Gebhard Böckle

A local-to-global principle for deformations of Galois representations

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].

Mod

As a sequel to our proof of the analog of Serres conjecture for function fields in Part I of this work, we study in this paper the deformation rings of

\ell

n

representations of arithmetic fundamental groups II: A conjecture of A. J. de Jong

-dimensional mod

Demuškin Groups with Group Actions and Applications to Deformations of Galois Representations

\ell

Cohomological Theory of Crystals over Function Fields and Applications

representations

Deformations of Galois Representations

\rho

Elliptic curves, Hilbert modular forms and Galois deformations

of the arithmetic fundamental group

Uniformizable families of

\pi_1(X)

t

where