# Contents

## Idea

The concept of higher local field is the generalization of that of local field to higher dimension. They are the subject of higher arithmetic geometry.

In particular two-dimensional local field are the analog of loop spaces in arithmetic geometry.

Their examples are $\mathbb{C}((t))$, $\mathbb{R}((t))$, $\mathbb{Q}_p((t))$, $\mathbb{F}_q((t))((u))$, etc. All these are naturally associated to a full flag of subschemes on algebraic and arithmetic surfaces.

These fields are not locally compact with respect to any reasonable topology on them. But they are topologically self-dual, similar to the classical local fields. For a collection of articles on all the main aspects of higher local fields known by 2000, see (Invitation to Higher Local Fields). For a survey of higher local fields and associated algebraic theories see (Morrow12).

One of the main new developments is the theory translation invariant measure and integration on higher local fields, a higher Haar measure, see (Fesenko03), (Fesenko06), (Morrow10). This measure takes values not in $\mathbb{C}$ but in $\mathbb{C}((X))$. Fourier transform with respect to this rigorously defined measure has many features similar to those of the Feynman path integral, the latter still does not have a rigorous math justification, see (Fesenko 06, sect. 18).

## References

• Wikipedia, Higher local field

• Ivan Fesenko, M. Kurihara (eds.), Invitation to Higher Local Fields, Geometry and Topology Monographs vol 3, Warwick 2000, 304 pp. (web)

• Ivan Fesenko, Analysis on arithmetic schemes. I,

Docum. Math. Extra Volume Kato (2003) 261-284; (web)

• Ivan Fesenko, Measure, integration and elements of harmonic analysis on generalized loop spaces,Proceed. St. Petersburg Math. Soc., Vostokov Festschrift, vol. 12 (2005), 179199;

English translation in AMS Transl. Series 2, vol. 219, 149–164, 2006 (pdf)

• M. Morrow, Integration on valuation fields over local fields, arXiv:0712.2172, Tokyo J. Math. 33(2010), 235-281

• M. Morrow, An introduction to higher local fields and adeles, arXiv:1204.0586

For more on applications of higher local fields to fundamental open problems in arithmetic geometry see also

• Ivan Fesenko, Adelic approch to the zeta function of arithmetic schemes in dimension two, Moscow Math. J. 8 (2008), 273–317 (pdf)

Last revised on August 18, 2014 at 03:30:44. See the history of this page for a list of all contributions to it.