Cohomology of Number Fields

Test elements in pro-p groups with applications in discrete groups

“…One of the main features of Demushkin groups is that every subgroup of finite index of a Demushkin group is also a Demushkin group and every subgroup of infinite index is free pro-p (cf. [23] or [17]). Since Demushkin groups are of cohomological dimension 2, using the multiplicativity of the Euler-Poincaré characteristic one can show that Theorem 6.7.…”

“…Demushkin groups play an important role in algebraic number theory. For instance, if k is a p-adic number field containing a primitive p-th root of unity and k(p) is the maximal p-extension of k, then Gal(k(p)/k) is a Demushkin group (see [17]). Demushkin groups also appear in topology since pro-p completions of orientable surface groups are Demushkin groups.…”

Arithmetic of Del Pezzo surfaces

“…Furthermore, if k is a global field, then H 3 Gal(k s /k), k s * = 0; this fact is due to Tatesee [NSW08,8.3.11(iv), 8.3.17].…”

Geometry of the eigencurve at critical Eisenstein series of weight 2

“…This formula (rather its analog for finite coefficients) is due to A.Wiles and can be found in [15,Theorem 8.7.9], the version used here can be found in [2,Proposition 2.7].…”