This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets the leading coefficients of Artin L-functions around $s=0$ as certain regulators. Let $F/K$ be a Galois extension of number fields and $\rho$ an Artin representation of the absolute Galois group of $K$ factoring through $F$. Let $L(F/K,\rho,s)$ be the Artin L-function attached to $\rho$ and let $c(\rho)$ be the leading coefficient of its Taylor expansion around $s=0$. Then Tate's formulation of Stark's conjecture does the intuitive thing and "picks out the $\rho$-isotypical bit" of the units $\mathcal{O}_F^\times$ and measures its volume in a suitable sense. But there are various problems with this intuition and it doesn't quite work the way one would like to (see below). The result is that one only gets a complex number Reg$(\rho)$ that is well defined up to a rational multiple. The conjecture is that this complex number accounts for the transcendental part of $c(\rho)$, more precisely that Reg$(\rho)/c(\rho)$ is algebraic and that Galois acts on this quotient through its action on the Artin characters.
There is a different circle of conjectures, due to Beilinson, Bloch, Burns, Flach, Fontaine, Kato and Perrin-Riou, subsumed under the bracket of "Tamagawa number conjectures", which gives exact predictions for special values of much more general motivic L-functions. Some of these conjectures work up to units, but some of them give actual numbers. The regulators involved are rather more complicated motivic things and, being a hands-on person, I find it difficult to discern the arithmetic in them (I know, this is embarassing to admit for someone who works on Galois module structures).
Question:
Does anyone know of an explication of the Tamagawa number conjectures in the setting of Artin L-functions, which would interpret the exact value of $c(\rho)$ (possibly up to units) in terms of the objects that one intuitively expects to enter into such a formulation? Special cases would already help.
In an ideal world, I would like a formulation, which makes it visible how the "$\rho$-isotypical parts" of the units and of the class numbers are picked out and how one measures the volume of a suitable lattice in the $\rho$-isotypical part of the units. Two extra demands that I would have of such a formulation are
- It should be easily visible that for trivial $\rho$, we recover the analytic class number formula over $K$ and
- that the product of these conjectures for all irreducible $\rho$ that factor through $F$, raised to the powers dim$(\rho)$, gives the analytic class number formula over $F$.
If you don't know of a reference, could you give a hint, how one goes about picking the right sublattice of the units?
Difficulties: I see two big difficulties in what I am asking for, so
if you can't answer the full question, maybe you can address the difficulties one by one or give me examples of special cases where one or the other has been solved?
Firstly, if $\rho$ is not defined over $\mathbb{Q}$, then it's not at all clear, how to define the right volume on the sublattice we pick out. My naive picture is that volumes are determinants of some non-degenerate pairing. If I restrict a non-degenerate pairing to the $\rho$-component of a lattice (I'm happy to base change the lattice to the ring of definition of $\rho$, even at the risk of losing some information) and $\rho$ is not self-dual, then the pairing will become degenerate, so the determinant that I would expect in the formulation of a precise Stark conjecture would be 0.
Secondly, even if all Artin representations are defined over $\mathbb{Q}$, e.g. is the Galois group is $C_2$, then although it is easy to say what we mean by the $1$-isotypical component and the $-1$-isotypical component of a lattice, the product of the volumes of these will almost never be the volume of the whole lattice, since their direct sum will have finite non-trivial index in the whole lattice.
Many thanks in advance!
