A local-to-global principle for deformations of Galois representations
A local-to-global principle for deformations of Galois representations
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伽罗瓦表示变形的局部到全局原理
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发表时间：
1999
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通讯作者：
G. Böckle
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作者：
G. Böckle
摘要给定一个绝对不可约的伽罗瓦表示：GE → GLN（k），E是数域，k是特征l > 2的有限域，E的有限位置集Q包含l和∞以上的所有位置以及∞的所有分支，已经定义了许多表示这种表示的变形的严格等价类的函子，例如Mazur或Wiles在[15]或[26]中，与各种条件的行为的变形的地方在Q和条件的变形是unramified以外的Q。已知这些函子是可表示的。如上所述，我们的目标是提出一个相当普遍的一类满足局部变形条件的全局变形函子，并研究在什么条件下全局变形函子由局部变形函子决定。我们将给出精确的条件，在此条件下，Q中所有位置的局部函子都足以描述全局函子，首先是粗略的形式，然后是使用辅助素数的精细形式，如Taylor和Wiles在[24]中所做的那样。这有几个后果。最强的是，人们可以得出环理论的结果，普遍变形空间的马祖尔，如果一个使用的结果钻石和怀尔斯，比照。[11][26]如果一个人对当地的情况有很好的了解。此外，更容易理解在增加分歧的情况下会发生什么，正如波士顿和罗摩克里希纳在[6]和[20]，[21]中所做的那样。最后，我们将通过直接考虑某些亲-l伽罗瓦群的表示和重新考虑波士顿的素数-伴随原理来重新解释驯服表示情况下的结果，参见。[5]的文件。
Abstract Given an absolutely irreducible Galois representation : GE → GLN (k), E a number field, k a finite field of characteristic l > 2, and a finite set of places Q of E containing all places above l and ∞ and all where ∞ ramifies, there have been defined many functors representing strict equivalence classes of deformations of such a representation, e.g. by Mazur or Wiles in [15] or [26], with various conditions on the behaviour of the deformations at the places in Q and with the condition that the deformations are unramified outside Q. Those functors are known to be representable. For as above, our goal is to present a rather general class of global deformation functors that satisfy local deformation conditions and to investigate for those, under what conditions the global deformation functor is determined by the local deformation functors. We will give precise conditions under which the local functors for all places in Q are sufficient to describe the global functor, first in a coarse form, then in a refined form using auxiliary primes as done by Taylor and Wiles in [24]. This has several consequences. The strongest is that one can derive ring theoretic results for the universal deformation space by Mazur if one uses results of Diamond and Wiles, cf. [11] and [26], and if one has a good understanding of all local situations. Furthermore it is easier to understand what happens under increasing the ramification as done by Boston and Ramakrishna in [6] and [20], [21]. Finally we shall reinterpret the results in the case of a tame representation by directly considering presentations of certain pro-l Galois groups and revisiting the prime-to-adjoint principle of Boston, cf. [5].