An excursion into the \(p\)-adic world (Q2918939)

This article provides readers with a brief introduction to topics covered in \textit{J.-P. Serre}'s book [Trees. Springer Monographs in Mathematics. Berlin: Springer. (2003; Zbl 1013.20001)]. It begins by explaining \(p\)-adic absolute values, \(p\)-adic lattices \(L\) in the plane, the action of \(\mathrm{GL}_2(\mathbb Q)\) on these lattices, and equivalence classes of lattices with respect homothety. By defining when two lattices are ``neighbors'' one is led to the definition of a graph whose vertices are (equivalence classes of) lattices, which are connected by an edge if they are neighbors of each other. The main result is that these graphs \(X(p)\) are trees, and that it is possible to study the group \(\mathrm{SL}_2(\mathbb Q)\) by investigating its action on this tree.NEWLINENEWLINEFor the entire collection see [Zbl 1222.00035].