Conversely, let (km Sample Clauses

Conversely, let (km m∈N be any sequence in Z and let (fm)m∈N be a sequence of p-adic modular forms fm of weight km and level Γ(N ) over Zp such that f1 /≡ 0 mod p and fm+1 ≡ fm mod pm in q-expansion. Then the sequence (km)n∈N converges in End(Z×p ) to a weight χ : Z×p → Z×p and there is a unique p-adic modular form f of weight χ and level Γ(N ) over Zp such that fm ≡ f mod pm in q-expansion. A central result in the theory of p-adic modular forms is that classical Hecke eigenforms live in p-adic families of overconvergent p-adic eigenforms, as was first observed by Xxxx for ordinary modular forms [28], and later extended by Xxxxxxx [19], whose work with Xxxxx culminated in the construction of the eigencurve [18]. This is a rigid space which roughly can η be described as follows: Let W = Spf(Zp[[Z ]])rig be the rigid space over Qp whose X-points for any affinoid rigid space X over Qp parametrise p-adic weights κ : Z×p → O(X)×. Then the eigencurve is a rigid space w : E → W parametrising and interpolating overconvergent p-adic eigenforms of finite slope and level Γ(N ), relatively over the p-adic weight space W . The geometry of the eigencurve encodes deep questions about congruences of eigenforms and Galois representations, and the eigencurve is therefore nowadays a central topic to the study of p-adic modular forms. Despite some recent progress, many of its geometric features are still mysterious. We refer to [33] for a brief survey. We can picture the weight space W as being a disjoint union of open unit discs: Each point κ ∈ W corresponding to a weight κ : Z×p → K× valued in some non-archimedean field extension K of Qp can be assigned an absolute value |κ| := maxa∈Zp× |κ(a) − 1| which we can use to define subdiscs and annuli in W of prescribed radii in each component of W . Near the centre of the weight space discs, the geometry of the eigencurve is expected to be very complicated [12]. Near their boundary, on the other hand, the eigencurve is thought to behave in a much more “regular” fashion. Motivated by computational results, Xxxxxxx made the following conjecture: