The following is taken from notes distributed by Matthew Morrow at the Midwest Number Theory Conference for Graduate Students and Recent PhD’s 2010:

A two-dimensional local field F is a complete discrete valuation field whose residue field is a local field.

Some examples of two-dimensional local fields are

(i) For any local field $K$, consider $F=K((t))$. This is complete under the $t$-adic valuation and has residue field $K$.

(ii) Let $F$ be the $p$-adic completion of $\mathbb{Z}_p((t))$. $F$ is a characteristic zero complete DVR in which $p$ is prime, and its residue field is $\mathbb{F}_p((t))$.

Suppose that $X$ is a two-dimensional scheme of finite type and let $x\in X$ be a closed point and $y\subset X$ an irreducible curve containing $x$. Let $A=\mathcal{O}_{X,x}$ and let $\mathfrak{p}$ be the height one prime in $A$ which is the local equation of $y$ at $x$. Consider the following sequence of localisations and completions:

$A \rightsquigarrow \hat{A} \rightsquigarrow \hat{A}_{\mathfrak{p}\hat{A}} \rightsquigarrow \widehat{\hat{A}_{\mathfrak{p}\hat{A}}} \rightsquigarrow {\rm Frac}\widehat{\hat{A}_{\mathfrak{p}\hat{A}}}=: F_{x,y}$

It follows from the excellence of $A$ that $\mathfrak{p}\hat{A}$ is a radical ideal of the completion $\hat{A}$. We then localise and complete at $\mathfrak{p}\hat{A}$ and again use excellence to deduce that 0 is a radical ideal in the resulting ring, i.e., $\widehat{\hat{A}_{\mathfrak{p}\hat{A}}}$ is reduced. The total field of fractions $F_{x,y}$ is therefore isomorphic to a finite direct sum of fields, and each is a two-dimensional local field.

The two-dimensional adelic philosophy, originally due to A. Parshin, is that we should study $X$ via the family of two-dimensional local fields $(F_{x,y})_{x\in y\subset X}$.

As in dimension one, we can wrap together the local data using a ring of adeles.

$\mathbb{A}_X = \prod_{x\in y\subset X} ' F_{x,y},$

where the restricted product condition is more complicated. In fact, there are several different candidates for the rind of adeles, some better for geometry, others for arithmetic.

K. Kato, S. Saito, S. Bloch, and others contributed to the development of two-dimensional class field theory for $X$, describing the abelian extensions of $K(X)$ in terms of idele groups constructed from the Milnor $K$-theory of the fields $F_{x,y}$.

For $X$ a variety, properties of its zeta function follow from the theory of étale cohomology, but current techniques for treating arithmetic varieties usually only work in certain special cases. However, when X is (the spectrum of) the ring of integers of a number field $K$ then we recover the Dedekind zeta function of $K$, which is known to have meromorphic continuation and a functional equations, with the most insightful proof being that of Tate and Iwasawa.

With that in mind, the possibility of generalising Tate and Iwasawa’s techniques to higher dimensions has been dreamt of by several mathematicians over the years. The problem is that higher dimensional local fields are not locally compact in any reasonable topology, and therefore there is no existing theory of integration and harmonic analysis.

We can never hope to have a reasonable real-valued integration theory: two-dimensional local fields are just too large. A way around this problem is to add an extra degree of freedom and let our measure take values in $\mathbb{R}(T)$.