I'm trying to learn about eigenvarieties and I am confused as to whether or not the spectral variety is affinoid.
It seems to me that being the zero locus of a power series should make you affinoid, but I cannot find this stated anywhere, so I'm a bit confused.
No they're not in general affinoid. The problem is that the zero locus of the power series is computed within a space which is almost never affinoid -- for example in the modular curve case the ambient space would be the product of an open disc (not affinoid) and affine 1-space minus the origin (not affinoid either). The open disc is parametrising weights and the affine space $U_p$ eigenvalues so there's not much you can do. Of course it is a union of affinoids -- but then again, so is every rigid space. An analogue would be that a closed subscheme of an affine scheme is affine, but a closed subscheme of a random scheme is just random and certainly may not be affine.
