# nLab higher dimensional arithmetic geometry

This entry is about traditional arithmetic geometry over higher local fields. For “E-∞ arithmetic geometry” see there.

# Contents

## Idea

the study of arithmetic geometry which concentrates on arithmetic schemes of higher dimensions and uses associated higher structures such as higher local fields, higher adelic structures, (commutative) higher class field theory and hence Milnor K-theory is often called higher arithmetic geometry.

The zeta functions in higher dimensional arithmetic geometry are called arithmetic zeta functions.

## References

• Alexander Beilinson, Residues and adeles, Funct. Anal. and Appl. 14(1980), 34-35

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

• Ivan Fesenko, Geometric adeles and the Riemann–Roch theorem for 1-cycles on surfaces (pdf)

• Ivan Fesenko, Measure, integration and elements of harmonic analysis on generalized loop spaces, Proc. of

St Petersburg Math. Soc. 12(2005), 179–199; English transl. in AMS Transl. Series 2, 219 (2006), 149–164 (pdf)

• Ivan Fesenko, G. Ricotta, M. Suzuki, Mean-periodicity and zeta functions, Ann. L’Inst. Fourier, 62(2012), 1819-1887 (pdf)

• K. Kato, Sh. Saito, Two-dimensional class field theory, Adv. Stud. Pure Math., vol. 2, 1983, 103-152

• K. Kato and Sh. Saito, Global class field theory of arithmetic schemes, Contemp. Math., vol. 55 part I, 1986, 255-331

• M. Morrow, Grothendieck’s trace map for arithmetic surfaces via residues and higher adeles, Algebra and Number Theory J., 6-7 (2012), 1503-1536

• A. Parshin, On the arithmetic of two dimensional schemes. I, Distributions and residues, Math. USSR Izv. 10(1976), 695-729

• A. Parshin, Abelian coverings of arithmetic schemes, Dokl. Akad. Nauk SSSR 243(1978), no. 4, 855-858

• M. Suzuki, Two dimensional adelic analysis and cuspidal automorphic representations of $GL(2)$, In Multiple Dirichlet Series, L-functions and automorphic forms, D. Bump, S. Friedberg, D. Goldfeld, eds., Progress in Math. 300, pp.339-361, Birkhaeuser 2012

• Amnon Yekutieli, An explicit construction of the Grothendieck residue complex, Asterisque 208, 1992.