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Not long back I asked a question about the existence of p-adic L-functions for number fields that are not totally real; and I was told that when the number field concerned has a nontrivial totally real or CM subfield, then there is a construction due to various people including Coates-Sinnott and Katz.

But my favourite number field at the moment is K = $\mathbb{Q}(\sqrt[3]{2})$, and sadly K contains no totally real or CM subfield, so for trivial reasons $L(n, \chi) = 0$ for every Groessencharacter $\chi$ of $K$ and every $n \le 0$. So in this case the above constructions just give zero. When I learnt this, I thought "that can't be the whole story, what about higher derivatives at 0"? Asking around, I was told about Stark's conjectures, which apparently predict that the leading term at $s = 0$ of the L-function of any GC of K should be the product of an explicit transcendental regulator and an algebraic number (which, if I've understood this right, should lie in the field $\mathbb{Q}$(values of $\chi$).)

My question is this: assuming Stark's conjecture, can we construct a distribution on the Galois group of the maximal unramified-outside-p abelian extension of K whose evaluation at any locally constant character of this group gives the algebraic part of the leading term at 0 of the L-series of the corresponding Groessencharacter?

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Conjecturally, the answer is yes, but the amount of work required is not trivial at all. The general set-up is roughly as follows: the special values of $L$-functions (in your case, for Tate motives) are predicted by the Tamagawa Number Conjecture, and by the Equivariant Tamagawa Number Conjecture (ETNC) when one wishes to incorporate the action of some Galois group (as you do). The ETNC links the leading term at 0 of the $L$-function to the determinants of some cohomological complexes. However, in order to do this coherently for any locally constant character (again, as you want to), one needs rather strong hypotheses on the complexes involved: namely, they need to be semisimple at $\rho$ if one wishes to interpolate at $\rho$. Here, the bad news start: showing that a complex is semisimple at $\rho$ amounts to Leopoldt's conjecture for Tate motives so is in general very hard. But things should be fine in your favourite case. In this way, you can construct (leading terms of) $p$-adic $L$-functions for Tate motives.

If you are ready to admit all conjectures, all of this has been known for quite some time, see for instance B.Perrin-Riou Fonctions $L$ $p$-adiques des représentations $p$-adiques section 4.2 Conjecture CP(M). For a more recent and more flexible formulation, I recommend D.Burns and O.Venjakob On the leading terms of Zeta Isomorphism... section 3.2.2 and 3.2.3.

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Thanks, that's a really superb answer! I was sure that this had to be known, but I didn't know where to look in the literature. I will certainly take a look at the references you give. –  David Loeffler Mar 29 '10 at 13:09
    
Thanks for the nice words, but I think you are giving too much credit. The crucial point here is "assuming all conjectures". Unless you can prove Stark's conjecture for $K$ (which I hear is accessible, if not known), all the above is pretty speculative. –  Olivier Mar 30 '10 at 8:21

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