## Tokyo Journal of Mathematics

### Integration on Valuation Fields over Local Fields

Matthew MORROW

#### Abstract

We present elements of a theory of translation-invariant integration, measure, and harmonic analysis on a valuation field with local field as residue field, extending work of I. Fesenko. Applications to zeta integrals for two-dimensional local fields are then considered.

#### Article information

Source
Tokyo J. Math., Volume 33, Number 1 (2010), 235-281.

Dates
First available in Project Euclid: 21 July 2010

https://projecteuclid.org/euclid.tjm/1279719589

Digital Object Identifier
doi:10.3836/tjm/1279719589

Mathematical Reviews number (MathSciNet)
MR2682892

Zentralblatt MATH identifier
1203.11081

#### Citation

MORROW, Matthew. Integration on Valuation Fields over Local Fields. Tokyo J. Math. 33 (2010), no. 1, 235--281. doi:10.3836/tjm/1279719589. https://projecteuclid.org/euclid.tjm/1279719589

#### References

• A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields, Amer. J. Math., 124 (2002), no. 5, 879–920.
• A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields II, Doc. Math. extra vol. Kazuya Kato's fiftieth birthday (2003), 5–72, (electronic).
• T. M. Apostol, Mathematical analysis, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., second ed., 1974.
• J. M. Borger, Conductors and the moduli of residual perfection, Math. Ann., 329 (2004), no. 1, 1–30.
• J. M. Borger, A monogenic Hasse-Arf theorem, J. Théor. Nombres Bordeaux, 16 (2004), no. 2, 373–375.
• D. Bump, Automorphic forms and representations, vol. 55 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1997.
• R. Cluckers, Igusa and Denef-Sperber conjectures on nondegenerate $p$- adic exponential sums, Duke Math. J., 141 (2008), no. 1, 205–216.
• R. Cluckers, Igusa's conjecture on exponential sums modulo $p$ and $p \sp 2$ and the motivic oscillation index, Int. Math. Res. Not. IMRN (2008), no. 4, Art.ID rnm118, 20, pp.
• J. Denef, The rationality of the Poincaré series associated to the $p$-adic points on a variety, Invent. Math., 77 (1984), no. 1, 1–23.
• I. Fesenko, A multidimensional local theory of class fields II (Russian), Algebra i Analiz, 3 (1991), no. 5, 168–189; translation in St. Petersburg Math. J. 3 (1992), no. 5, 1103–1126.
• I. Fesenko, Topological Milnor $K$-groups of higher local fields, in [Fesenk o1999?], 61–74.
• I. Fesenko, Analysis on arithmetic schemes I, Doc. Math. Kazuya Kato's fiftieth birthday (2003), 261–284 (electronic), available at, http://www.maths.nott.ac.uk/personal/ibf/.
• I. Fesenko, Measure, integration and elements of harmonic analysis on generalized loop spaces, in Proceedings of the St. Petersburg Mathematical Society Vol. XII, vol. 219 of Amer. Math. Soc. Transl. Ser. 2 (2006), Providence, RI, Amer. Math. Soc., 149–165, available at, http://www.maths.nott.ac.uk/personal/ibf/.
• I. Fesenko and M. Kurihara, Eds., Invitation to higher local fields, vol. 3 of Geometry & Topology Monographs, Geometry & Topology Publications, Coventry, 2000. Papers from the conference held in Münster, August 29–September 5, 1999.
• I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, vol. 121 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, second ed., 2002. With a foreword by I. R. Shafarevich.
• I. B. Fesenko, Adelic approach to the zeta function of arithmetic schemes in dimension two, Mosc. Math. J., 8 (2008), no. 2, 273–317, 399–400.
• I. B. Fesenko, Analysis on arithmetic schemes II, 2008, expanded version of [Fesenko2008?], available at, http://www.maths.nott.ac.uk/personal/ibf/.
• I. Gelfand, D. Raikov and G. Shilov, Commutative normed rings, Chelsea Publishing Company, Bronx, New York, 1964.
• R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, vol. 260 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1972.
• P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950.
• E. Hrushovski and D. Kazhdan, Integration in valued fields, in, Algebraic geometry and number theory, vol. 253 of Progr. Math., Birkh äuser Boston, Boston, MA, 2006, 261–405.
• E. Hrushovski and D. Kazhdan, The value ring of geometric motivic integration, and the Iwahori Hecke algebra of $\rm SL\sb 2$, with an appendix by Nir Avni, Geom. Funct. Anal., 17 (2008), no. 6, 1924–1967.
• K. Iwasawa, 'Letter to J. Dieudonné', in, Zeta functions in geometry (Tokyo, 1990), vol. 21 of Adv. Stud. Pure Math., Kinokuniya, Tokyo, 1992, 445–450.
• G. W. Johnson and M. L. Lapidus, The Feynman Integral and Feynman's Operational Calculus, Oxford University Press, 2000.
• K. Kato, Swan conductors for characters of degree one in the imperfect residue field case, in, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987), vol. 83 of Contemp. Math., 101–131, Amer. Math. Soc., Providence, RI, 1989.
• K. Kato, Class field theory, $D$-modules, and ramification on higher-dimensional schemes I, Amer. J. Math., 116 (1994), no. 4, 757–784.
• H. H. Kim and K.-H. Lee, Spherical Hecke algebras of $\rm SL\sb 2$ over $2$-dimensional local fields, Amer. J. Math., 126 (2004), 1381– 1399.
• H. H. Kim and K.-H. Lee, An invariant measure on $\rm GL\sb n$ over $2$-dimensional local fields, University of Nottingham Mathematics preprint series, (2005).
• C. J. Moreno, Advanced Analytic Number Theory: L-Functions, vol. 115 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2005.
• M. Morrow, Integration on product spaces and $\mbox{GL}_n$ of a valuation fields over a local field, Communications in Number Theory and Physics, vol. 2 (2008), 3, 563–592.
• M. Morrow, Fubini's theorem and non-linear changes of variables over a two-dimensional local field, arXiv:math.NT /0712.2177, 2008.
• M. Morrow, Investigations in two-dimensional arithmetic geometry, Ph.D. thesis, School of Mathematical Sciences, University of Nottingham, 2009, available at http://www.maths.nottingham.ac.uk/personal/ pmzmtm/.
• A. N. Parshin, Higher dimensional local fields and $L$-functions, in [Fesenk o1999?], 199–213, 2000.
• W. Rudin, Real and complex analysis, McGraw-Hill Book Co., New York, third ed., 1987.
• J. T. Tate, Fourier analysis in number fields, and Hecke's zeta- functions, in, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, 305–347.
• A. Weil, Fonction zêta et distributions, in Séminaire Bourbaki, Vol. 9 (1995), exp. no. 312, Soc. Math. France, Paris, 523–531.
• I. B. Zhukov, An approach to higher ramification theory, in [Fesenk o1999?], 143–150, (electronic).
• I. B. Zhukov, On ramification theory in the case of an imperfect residue field, Mat. Sb., 194 (2003), 3–30.