Duke Mathematical Journal

Expander graphs, gonality, and variation of Galois representations

Jordan S. Ellenberg, Chris Hall, and Emmanuel Kowalski

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Abstract

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.

Article information

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

Dates
First available in Project Euclid: 4 May 2012

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1336142075

Digital Object Identifier
doi:10.1215/00127094-1593272

Zentralblatt MATH identifier
06047810

Mathematical Reviews number (MathSciNet)
MR2922374

Subjects
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

Citation

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


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