K-theory of Minkowski question-mark function

TL;DR

The paper addresses realizing the Minkowski question-mark function $?$ within the $K$-theory of Cuntz-Pimsner algebras and cluster $C^*$-algebras. It constructs an injective map $i: K_0(\mathcal{O}_{A_{\infty}})\to K_0(\mathbb{A}_{\Theta})$ whose pointwise action coincides with $?$ and expresses this via Jacobi-Perron data $A_{\infty}$ derived from $\Theta$. This framework yields an explicit description of the Frobenius endomorphism at the infinite prime as $Fr_{\infty}^i\cong A_{\infty}^i$, linking local zeta-factor theory in the Serre–Deninger tradition to noncommutative invariants. The result forges a novel connection between continued-fraction dynamics, cluster/noncommutative geometry, and arithmetic geometry, providing a K-theoretic realization of $?$ and a concrete conduit to Frobenius actions at infinity.

Abstract

It is proved that the Minkowski question-mark function comes from the K-theory of Cuntz-Pimsner algebras. We apply this result to calculate the action of Frobenius endomorphism at the infinite prime. Such a problem was raised by Serre and Deninger in the theory of local factors of zeta functions of projective varieties.

Figures (2)

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Theorems & Definitions (20)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 2.1
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Example 3.1