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Consider a classical cusp eigenform f=∑n=1∞an(f)qn of weight k≥2 for Γ0(N) with a Dirichlet character ψ mod N, and let Lf(s,χ)=∑n=1∞χ(n)an(f)n-s denote the L-function of f twisted with an arbitrary Dirichlet character χ. For a prime number p≥5, consider a family of cusp eigenforms f(k′) of weight k′, k′↦{f(k′)= ∑n=1∞an(f(k′))qn} containing f=f(k), such that the Fourier coefficients an(f(k′)) are given by certain p-adic analytic functions k′↦an(f(k′)). The purpose of this paper is to construct a two variable p-adic L function attached to Coleman’s family {f(k′)} of cusp eigenforms of a fixed positive slope σ=vp(αp)>0 where αp=αp(k′) is an eigenvalue (which depends on k′) of the Atkin operator U=Up. Our p-adic L-function interpolates the special values Lf(k′)(s,χ) at points (s,k′) with s=1,2,...,k′-1. We give a construction using the Rankin-Selberg method and the theory of p-adic integration on a profinite group Y with values in an affinoid K-algebra A, where K is a fixed finite extension of Qp. Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. In their turn, such measures come from Eisenstein distributions with values in certain Banach A-modules M†=M†(N;A) of families of overconvergent forms over A. [ABSTRACT FROM AUTHOR]

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p≥0, and 𝔤=Lie G. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let 𝒩=𝒩(𝔤) denote the nilpotent variety of 𝔤, and ℭnil(𝔤):={(x,y)∈𝒩×𝒩 | [x,y]=0}, the nilpotent commuting variety of 𝔤. Our main goal in this paper is to show that the variety ℭnil(𝔤) is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme ℋn⊂Hilbn(ℙ2) is irreducible over any algebraically closed field. [ABSTRACT FROM AUTHOR]