On the kernel of the reciprocity map of simple normal crossing varieties over finite fields

TL;DR

This paper investigates the kernel of the reciprocity map for simple normal crossing varieties over finite fields, providing an example where the map is not injective even after finite scalar extensions.

Contribution

It presents the first example of a simple normal crossing surface with a non-injective reciprocity map over finite fields, highlighting limitations of reciprocity in such varieties.

Findings

Constructed a simple normal crossing surface with non-injective reciprocity map

Showed the reciprocity map is not injective for any finite scalar extension

Provided insights into the behavior of reciprocity maps over finite fields

Abstract

In this paper, we study the kernel of the reciprocity map of certain simple normal crossing varieties over a finite field and give a example of a simple normal crossing surface whose reciprocity map is not injective for any finite scalar extension.