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Each number field \(L\) comes with its cyclotomic \({\bf Z}_p\)-extension \(L_\infty\)
(with \(p\) any fixed odd prime number),
and to this one can associate a module \(X\) over the Iwasawa algebra. The so-called
characteristic series already gives a lot of information on \(X\) (for instance it gives
the \(\lambda\)-invariant). For a long time now the equivariant situation has been
studied. Here \(L/k\) is a CM Galois extension, abelian in this talk for simplicity,
and the characteristic series is replaced by a so-called Fitting ideal in the
Iwasawa algebra \(\Lambda={\bf Z}_p[[Gal(L_\infty/k)]]\). After tensoring
with \(\bf Q\) (a process in which information
is lost) and taking character parts, this gives back characteristic
series. In a way the description of the Fitting ideal falls into two parts:
the arithmetical part coming from \(p\)-adic L-functions, and the algebraic part,
which gives certain ``correcting" ideals by which one has to multiply
the principal ideals generated by series associated to \(p\)-adic L-functions,
in order to obtain the true Fitting ideal. In this talk we will try to show
how one can use the theory of Tate sequences to get a grip on the
algebraic part of the problem. The arithmetic part depends very much
on the specific extension \(L/k\), and it is one of our main findings that
the algebraic part only depends on the group \(Gal(L/k)\). At the time
being, we have to impose a restriction on
\(L/k\): we need that only
places above \(p\) are ramified. This is ongoing
joint work with Kurihara, which also generalizes preceding work of Kurihara.
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