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We present a method to control gonality of nonarchimedean curves based ongraph theory. Let k denote a complete nonarchimedean valued field. We firstprove a lower bound for the gonality of a curve over the algebraic closure of kin terms of the minimal degree of a class of graph maps, namely: one shouldminimize over all so-called finite harmonic graph morphisms to trees, thatoriginate from any refinement of the dual graph of the stable model of thecurve. Next comes our main result: we prove a lower bound for the degree ofsuch a graph morphism in terms of the first eigenvalue of the Laplacian andsome "volume" of the original graph; this can be seen as a substitute forgraphs of the Li-Yau inequality from differential geometry, although we alsoprove that the strict analogue of the original inequality fails for generalgraphs. Finally, we apply the results to give a lower bound for the gonality ofarbitrary Drinfeld modular curves over finite fields and for general congruencesubgroups Gamma of Gamma(1) that is linear in the index [Gamma(1):Gamma], witha constant that only depends on the residue field degree and the degree of thechosen "infinite" place. This is a function field analogue of a theorem ofAbramovich for classical modular curves. We present applications to uniformboundedness of torsion of rank two Drinfeld modules that improve upon existingresults, and to lower bounds on the modular degree of certain elliptic curvesover function fields that solve a problem of Papikian.

A combinatorial Li-Yau inequality and rational points on curves