Hi all,
I am sorry to ask a stupid question but I am really confused right now. KitagawaMazur constructed a $2$variable padic Lfunction attached to Hida families of modular forms. For their construction to work they needed some nice conditions for the universal deformation rings like Gorenstein property. In recent paper by Mok using similar ideas of Panchishkin constructs a $2$variable padic Lfunction for Hilbert modular forms. The conditions in this paper is very mild. It just uses the nonvanishing result of Rohrlich. I think I can use similar arguments to construct this padic Lfunction for modular forms. How is padic Lfunction(MokPanchishkin) different/similar from/to KitagawaMazurs's?
I am sorry if this is a bad question but I am really confused right now. It may be entirely possible that I made a mistake in my computation. Any help will be greatly appreciated.
Thanks.



As explained in a comment, I disagree with Olivier's answer. Let me try another one. Let $C$ denote the eigencurve, and $x$ a point in $C$, say with integral weight $k$ (I don't require the point $x$ to be classical, and the weight $k$ can be negative if you like). Suppose our objective is to prove the following statement:
Then the question arises: 1) On what condition on $x$ can we prove such a statement ? I claim 1a) that the method of Mazur (I have never seen it !), used also by Kitagawa prove such a statement for a point $x$ which satisfies an algebraic property that I will call A(x) (roughly that the dual of the generalized eigenspace attached to x in the space of modular symbols is free on the Hecke algebra acting on it). 1b) That this property A(x) is implied by a geometric property G(x), which can be for example that the eigencurve is smooth at x (but contrarily to a urban legend, Gorenstein at x is not enough). 1c) And, that no other known methods has ever proved the statement above for any $x$ that did not satisfy A(x) ! My belief is that actually A(x) is a necessary condition, not a technical ones. Let me justify 1c) by a partial review of the literature: In Mazur and Kitagawa, A(x) is explicitly assumed. In GreenbergStevens, Panchiskin, and (I believe) Mok, x is a classical point of noncrtical slope (even slope 0 in the GreenbergStevens case) and then A(x) follows from the Hida/Coleman control's theorem which states that the generalized eigenspace considered in A(x) is in fact an eigenspace. In my 2011 inventiones paper, x is a critical classical point but I prove G(x), which implies A(x). So I don't think that Panchiskin (or Mok, or myself for that matter) does anything that really goes beyond MazurKitagawa, except of course that it does that on a Coleman's family (a.k.a my open set U in the eigencurve) while MazurKitagawa worked on Hida's families, the only one that existed then. PS: I have stated my opinions in strong and perhaps too precise terms, in order to be more easily corrected or refuted if needed. I may be missing something  or even everything. 


Assuming you are writing about Mok's Compositio 2009 article, the answer is easy: it's a question of quantifier ordering. There are two statements which you could call the interpolation property for a several variables $p$adic $L$function $L_{p}$ (which I assume to be living in $R[[X]]$ where $R$ is some ring and $X$ is the cyclotomic variable). If $x$ is a classical point of $R$, I denote by $L_{p}^{cyc}(V_{x},)$ the usual cyclotomic $p$adic $L$function of the Galois representation $V_{x}$. (1) For all classical points of $R$, there exists a period $\Omega_{x}$ such that for all character $\chi$, $L_{p}(x,\chi)=L_{p}(V_{x},\chi)/\Omega_{x}$. (2) There exists $\Omega$ in $R$ such that for all classical points $x$ of $R$ and for all character $\chi$, $L_{p}(x,\chi)=L_{p}(V_{x},\chi)/\Omega_{x}$. Type (1) results guarantee interpolation only locally at a classical point and thus typically require only the rather weak condition of nonvanishing of $L$values. On the other hand, they provide only local informations, so they are particularly suitable for problems which are local at a classical point: the main example being the problem of trivial zeroes. This is why type (1) is what GreenbergStevens and Mok construct. Type (2) results are much stronger and much more precise but they typically require $p$adic interpolation of comparison theorem between Betti and De Rham cohomology or De Rham and étale cohomology. Under current technology, the proofs of these theorems relies on techniques formally similar to the complete intersection and freeness criterion of TaylorWiles and Fujiwara so tend to require comparable hypotheses. It seems likely to me that Kisin's amelioration of these techniques could have a bearing on these questions but I am not sure how this could be done. You could also read the recent works of T.Ochiai (appeared in Documenta) and M.Dimitrov (to appear in American Journal of Math) on the topic: they contain very lucid explanations of the relevant points (both easily available online). 

