mathematics 149 (1999), 905-919) for totally real fields in which p is unramified. This can be used to prove certain cases of the strong Artin conjecture over totally real fields.
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We extend the modularity lifting result of the arXiv:1111.2804 to allow Galois representations with some ramification at p. We also prove modularity mod 2 and 5 of certain Galois representations. We use these results to prove many new cases of the strong Artin conjecture over totally real fields in which 5 is unramified. As an ingredient of the proof, we provide a general result on the automatic analytic continuation of overconvergent p-adic Hilbert modular forms of finite slope which substantially generalizes a similar result in arXiv:1111.2804.
Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.
In this paper, we prove a minimal modularity lifting theorem for Galois representations (conjecturally) associated to Siegel modular forms of genus two which are holomorphic limits of discrete series at infinity.
We prove a non-minimal modularity lifting theorem for ordinary Galois representations over imaginary quadratic fields, conditional on a local-global compatibility conjecture for ordinary torsion classes.
We prove a modularity lifting theorem for potentially Barostti-Tate representations over totally real fields, generalising recent results of Kisin. Unfortunately, there was an error in the original version of this paper, meaning that we can only obtain a slightly weaker result in the case where the representations are potentially ordinary; an erratum has been added explaining this error.
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.
We prove an integral R = T theorem for odd two dimensional p-adic representations of the absolute Galois group which are unramified at p, extending results of [CG] to the non-minimal case. We prove, for any p, the existence of Katz modular forms modulo p of weight one which do not lift to characteristic zero.
Commutative algebra inspired by modularity lifting
本文综述了受怀尔斯、泰勒、怀尔斯、迪亚蒙德、伦斯特等人椭圆曲线模性研究启发的最新交换代数结果。
We prove modularity lifting theorems for l-adic Galois representations of any dimension satisfying a unitary type condition and a Fontaine-Laffaille type condition at l. This extends the results of Clozel, Harris and Taylor, and the subsequent work by Taylor. The proof uses the Taylor-Wiles method, as improved by Diamond, Fujiwara, Kisin and Taylor, applied to Hecke algebras of unitary groups, and results of Labesse on stable base change and descent from unitary groups to GL_n.
The explicit Local Langlands Correspondence for $\mathrm{GSp}_4$, $\mathrm{Sp}_4$ and stability (with an application to Modularity Lifting)
本研究给出 $\mathrm{GSp}_4$ 与 $\mathrm{Sp}_4$ 显式局部朗兰兹对应的纯局部证明，通过 $L$-包稳定性等性质完成唯一刻画，并将其应用于模性提升问题。
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。
In a previous paper [CG], we showed how one could generalize Taylor-Wiles modularity lifting theorems [Wil95, TW95] to contexts beyond those in which the automorphic forms in question arose from the middle degree cohomology of Shimura varieties, in particular, to contexts in which the relevant automorphic forms contributed to cohomology in exactly two degrees. In this sequel, we extend our method to the general case in which Galois representations are expected to occur in cohomology, contingent on the (as yet unproven) existence of certain Galois representations with the expected properties. A…
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{…
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 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 extend the lifting methods of our previous paper to lift reducible odd representations $\barρ:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when combined with automorphy lifting results pioneered by Wiles in the number field case and the results on the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod $p$ Galois representations in both reducible and irreducible cases. In the reducible case this allows one to s…
Modularity and Heights of CM cycles on Kuga-Sato varieties
本文在任意级模曲线的Kuga-Sato簇上证明了CM闭链的高权一般Gross-Zagier公式，并证明CM闭链生成的Hecke模是半单模，为Beilinson-Bloch猜想提供证据。
This brief note only contains a modest contribution: we just fix some inaccuracies in the proof of the prime level weight 2 case of Serre's conjecture given in Khare's preprint "On Serre's modularity conjecture for 2-dimensional mod p representations of G_Q unramified outside p", for the case of trivial character. More precisely, the modularity lifting result needed at a crucial step is the one for the case of a deformation corresponding to a $p$-adic semistable (in the sense of Fontaine) Galois representation attached to a semistable abelian variety, but in Khare's preprint it is applied a le…
We study an analogue of Serre's modularity conjecture for projective representations $\overlineρ: \operatorname{Gal}(\overline{K} / K) \rightarrow \operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = \mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.
