# Graph of even points on an arithmetic curve

## TL;DR

This paper explores the structure of points on an arithmetic curve over a global function field, revealing a connected graph with dimension 2 and extending quadratic reciprocity and square theorems to this setting.

## Contribution

It introduces a novel graph structure for points with 2-divisible classes in the Picard group and generalizes classical reciprocity laws and square theorems to global function fields.

## Key findings

- The graph of 2-divisible points is connected with dimension 2.
- The incidence relation generalizes quadratic reciprocity law.
- An analog of the global square theorem is developed.

## Abstract

We show that the points of a global function field, whose classes are 2-divisible in the Picard group, form a connected graph, with the incidence relation generalizing the well known quadratic reciprocity law. We prove that for every global function field the dimension of this graph is precisely 2. In addition we develop an analog of global square theorem that concerns points 2-divisible in the Picard group.