p-adic L-functions of modular forms: why the condition $v_p(\alpha)<k-1$?

Question

Let $f$ be a modular form (cuspidal, new, eigenform) of weight $k$ and level $N$. Let $p$ be a prime number not dividing $N$. In order to construct a $p$-adic $L$-function $L_p(f, s)$ interpolating the usual complex $L$-function, people choose a root $\alpha$ of the polynomial

$T^2-a_p X+p^{k-1}=0$

($a_p$ being the $p$-th Fourier coefficient) with the condition $v_p(\alpha)<k-1$.

Q1. Why is this necessary for the construction?

Q2. What happens when none of the roots satisfy this condtion? Is it still possible to define a p-adic L-function?

Answer 1

Let me answer your questions in the opposite order.

(2) This question is vacuous: it cannot happen that both roots have valuation $> k-1$, because the product of the roots is $p^{k-1}$. So either both roots have valuations $< k-1$ (which happens if $v_p(a_p) > 0$), or one has valuation exactly $k-1$ and the other has valuation 0 (which happens if $v_p(a_p) = 0$, the "ordinary" case).

(1) This is necessary for the usual (modular-symbol) construction, which can be summarized as follows: we want to find a p-adic analytic function interpolating a given set of values (the critical $L$-values of $f$ and its twists by $p$-power Dirichlet characters); if $v_p(\alpha) < k-1$, then this set of numbers "grows fairly slowly", and a general theorem in p-adic analysis shows that there is a unique "slowly-growing" analytic function interpolating them and we define this to be the $p$-adic $L$-function. When $v_p(\alpha) = k-1$, then the set of values we're interpolating grows too fast, so any function interpolating these values must grow so quickly that it is no longer uniquely determined by these values.

There are more sophisticated constructions (using Euler systems, eigenvarieties, overconvergent modular symbols, etc) which can be used to supply a $p$-adic $L$-function in the "critical-slope" case (when $v_p(a_p) = 0$ and $\alpha$ is the non-unit root).

For more on this see e.g. the papers "Critical slope p-adic L-functions" by Pollack and Stevens, "Wach modules and critical slope p-adic L-functions" by myself and Zerbes, "Critical p-adic L-functions" by Bellaïche.