Duke Mathematical Journal

Expander graphs, gonality, and variation of Galois representations

Jordan S. Ellenberg, Chris Hall, and Emmanuel Kowalski

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We show that families of coverings of an algebraic curve where the associated Cayley–Schreier graphs form an expander family exhibit strong forms of geometric growth. We then give many arithmetic applications of this general result, obtained by combining it with finiteness statements for rational points of curves with large gonality. In particular, we derive a number of results concerning the variation of Galois representations in one-parameter families of abelian varieties.

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Duke Math. J. Volume 161, Number 7 (2012), 1233-1275.

First available: 4 May 2012

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Primary: 14G05: Rational points 14D10: Arithmetic ground fields (finite, local, global) 05C40: Connectivity 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Secondary: 14K15: Arithmetic ground fields [See also 11Dxx, 11Fxx, 11G10, 14Gxx] 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.) 35P15: Estimation of eigenvalues, upper and lower bounds


Ellenberg, Jordan S.; Hall, Chris; Kowalski, Emmanuel. Expander graphs, gonality, and variation of Galois representations. Duke Mathematical Journal 161 (2012), no. 7, 1233--1275. doi:10.1215/00127094-1593272. http://projecteuclid.org/euclid.dmj/1336142075.

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