# Periods for 2-variable p-adic L-functions

Hi all,
I am sorry to ask a stupid question but I am really confused right now. Kitagawa-Mazur constructed a $2$-variable p-adic L-function 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 p-adic L-function for Hilbert modular forms. The conditions in this paper is very mild. It just uses the non-vanishing result of Rohrlich. I think I can use similar arguments to construct this p-adic L-function for modular forms. How is p-adic L-function(Mok-Panchishkin) different/similar from/to Kitagawa-Mazurs'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.

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The method of Mazur and Kitagawa has indeed strong conditions to construct a meaningful 2-variables $p$-adic $L$-functions. I know next to nothing about the method of Panchiskin and its offspring, such as Mok's method. Hence I'd really like to know the answer to this question: +1. –  Joël Feb 6 '13 at 0:26
Dear Arijit, since you seem to have some ideas on this method, can you answer the following questions. 1) Does this methods construct a two-variable p-adic L-function over the full eigencurve, or just some part of it (and which one)? 2) How is this two-variable p-adic L-functions related to the one variable p-adic L-functions of modular forms of Visik and Amice-Velu (they are also re-defined in Mazur-Tate-Teitelbaum)? Let me precise question 2). One expects that the two-variable p-adic L-function, when the first variable is fixed equal to a point corresponding to a suitable classical... –  Joël Feb 6 '13 at 0:32
p-stabilized modular form $f$, is equal as a font ion of the second variable, and up to a scalar $c(f)$ depending to the choices of periods, to the Visik-Amice-Velu p-adic L-function of f. So if something like that is proved in the Panchiskin-Mok method, 2a) what are the classical point for which such a result is known ? 2b) what do we know about the scalar $c(f)$? can it be chosen a $p$-adic unit ? –  Joël Feb 6 '13 at 0:35
Dear Professor Bellaiche, Thanks a lot for your quick comments. I would like to point out that the paper of Panchishkin I mentioned constructs p-adic L-functions for Coleman families of positive slope and yes he does need the slope to be positive. The p-adic L-function is not constructed on the whole eigencurve. Fix a wt in your weight space and look at an affinoid neighborhood around that point. Assume that $L_{f(k')}(k'-1,\psi)\neq0$ for all $k'$ in that neighborhood and $\psi$ is a non-trivial Dirichlet character mod $p$ but this will follow from Rohrlich's theorem. –  Arijit Feb 6 '13 at 9:13
Oops I forgot to mention $k'$ is bigger than $2$slope $+2$. So in general this construction can not extend to the whole eigencurve. But I think one can patch them up in small neighborhoods to extend the function but definitely not to the whole eigencurve. As for the periods it is just coming from the Petersson with complex conjugate of $f$ and Atkin-Lehner involution of $f$. Sorry I am being sloppy but by $f$ I mean a suitable $p$-stabilization of $f$. And this method really uses that the slope is positive. In my next few comment I will discuss Mok's method later tonight. –  Arijit Feb 6 '13 at 9:25

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:

There exists an open neighborhood $U$ of $x$ in $C$ where classical points of non-critical slope are Zariski-dense and a two variable analytic function $L_p(u,s)$, where $u$ runs in $U$, and $s$ is the natural variable of a $p$-adic $L$-function, such that for every classical point $y$ in $U$ of non-critical slope corresponding to a classical $p$-stabilized modular form $f_y$, one has $$L_p(y,s) = c(y) L_p(f_y,s)$$ where $c(y)$ is a non-zero constant (but depending on $y$ !) Here $L_p(f_y,s)$ is the p-adic L-function of f_y defined in AMice-Velu and Visik (and also Mazur-Tate-Teiltelbaum).

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 Greenberg-Stevens, Panchiskin, and (I believe) Mok, x is a classical point of non-crtical slope (even slope 0 in the Greenberg-Stevens 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 Mazur-Kitagawa, except of course that it does that on a Coleman's family (a.k.a my open set U in the eigencurve) while Mazur-Kitagawa 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.

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Dear Professor Belaiche, I think the property G(x) will probably fail at points where the cuspidal component meets the eisentein component. But Panchishkin-Mok method should work around neighborhoods of such points whereas Kitagawa-Mazur method will fail. I may completely be wrong here and that is a source of my confusion. But what I do believe that in a neighborhood and in favorable conditions when both of them exist, the $2$ functions can be compared. –  Arijit Feb 6 '13 at 17:05
Also one quick remark: Do you also want $c(y)$ to be an analytic function of $y$? –  Arijit Feb 6 '13 at 17:09
Dear Joël, The point is that different hypotheses will yield different conclusions about the nature of $c$. Does $c$ vary analytically or not on $U$ for instance? You are right to say that $A(x)$ is a necessary condition, but as usual, the devil is in the details. You say that $A(x)$ means that the space of modular symbols is free on the Hecke algebra, but which Hecke algebra? If it is the Hecke algebra localized at $x$, then you get Greenberg/Stevens, Mok style results. If it is the Hecke algebra before localization, then you get Kitagawa style results (type (1) and (2) of my answer). –  Olivier Feb 6 '13 at 17:13
Of course, the freeness over the localized Hecke algebra is much easier to prove (the real freeness being false under suitable hypotheses) and this is why Greenberg/Stevens type results require much weaker hypotheses. –  Olivier Feb 6 '13 at 17:17
Thank you Professor Fouquet. These comments do clear things a lot. –  Arijit Feb 6 '13 at 17:25

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 non-vanishing 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 Greenberg-Stevens 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 Taylor-Wiles 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).

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Merci! I could have just asked you this yesterday. –  Arijit Feb 6 '13 at 9:27
@Olivier: I guess you do not want the small subscript $\chi$ in $\Omega_\chi$ in point (2). @Arijit: About the period comparison, you can also check the end of the introduction of T. Ochiai's paper in American J. of Math. –  Filippo Alberto Edoardo Feb 6 '13 at 12:00
@Filipo Which paper of Ochiai are you talking about? Is it the one in which he constructs an Euler system. It is indeed not clear(at least to me) how to compare the periods coming from Euler system and the periods in Kitagawa-Mazur. –  Arijit Feb 6 '13 at 12:28
@Arijit: this is not a trivial matter. The periods coming from Euler systems are, by their very construction, de Rham periods while Kitagawa-Mazur use modular symbols and their periods live in the Betti side of the story. I think Ochiai discusses this somewhere (the paper I was refering to before is the "Coleman map for nearly ordinary deformations") but I do not remember where... –  Filippo Alberto Edoardo Feb 6 '13 at 14:42
This is interesting, but I don't really understand how it answers the question (perhaps I am missing something). Arijit notes that the Panchiskin-Mok's method seems to have much milder hypotheses than the Kitagawa-Mazur method, and asks if there is a price to pay for the generality of Panchiskin-Moks by subtly weaker result. Then you say that there are two types of possible results (1) and (2), (2) being much stronger. Now Kitagawa-Mazur's method give results of type (1), don't they? (if not, then I am not understanding what you exactly mean by (2)). So how can that answer the question ? –  Joël Feb 6 '13 at 16:05