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Gunther Cornelissen


Speaker: Gunther Cornelissen
Title: Graphs, Spectra, and Diophantine Equations
Affiliation: Utrecht University
Date: Friday, March 28, 2014
Place and Time: Room 101, Love Building, 3:35-4:30 pm
Refreshments: Room 204, Love Building, 3:00 pm

Abstract. To solve diophantine equations, it helps to consider their geometry: in advantageous cases, geometric invariants determine properties of the solutions of the original equation. For example, if the diophantine equation defines a curve, a famous theorem of Faltings says that if the geometric (actually, even topological) invariant called "genus" is larger than 1, then the number of solutions of the diophantine equation is finite. In this talk, we are concerned with another invariant of a curve, namely, its "gonality": the minimal degree of a map from the curve to a line. Again, gonality relates to finiteness of the set of solutions of the diophantine equation, this time even varying over infinitely many different fields. Some curves can be degenerated into graphs. We establish a relation between the gonality of the curve and the "stable gonality" of the graph, where the latter is a new concept. Then, we prove that stable gonality of a graph is lower bounded by spectral invariants of the graph. This is a graph theoretical analogue of a theorem of Li and Yau from differential geometry, proven using some probability distributions on the graph vertices. The final outcome is that for some diophantine equations, combinatorial properties of their associated graphs give uniform finiteness statements for the solutions of the equation. We can illustrate this in concrete examples and apply the result to gonality of modular curves and to the degree of modular parametrizations of elliptic curves over function fields. (Joint work with Fumiharu Kato and Janne Kool.)