The cyclosyntomic regulator of a number field

TL;DR

This paper introduces a q-deformation of the p-adic regulator for number fields, called the cyclosyntomic regulator, utilizing prismatic cohomology and norm maps, with explicit computations at roots of unity.

Contribution

It constructs the cyclosyntomic regulator using refined norm maps in prismatic cohomology, extending classical regulators with a new q-deformation approach.

Findings

Constructed the cyclosyntomic regulator using prismatic cohomology.

Computed regulator values at units of the form 1 - ζ, with ζ a root of unity.

Introduced a refinement of Sulyma's norm maps interpolating classical and Frobenius maps.

Abstract

We construct a q-deformation of the p-adic regulator of a number field, called the cyclosyntomic regulator, building on the Habiro ring of Garoufalidis-Scholze-Wheeler-Zagier. The key new ingredient in our construction is a refinement of Sulyma's norm maps in prismatic cohomology, which interpolate between classical powers and Frobenius maps at various prime numbers p. Furthermore, we compute the values of the cyclosyntomic regulator at units of the form $1-\zeta$, where $\zeta$ is a root of unity.