Even points on an algebraic curve
Let \(\mathbb{F}_q\) be a finite field of odd characteristic and \(K \) an algebraic function field of one variable over \(\mathbb{F}_q\). We denote by \(X\) the set of classes of valuations of \(K\). For a point \(p \in X\), we denote by \([p]\) its class in the Picard group Pic\(X\) of \(X\). The point \(p\) is called {\em even} if its class in the Picard group is 2-divisible. In this paper, it is proved that the graph of even points of \(K\) is always connected and its diameter is precisely 2. Furthermore, it is shown that the set of even points have positive density, and the quotient group Pic\(X/2\mathrm{ Pic }X \) is naturally isomorphic to the group of square classes of K that have even valuations everywhere on X. Moreover, an analogue of global square theorem that concerns points 2-divisible in the Picard group is presented.
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