Proceedings of the Japan Academy, Series A, Mathematical Sciences

Remarks on zeta functions and K-theory over ${\mathbf F}_1$

Deitmar Anton

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We show that the notion of zeta functions over the field of one element $\F_1$, as given in special cases by Soulé, extends naturally to all $\F_1$-schemes as defined by the author in an earlier paper. We further give two constructions of K-theory for affine schemes or $\F_1$-rings, we show that these coincide in the group case, but not in general.

Article information

Proc. Japan Acad. Ser. A Math. Sci. Volume 82, Number 8 (2006), 141-146.

First available in Project Euclid: 6 November 2006

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Zentralblatt MATH identifier

Primary: 11G25: Varieties over finite and local fields [See also 14G15, 14G20]
Secondary: 11S40: Zeta functions and $L$-functions [See also 11M41, 19F27] 11S70: $K$-theory of local fields [See also 19Fxx] 14G15: Finite ground fields

Zeta function field of one element


Anton, Deitmar. Remarks on zeta functions and K-theory over ${\mathbf F}_1$. Proc. Japan Acad. Ser. A Math. Sci. 82 (2006), no. 8, 141--146. doi:10.3792/pjaa.82.141.

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