The 2-adic eigencurve at the boundary of weight space

By Kevin Buzzard, L. Kilford. Compositio Mathematica, 2005

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Symmetric power functoriality for holomorphic modular forms, II (2020, methodology): “[LWX17, Conjecture 1.2]). We specialise to the case p= 2, in which case Buzzard–Kilford give a beautifully simple and explicit description of the geometry of Ep“close to the boundary of weight space” [BK05]. More precisely, E2 is supported above a single connected component W+ 2 ⊂W2, which we may id”
L -functions and Galois Representations: Eigenvarieties (2007, background): “See however [7] for explicit calculations in the case p = 2, and note that these calculations may well generalise to odd primes p.”
The eigencurve over the boundary of weight space (2014, methodology): “this point is T xk˜ = exp(kq) ˜(exp(kq)) 1. For r2(0;1), we use C>r to denote the preimage wt 1(W>r). The following is a folklore conjecture suggested by a computation of Buzzard and L. Kilford [BK05] which addresses a question asked by Coleman and Mazur [CM98]. Conjecture 1.2 (Coleman{Mazur{B”