non-commutative iwasawa theory

Question

In commutative Iwasawa theory, the main conjecture states that the p-adic L-function generates the characteristic ideal of an algebraic object. Non-commutative Iwasawa theory seems to mimik this - except that the existence of the object on the analytic side (to my knowledge) is still conjectural. My question is: why is it so hard to define a "non-commutative" p-adic L-function? And on a more technical note: In Coates-Fukaya-Kato-Sujatha-Venjakob, this conjectural function only exists for primes of good ordinary reduction. This seems to imply that in the excluded cases, things go horribly wrong. Does anybody know what or why?

Answer 1

First a short answer. I don't think one can say that the commutative analytic side is known, as you do. It is fully known only in the cyclotomic $\mathbb{Z}_{p}$ situation, assuming the ETNC and in the crystalline case (see below). The answer to your "why is it so hard" question is: because $p$-adic $L$-functions are linked with $B_dR$ and so require subtle knowledge of $p$-adic Hodge theory which is lacking in the non-commutative case (and also mostly in the general commutative case). The answer to your technical question is: because in the ordinary case, there is a concrete incarnation of $D_{dR}$ which allows for a definition of the required trivialization. Outside the ordinary case, no such concrete incarnation is known and so things are two orders of magnitude harder, already in the commutative case (again, see below for some more details).

In both the commutative and non-commutative situation, if you want to construct a $p$-adic $L$-function in the style of Kato, Perrin-Riou, Colmez (basically in the cyclotomic $\mathbb{Z}_{p}$-extension case) and Fukaya-Kato, CFKSV (in the non-commutative but still number field tower case) you need two things: one is an equivariant basis of the fundamental line (more or less the determinant of the motivic cohomology of your motive, or more concretely some sort of Euler system), the other is what is usually called a reciprocity law or Coleman map or $\epsilon$-isomorphism to transform this into a "function" or "measure" or "element in a localized $K_{1}$" (depending what you mean by $p$-adic $L$-function).

Slightly more precisely and in a more technical language, in order to formulate a conjectural setting allowing for the existence of a $p$-adic $L$-function, you need a trivialization of some complexes of Galois cohomology which is compatible with "evaluation at characters" (without this compatibility, there can be no interpolation property worth its name). This trivialization involves very subtle properties of the ring $B_{dR}$.

Now, in the cyclotomic $\mathbb{Z}_{p}$-extension case, such a trivialization is known to exist for any crystalline motive thanks to very deep results of many people, but most notably Perrin-Riou, Kato-Kurihara-Tsuji, Colmez and Benois-Berger. So in that case, if you $assume$ the Equivariant Tamagawa Number Conjecture, then the $p$-adic $L$-function is well-defined. This is only in this (very limited) sense that the analytic object you refer to in your question is well-defined.

In the general commutative case, very little is known, because one has to consider the $D_{dR}$ of Galois representations with coefficients in rings of large dimension and this is extremely hard though spectacular progresses are made each day in that respect. In the good ordinary tower of number fields case, be it commutative or non-commutative, the required trivialization is known because in that case there is a "concrete incarnation" of $D_{dR}$.

Finally, in the general non-commutative case, almost nothing is known because, as far as I know, the collective knowledge we have on the behaviour of $D_{dR}$ for non-commutative rings of large dimension is very far from what would be required only to formulate a question.

Answer 2

On a lower level and restricted to elliptic curves, there are at least two problems on the analytic side. First of all, if $\rho$ is an irreducible Artin representation of dimension $>1$ of a $p$-adic Lie group, then usually no one knows if $L(E,\rho,s)$ has an analytic continuation and so one can not yet interpolate the values at $s=1$. If one has the values, one needs congruences between them. So we need more complicated automorphic gadjets than nice modular forms of weight 2. Likewise, if we wish to construct it from an "Euler system", whatever that will be in the non-commutative setting.

For the supersingular case not even the algebraic side has ever been looked at to my knowledge. The usual conjectures like that the Selmer group should lie in $\mathfrak M_H$ are probably not true since the Tate-Shafarevich group is expected to grow a lot in these extensions. As for the cyclotomic extension, a single $p$-adic L-function will not be enough to discribe the situation here. First, we need a good cyclotomic $\pm$-theory for all number fields. (Or do we have that now ?)