Modularity of trianguline Galois representations
本文通过伪刚性空间上的三角线模理论，证明了特定三角线伽罗瓦表示的模性提升定理，并构造了修补的四元数本征簇。
On the modularity of elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension of some real quadratic fields
本文遵循Thorne的思想，利用自守性提升定理与Iwasawa理论，在给定假设下证明了某些实二次域的$\mathbb{Z}_p$-扩张上椭圆曲线的模性。
We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p,p) type group schemes over the rational integers. We prove that a Galois representation as above of Artin cond…
Typos in the abstract have been corrected. Let $ρ_n$ be an ordinary weight two representation of absolute Galois group of the rationals to $GL_2(\mathcal O/π^n)$. Here $\mathcal O$ is a ramified DVR with uniformiser $π$. If $ρ_n$ satisfies mild hypotheses we lift it to a characteristic zero $\mathcal O$-valued geometric weight two representation. The earlier methods could handle only the unramified case. We show that the deformation ring of a residual representation arising from a newform can be arranged to be a prescribed DVR provided we choose a suitable auxiliary level. We extend earlier pr…
Resolutions of spaces of crystalline representations and modularity
本文研究晶体现象空间的解析，证明在Hodge-Tate权重间隙小于p时，部分解析的存在性，并在最小正则权重情况下展示解析的正常性，从而推导出自动性提升、塞雷猜想的权重部分及三维最小正则权重下的布里乌-梅兹ard猜想。
Let $p>2$ be prime, and let $F$ be a totally real field in which $p$ is unramified. We give a sufficient criterion for a mod $p$ Galois representation to arise from a mod $p$ Hilbert modular form of parallel weight one, by proving a "companion forms" theorem in this case. The techniques used are a mixture of modularity lifting theorems and geometric methods. As an application, we show that Serre's conjecture for $F$ implies Artin's conjecture for totally odd two-dimensional representations over $F$.
Automorphic side of Taylor-Wiles method for orthogonal and symplectic groups
本文针对全实域上的定特殊正交群或辛群，构造Taylor-Wiles素数并发展其自守侧，证明极小R=ℤ[T]定理，推导伴随表示的Bloch-Kato猜想，为相关Langlands纲领提供新证据。
We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod $p$ Hilbert modular forms, by making use of modularity lifting theorems and computations in $p$-adic Hodge theory.
We explain in this letter how using a recent Modularity Lifting Theorem proved by Lue Pan the proofs of Serre's Modularity Conjecture over $\mathbb{Q}$ given by Khare-Wintenberger and the author can be greatly simplified. The main difference with the previous proofs is that neither the process of "weight reduction" developed by Khare in his proof of the level 1 case nor the alternative method of "weight reduction via Galois conjugation" developed by the author are required: weight reduction can now be obtained just with two applications of Pan's result, together with a Lemma that guarantees th…
Under mild hypotheses, we prove that if F is a totally real field, k is the algebraic closure of the finite field with l elements and r : G_F --> GL_2(k) is irreducible and modular, then there is a finite solvable totally real extension F'/F such that r|_{G_F'} has a modular lift which is ordinary at each place dividing l. We deduce a similar result for r itself, under the assumption that at places v|l the representation r|_{G_F_v} is reducible. This allows us to deduce improvements to results in the literature on modularity lifting theorems for potentially Barsotti-Tate representations and th…
Constructing Arithmetic Siegel Modular Forms: Theta Lifting and Explicit Methods for Real Multiplication Abelian Surfaces
该研究提出显式可计算蓝图，利用酉对偶对$(\text{U}(2,2), \text{Sp}_4)$的西塔对应构造实乘法阿贝尔曲面的向量值西格尔模形式，通过局部施瓦茨函数、分歧素数测试向量构造及计算流程，为朗兰兹函子性实例提供路线图。
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.
Serre weight conjectures for $\mathrm{GSp}_4$
本研究证明了满足明确一般性条件的驯分歧$\u200b\mathrm{GSp}_4\u200b$值伽罗瓦表示的Serre猜想权部分，改进了此前需非明确一般性假设的结果，并推导了相应的模性提升定理。
Let $F$ be a CM number field. We prove modularity lifting theorems for regular $n$-dimensional Galois representations over $F$ without any self-duality condition. We deduce that all elliptic curves $E$ over $F$ are potentially modular, and furthermore satisfy the Sato--Tate conjecture. As an application of a different sort, we also prove the Ramanujan Conjecture for weight zero cuspidal automorphic representations for $\mathrm{GL}_2(\mathbf{A}_F)$.
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 say that a two dimensional p-adic Galois representation of a number field F is weight two if it is de Rham with Hodge-Tate weights 0 and -1 equally distributed at each place above p; for example, the Tate module of an elliptic curve has this property. The purpose of this paper is to establish a variety of results concerning odd weight two representations of totally real fields in as great a generality as we are able. Most of these results are improvements upon existing results. Three of our main results are as follows. (1) We prove a modularity lifting theorem for odd weight two representat…
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…
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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