Lifting Global Representations with Local Properties

Let k be a global field, with Galois group Gk and Weil group Wk relative to a choice of separable closure ks/k. Let Γ be either Gk or Wk, and H a linear algebraic group over F = C or Qp with p 6= char(k). Let ρ : Γ→ H(F ) be a (continuous) representation, always understood to be ramified at only finitely many places (as is automatic for commutative H, but not for H = GL2 with k = Q and F = Qp, even assuming semistability at p; see [KR, Thm. 25(b)].) Let f : H ′ → H be a quotient map between linear algebraic F -groups, with Z := ker f central of multiplicative type; e.g., an isogeny between connected H and H ′. Consider the problem of lifting a given ρ to a representation ρ′ : Γ→ H ′(F ). In the absence of local lifting obstructions, the global obstruction lies in a Tate–Shafarevich group that can be analyzed via Tate duality. But we want more, namely to preserve local properties of ρ at finitely many places and construct “optimal” (and explicit) counterexamples. For F = Qp and char(k) = 0, by [W2] the local lifting obstruction at v|p when requiring semistability at v is that ρ|Iv admits a Hodge–Tate lift; for finite Z this amounts to lifting 1-parameter subgroups (see Theorem 6.2 and Corollary 6.7). The study of finite Z rests on killing obstructions using central pushouts along an inclusion of Z into a torus, so the special case H = H ′ = Gm (forcing f(t) = t n for some nonzero n ∈ Z) controls the general case. Class field theory suggests that this local-global problem for characters (i.e., if χ : Wk → F× is an nth power on Wkv for all v then is χ an nth power?) is “dual” to the classical Grunwald–Wang problem: are the nth powers in k× characterized by local conditions away from a fixed finite set S of places of k? The Grunwald–Wang theorem (see Appendix A) characterizes the triples (k, S, n) for which there are counterexamples to the classical problem, and describes the counterexamples explicitly. There is a “universal formula” for counterexamples to the classical Grunwald–Wang problem, but apparently no “universal formula” for counterexamples to the local-global problem for characters, nor any direct link between counterexamples to the two problems; see Remark 2.2. However, the two problems have finite obstruction spaces that are dual to each other (thereby identifying the pairs (k, n) to consider for counterexamples to the latter problem):