Gonality, a Li-Yau inequality for graphs and applications to Drinfeld modular curves

Back in 1996, Abramovich has proven a lower bound for the
gonality of a modular curve for congruence groups which is linear in
the index of the congruence group in the full modular group. He used
in his prove a Li-Yau inequality, which related the degree of a map
with the smallest nonzero eigenvalue of the Laplace operator and the
volume of the curve. In this talk I will present a similar result for
Drinfeld Modular curves. In particular, I will introduce a new notion
of gonality of a graph, and derive a lower bound for this gonality in
terms of the eigenvalues of the discrete Laplace operator on the dual
intersection graph and the number of vertices. If time permits, I will
discuss some more applications of this bound. This talk is based on
joined work with Gunther Cornelissen and Fumiharu Kato.