Motives for modular forms
Motives for modular forms
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模块化形式的动机
DOI：
10.1007/bf01231194
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发表时间：
1990
影响因子：
3.1
通讯作者：
A. Scholl
中科院分区：
文献类型：
--
作者：
A. Scholl
设M是秩为n的数域F上的一个纯动机，其系数在T∧C；M可以被认为是光滑射影变量X在F上的上同调的一个直接因子，它被定义在T上的代数对应所切断。是朗兰兹提出，在两种语言之间应该有一种客观的对应关系。绝对不可约的动机和代数的动机。GLn（AF）的自同构表示Π，使得它们的不变量‘ s∈C中的L-函数’，L（M, s）和L（Π， s）匹配。直到20年前，大多数积极的结果都是关于Π 7→M = MΠ的方向，灵感来自德列涅在[4]的见解；但是在相反的“更硬”的方向上，在Wiles/TaylorWiles关于Q上半稳定椭圆曲线的模性和Breuil-Conrad-Diamond-Taylor关于Shimura-Taniyama-Weil猜想中，有一些惊人的发展→Π = ΠM（通常称为“伽罗瓦表示的模块化”）。最后，Kisin和Emerton（独立地）提出了GL2 / q的FontaineMazur猜想。人们甚至可以表述和证明“朗兰兹对应”的mod p和局部p-adic类似物的一些情况，而目前由Calegari和Geraghty领导的研究似乎非常有希望大大提高我们对M ‘ s和Π ’ s之间双反的许多新情况的理解。撇开这一惊人的发展不谈，本次研讨会的目标是揭示对应关系Π 7→M在n = 2的情况下，F是一个完全真实的场；更准确地说，F = Q的[4]和[8]，一般F的[1]。
Let M be a pure motive over a number field F of rank n with coefficients in T ⊂ C; M may be thought of as a direct factor of the cohomology of a smooth projective variety X over F cut out by an algebraic correspondence defined over T . It was Langlands who propounded that there should be a bijective correspondence between (resp. absoultey irreducible) motives M and algebraic (resp. cuspidal) automorphic representations Π of GLn(AF ), in such a way that their invariants, ‘L-functions in s ∈ C’, L(M, s) and L(Π, s) match up. Until twenty years ago, most of positive results had been about the direction Π 7→ M = MΠ inspired by Deligne’s insights in [4]; but there have been some spectacular development in the opposite ‘harder’ direction M 7→ Π = ΠM (commonly knowns as ‘modularity of Galois representations’) in Wiles/TaylorWiles on modularity of semi-stable elliptic curves over Q and Breuil-Conrad-Diamond-Taylor on the Shimura-Taniyama-Weil conjecture, culminating in Kisin and Emerton (independently) on the FontaineMazur conjecture for GL2 over Q. One can even formulate and prove some cases of mod p and local p-adic analogues of ‘Langlands correspondences’, while the research currently led by Calegari and Geraghty seems very promising to improve considerably our understanding of significantly many new cases of bijections between M ’s and Π’s. With that spectacular development aside, the goal of this seminar is to unravel the correspondence Π 7→M in the case n = 2 and F is a totally real field; more precisely, we shall read [4] and [8] for F = Q, and [1] for general F .