Expander graphs, gonality, and variation of Galois representations

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 po...

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Bibliographic Details
Published in:Duke mathematical journal 2012-05, Vol.161 (7), p.1233-1275
Main Authors: Ellenberg, Jordan S., Hall, Chris, Kowalski, Emmanuel
Format: Article
Language:English
Subjects:
05C40
05C50
11Fxx
11G10
14D05
14D10
14G05
14Gxx
14K15
35P15
Arithmetic ground fields (finite
Arithmetic ground fields [See also 11Dxx
Connectivity
eigenvalues
Estimation of eigenvalues
etc.
global
Graphs and linear algebra (matrices
local
monodromy
Rational points
Structure of families (Picard-Lefschetz
upper and lower bounds
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Summary: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.
ISSN:0012-7094
1547-7398
DOI:10.1215/00127094-1593272