Rationality of dlog \mathbbA¹-zeta functions

2023/11/12 by Xiaowen Hu, Hu, Xiaowen
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Coding theory and cryptography

paper · pdf · doi:10.48550/arxiv.2311.06793

Abstract

For every smooth proper scheme over a finite field \mathbbF_q, Bilu, Ho, Srinivasan, Vogt, and Wickelgren introduced the dlog zeta function with coefficients in the Grothendieck-Witt ring GW(\mathbbF_q), enriching the dlog of the classical Weil zeta function with coefficients in ℤ. They defined a notion of dlog rationality of such dlog zeta functions, which enriches the rationality of the Weil zeta function, and showed the dlog rationality for simple cellular schemes. In this paper, we show that for any smooth proper schemes over \mathbbF_q, the dlog zeta function is rational, but not necessarily dlog rational.

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