Moscow Mathematical Journal

Volume 14, Issue 4, October–December 2014 pp. 773–806.

Recursive Towers of Curves over Finite Fields using Graph Theory

Authors: Emmanuel Hallouin and Marc Perret
Author institution:Université Toulouse 2, 5, allées Antonio Machado, 31058 Toulouse cedex, France

Summary:

We give a new way to study recursive towers of curves over a finite field, defined à la Elkies from a bottom curve X and a correspondence Γ on X. A close examination of singularities leads to a necessary condition for a tower to be asymptotically good. Then, spectral theory on a directed graph, Perron–Frobenius theory and considerations on the class of Γ in NS(X × X) lead to the fact that, under some mild assumption, a recursive tower can have in some sense only a restricted asymptotic quality. Results are applied to the Bezerra–Garcia–Stichtenoth tower along the paper for illustration.

2010 Mathematics Subject Classification. 11G20, 14G05, 14G15, 14H20, 5C38, 5C50

Keywords: Curves over a finite field, curves with many points, graphs, towers of function fields, zeta functions

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