# Multiplicative zeta function and logarithmic point counting over finite fields

## TL;DR

This paper explores a multiplicative zeta function related to motives over finite fields, analyzing its properties and providing criteria for its analytic continuation, with applications to specific classes of varieties.

## Contribution

It introduces a multiplicative zeta function respecting tensor products of motives and establishes conditions for its analytic continuation, extending the understanding beyond classical Weil conjectures.

## Key findings

- Analytic continuation criteria for the multiplicative zeta function
- Examples include cellular varieties, abelian varieties, and certain curves
- No direct analogue of Weil conjectures for the multiplicative zeta function

## Abstract

The zeta function of a motive over a finite field is multiplicative with respect to the direct sum of motives. It has beautiful analytic properties, as were predicted by the Weil conjectures. There is also a multiplicative zeta function, which instead respects the tensor product of motives. There is no analogue of the Weil conjectures, and we give a sufficient criterion for an analytic continuation to exist. This happens, for example, for cellular varieties, abelian varieties, or genus g > 1 curves with a supersingular Jacobian.