Special values of $p$-adic $L$-functions.

This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes.

My understanding is that nowadays there are conjectures which essentially predict (perhaps up to a sign) the value $L(M,n)$, where $M$ is a motive, $L$ is its $L$-function, and $n$ is an integer. My understanding of the history is that (excluding classical works on rank 1 motives from before the war) Deligne realised how to unify known results about $L$-functions of number fields and the B-SD conjecture, in his Corvallis paper, where he made predictions of $L(M,n)$, but only up to a rational number and only for $n$ critical. Then Beilinson extended these conjectures to predict $L(M,n)$ (or perhaps its leading term if there is a zero or pole) up to a rational number, and then Bloch and Kato went on to nail the rational number.

Nowadays though, many motives have $p$-adic $L$-functions (the toy examples being number fields and elliptic curves over $\mathbf{Q}$, perhaps the very examples that inspired Deligne), and these $p$-adic $L$-functions typically interpolate classical $L$-functions at critical values, but the relationship between the $p$-adic and classical $L$-function is (in my mind) a lot more tenuous away from these points (although I think I have seen some formulae for $p$-adic zeta functions at $s=0$ and $s=1$ that look similar to classical formulae related arithmetic invariants of the number field).

So of course, my question is: is there a conjecture predicting the value of $L_p(M,n)$, for $n$ an integer, and $L_p$ the $p$-adic $L$-function of a motive? Of course that question barely makes sense, so here's a more concrete one: can one make a conjecture saying what $\zeta_p(n)$ should be (perhaps up to an element of $\mathbf{Q}^\times$) for an integer $n\geq2$ and $\zeta_p(s)$ the $p$-adic $\zeta$-function? My understanding of Iwasawa theory is that it would only really tell you information about the places where $\zeta_p(s)$ vanishes, and not about actual values---Iwasawa theory is typically only concerned with the ideal generated by the function (as far as I know). Also, as far as I know, $p$-adic $L$-functions are not expected to have functional equations, so the fact that we understand $\zeta_p(s)$ for $s$ a negative integer does not, as far as I know, tell us anything about its values at positive integers.

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I'd like to write a better response, but I must be brief.

For now, let me offer some places to read. Long story short, it is predicted that there's a relationship between special values of $p$-adic $L$-functions and syntomic regulators (which are the analogue of Beilinson's regulators in the $p$-adic world).

1. The beautiful paper of Manfred Kolster and Thong Nguyen Quong Do is, I think, a very readable resource.

2. The best results I know in this direction are Besser's papers here and here, which use rigid syntomic cohomology.

3. Besser's overview talk at the conference in Loen (notes available here) was a real joy.

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Actually, p-adic L-functions are expected to satisfy functional equations compatible with the classical ones. For M an ordinary motive, Coates and Perrin-Riou conjectured the interpolation property at critical integers and the expected functional equation in some papers in the early nineties (see for example this). In particular, the Kubota-Leopoldt p-adic L-functions interpolate all critical values of the classical Dirichlet L-functions (up to a period and a multiple). For modular forms, Mazur-Tate-Teitelbaum, in their 1986 paper) prove a p-adic functional equation in section 17. In fact, the two-variable p-adic L-function of an ordinary family of modular forms satisfies a two-variable functional equation interpolating the one-variable functional equation at each weight (see for example Greenberg-Stevens' inventiones paper) (I'd post more mathscinet links but it appears to be down...).

As for the values of the p-adic L-function at non-critical integers, that's much more mysterious. Rubin has a computation outside of the critical points for a CM elliptic curve in section 3.3 of his paper in the "p-adic monodromy and BSD" proceedings. I think I've seen other cases, but generally it takes a lot of effort, I think.

(Also, regarding Iwasawa theory's concern with values of L-functions, it is true that the Main Conjecture is only an equality of ideals in some power series ring, but one can still hope to construct p-adic L-functions on the analytic side that do a nice job at interpolating, say up to a p-adic unit.)

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Here's a nice expository article by Colmez on Perrin-Riou's conjectures:

http://people.math.jussieu.fr/~colmez/851bourbaki.pdf

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Others have hinted at it, but let me emphasize the point. At least if you are happy to assume all conjectures (and perhaps that your motive has good reduction at $p$), the conjectural landscape for $p$-adic $L$-functions is as complete as that for usual $L$-functions. Namely: there is a conjectural description of the value of the cyclotomic $p$-adic $L$-function at any integer (in fact any character $\eta\chi_{cyc}^{s}$ with $\eta$ finite).

This can either be done in B.Perrin-Riou's style, see Fonctions $L$ $p$-adiques des représentations $p$-adiques, or in K.Kato's style, in which case it follows from the conjectures on special values of $L$-functions taking into account the action of a group algebra (the so-called equivariant conjectures). In fact, I exaggerate slightly here: at some special values, there could be an exceptional zero, in which case the leading term should incorporate an $\mathcal{L}$-invariant, and I don't think this has been (conjecturally) defined in all generality.

Also, as Rob H. wrote, $p$-adic $L$-function are indeed expected to satisfy a functional equation. This can be seen either from Perrin-Riou's conjectural construction from motivic elements, in which case the functional equation follows from the explicit reciprocity law of Perrin-Riou (and Colmez in the de Rham case) or via Iwasawa main conjectures, in which case it follows from duality results for cohomology complexes.

So everything you could wish for is conjectured. Not much, of course, is actually known.

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