Question

Let f be a modular form of weight k for $\Gamma_0(N)$. Let us assume that $p\not\vert$N. Then we can construct 2 p-adic L-functions corresponding to the 2 roots $\alpha$ and $\beta$ of the equation $x^2+a_px+p^{k-1}=0$ (assume we are not in critical slope case) and they are related by $L_{(p,\alpha)}(f,j)=\frac{(1-p^{j-1}/\alpha)(1-\beta/p^j)L_{(p,\beta)}(f,j)}{(1-p^{(j-1)}/\beta)(1-\alpha/p^j)}$. So one of them determines the other. So I have 2 questions: what happens in the critical slope case: is there some relation between p-adic L-function corresponding to the unit root and the critical slope one. The only reason one can even hope to get such a thing is because the roots have a relation between them and the usual functional equation for complex L-functions. My second question is how should one think of the Euler factors that appear?(maybe that should be my first question)

Answer 1

To answer the second question:

The interpolation factor is the determinant of $1-\varphi$ on $D_{\text{cris}}$ divided by its determinant on $D_{\text{cris}}^\ast(1)$. As to why this is what it should be, you can trace that back to Coates & Perrin-Riou's original paper (p-adic L-functions attached to motives over Q) where right above their definition of the interpolation factor (equation 4.11) they say "Following a suggestion of R. Greenberg". Deligne suggested an interpretation of the interpolation factor as modifying some $\varepsilon$-factors in a way completely analogous to the modifications of the Gamma-factors at $\infty$, this appears in the papers Coates wrote after the Coates–Perrin-Riou paper, of which Motivic p-adic L-functions in the Durham proceedings is the most definitive account. So that's a couple of ways of thinking about the Euler factors that appear in the interpolation property.

Update: @arijit: to answer the question you asked in the comments of David's answer, I think what you do is the following: In the ordinary case, the Selmer group one generally looks at is the one Greenberg defined, i.e. the "ordinary" Selmer group whose definition is based on the existence of the subrepresentation on which Frobenius acts via the unit root $\alpha$. I.e. there is a sub $W\subseteq V$ and the local condition of the Selmer group is

$$\ker\left(H^1(D_p,V)\rightarrow H^2(I_p,V/W)\right)$$

Accordingly, you should be looking at the $p$-adic $L$-function given by $\alpha$. Now, the filtered $(\varphi,N)$-module $D=D_{\text{cris}}(V)$, has two $\varphi$-stable subspaces: the one coming from the bona fide subrepresentation, namely $D_{\text{cris}}(W)$, and a non-admissible sub $D^\prime$ coming from the non-unit root. If you use this sub, you can define a local condition for a Selmer group completely analogously to the Greenberg defintion, but in the cohomology of $(\phi,\Gamma)$-modules. This should be related to the critical $p$-adic $L$-function. Basically, different "refinements" of the filtered $(\varphi,N)$-module (or "triangulations" of the $(\phi,\Gamma)$-module, or "$p$-stabilizations" of the automorphic representation) should correspond to different Selmer groups and different $p$-adic $L$-functions. Someone please correct me if I've simply made this up!

Answer 2

The formula you give relating $L_\alpha$ and $L_\beta$ is correct, but it is only valid for $1 \le j \le k-1$, so it only gives you finitely many values and hence it doesn't show that one L-function determines the other.

There is another similar formula relating the values of $L_\alpha$ and $L_\beta$ for twists of $f$ by p-power Dirichlet characters, though. Essentially these formulae are a by-product of the fact that the values of the p-adic L-functions are related to values of the corresponding complex L-function. In the critical slope case, the special values don't uniquely determine the L-function.

There is much more on these critical-slope L-functions in Pollack + Stevens' papers "Overconvergent modular symbols and p-adic L-functions", and "Critical slope p-adic L-functions", as well as other more recent preprints by Bellaiche and by myself and Zerbes.