Drawing Curves Over Number Fields

Neighboring powers

“…His approach is connected with the dessins d'enfants of Grothendieck, see e.g. [10]. Similarly one can show that there are at most two equivalence classes of solutions over Q for k = 5.…”

On the minimal number of critical points of smooth maps between closed manifolds

“…In [1] we treated separately the case N = S 2 , by making use of Belyi maps (see [26]), but the proof was rather sketchy. Morris Hirsch asked us for more details and later gave us the following simple proof using triangulations.…”

Some Elementary Ramanujan Graphs

“…The graph G(7) is the 1-skeleton of a beautiful triangulation of a surface of genus three investigated by Klein [4]. In general G(N ) is the 1-skeleton of a triangulation of the modular curve X(N), and as such has appeared in the work of Brooks [1,2] and (essentially) Shabat-Voevodsky [13].…”