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Zbl 1018.11059
Reid, Michael
The local Stark conjecture at a real place. (English)
[J] Compos. Math. 137, No.1, 75-90 (2003). ISSN 0010-437X; ISSN 1570-5846

Suppose $K/k$ is an Abelian extension of number fields with Galois group $G$ and $S$ a set of places of $k$ containing at least one place $v_0$ that splits totally in $K$ and satisfying some other conditions. Stark's conjecture postulates the existence of a ``Stark unit'' $\varepsilon$, more precisely: $\varepsilon$ is a unit outside $v_0$ and the logarithmic embeddings of $\varepsilon$ at the places over $v_0$ are predicted by first derivatives at zero of the L-functions $L_S(\chi,s)$. If $v_0$ is non-Archimedean, one may also say that the valuation vector of $\varepsilon$ at the places over $v_0$ is given by a Brumer-Stickelberger element. The local (or: refined) Stark conjecture is still more ambitious, making more precise predictions about the image of $\varepsilon$ in all $w$-adic completions of $K$, $w$ above $v_0$. Actually, $\varepsilon$ is, for this purpose, replaced by a so-called modified Stark unit $\varepsilon_{\bold q}$ for suitable primes $\bold q$ of $k$, and the $w$-adic information is encoded as information on the image of $\varepsilon_{\bold q}$ under the local Artin map at $v_0$ for a (variable) Abelian extension $L/k$ that contains $K$. \par This refined conjecture goes back to Gross. The reader should note that there are two different but of course related formulations in the papers J. Fac. Sci., Univ. Tokyo, Sect. I A 28, 979-994 (1981; Zbl 0507.12010) and ibid. 35, 177-197 (1988; Zbl 0681.12005), and it is the second formulation (the one that does not use derivatives of $p$-adic L-functions) that is used in the paper under review. \par For the base field $k=\bbfQ$, the refined conjecture is known to hold, as shown by Gross, working with the first formulation and using the Gross-Koblitz relation together with the Ferrero-Greenberg formula. \par Very interestingly, the refined conjecture makes perfect sense also if $v_0$ is Archimedean, and it is this case that is treated in the paper. The author proves the following two nice results: (1) If $L/k$ is CM and $K$ is the maximal real subfield, then the local Stark conjecture for the setup $L/K/k$ is true for every Archimedean $v_0$. (2) If $k$ is totally real and $L$ is Abelian over $k$ (note that we no longer require it to be CM), and the complex conjugations in Gal$(L/k)$ are ``not in general position'' (see the paper for details), and if $v$ is an Archimedean place of $k$ that becomes complex in $L$, then the validity of the local Stark conjecture at $v$ implies that the Stark unit for $K/k$, suitably chosen, is a square. This latter result generalizes work of Dummit and Hayes. \par The main tool is afforded by the 2-adic congruences on values of L-functions, due to Deligne and Ribet. It is remarkable and nice to watch how closely they fit the situation in the paper. At the end of the paper, a numerical example is discussed, in the situation of the second result; the (hypothetical) Stark unit turns out to be a square as expected. In fact the L-functions are evaluated to 70 decimal places, and so the unit that was found to satisfy these L-data is highly likely to really be the Stark unit.
[Cornelius Greither (Neubiberg)]

MSC 2000:: *11R42 Zeta functions and L-functions of global number fields
11R20 Other abelian and metabelian extensions
11S31 Class field theory for local fields

Keywords: 2-adic congruences; refined Stark conjecture; L-functions

Citations: Zbl 0507.12010; Zbl 0681.12005

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