Iwasawa theory

Iwasawa worked with so-called $\backslash Z\_p$-extensions: infinite extensions of a number field $F$ with Galois group $\backslash Gamma$ isomorphic to the additive group of p-adic integers for some prime p. (These were called $\backslash Gamma$-extensions in early papers.{{cite web |last1=Greenberg |first1=Ralph |title=Memories of Professor Iwasawa |url=https://sites.math.washington.edu/~greenber/mem.html |access-date=25 September 2021}}) Every closed subgroup of $\backslash Gamma$ is of the form $\backslash Gamma^\{p^n\},$ so by Galois theory, a $\backslash Z\_p$-extension $F\_\backslash infty/F$ is the same thing as a tower of fields

:$F=F\_0\; \backslash subset\; F\_1\; \backslash subset\; F\_2\; \backslash subset\; \backslash cdots\; \backslash subset\; F\_\backslash infty$

such that $\backslash operatorname\{Gal\}(F\_n/F)\backslash cong\; \backslash Z/p^n\backslash Z.$ Iwasawa studied classical Galois modules over $F\_n$ by asking questions about the structure of modules over $F\_\backslash infty.$

More generally, Iwasawa theory asks questions about the structure of Galois modules over extensions with Galois group a p-adic Lie group.

Let $p$ be a prime number and let $K=\backslash Q(\backslash mu\_p)$ be the field generated over $\backslash Q$ by the $p$th roots of unity. Iwasawa considered the following tower of number fields:

:$K\; =\; K\_\{0\}\; \backslash subset\; K\_\{1\}\; \backslash subset\; \backslash cdots\; \backslash subset\; K\_\{\backslash infty\},$

where $K\_n$ is the field generated by adjoining to $K$ the pⁿ⁺¹-st roots of unity and

:$K\_\backslash infty\; =\; \backslash bigcup\; K\_n.$

The fact that $\backslash operatorname\{Gal\}(K\_n/K)\backslash simeq\; \backslash Z/p^n\backslash Z$ implies, by infinite Galois theory, that $\backslash operatorname\{Gal\}(K\_\{\backslash infty\}/K)\; \backslash simeq\; \backslash varprojlim\_n\; \backslash Z/p^n\backslash Z\; =\; \backslash Z\_p.$ In order to get an interesting Galois module, Iwasawa took the ideal class group of $K\_n$, and let $I\_n$ be its p-torsion part. There are norm maps $I\_m\backslash to\; I\_n$ whenever $m>n$, and this gives us the data of an inverse system. If we set

:$I\; =\; \backslash varprojlim\; I\_n,$

then it is not hard to see from the inverse limit construction that $I$ is a module over $\backslash Z\_p.$ In fact, $I$ is a module over the Iwasawa algebra $\backslash Lambda=\backslash Z\_p\backslash Gamma$. This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this description it is possible to recover information about the p-part of the class group of $K.$

The motivation here is that the p-torsion in the ideal class group of $K$ had already been identified by Kummer as the main obstruction to the direct proof of Fermat's Last Theorem.

From this beginning in the 1950s, a substantial theory has been built up. A fundamental connection was noticed between the module theory, and the p-adic L-functions that were defined in the 1960s by Kubota and Leopoldt. The latter begin from the Bernoulli numbers, and use interpolation to define p-adic analogues of the Dirichlet L-functions. It became clear that the theory had prospects of moving ahead finally from Kummer's century-old results on regular primes.

Iwasawa formulated the main conjecture of Iwasawa theory as an assertion that two methods of defining p-adic L-functions (by module theory, by interpolation) should coincide, as far as that was well-defined. This was proved by {{harvtxt|Mazur|Wiles|1984}} for $\backslash Q$ and for all totally real number fields by {{harvtxt|Wiles|1990}}. These proofs were modeled upon Ken Ribet's proof of the converse to Herbrand's theorem (the so-called Herbrand–Ribet theorem).

Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler systems, described in {{harvtxt|Lang|1990}} and {{harvtxt|Washington|1997}}, and later proved other generalizations of the main conjecture for imaginary quadratic fields.

The Galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied. In each case, there is a main conjecture linking the tower to a p-adic L-function.

In 2002, Christopher Skinner and Eric Urban claimed a proof of a main conjecture for GL(2). In 2010, they posted a preprint {{harv|Skinner|Urban|2010}}.

• Ferrero–Washington theorem
• Tate module of a number field

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Citations

{{Reflist}}

• {{citation | last=de Shalit | first=Ehud | title=Iwasawa theory of elliptic curves with complex multiplication. p-adic L functions | series=Perspectives in Mathematics | volume=3 | location=Boston etc. | publisher=Academic Press | year=1987 | isbn=978-0-12-210255-4 | zbl=0674.12004 }}
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• Tadashi Ochiai: Iwasawa Theory and Its Perspective, Vol.2, Amer. Math. Soc., (Mathematical Surveys and Monographs V.280), ISBN 978-1-4704-5673-3 (2024).
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• {{Springer|title=Iwasawa theory|id=i/i130090}}

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