nLab higher local field

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)