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Character sums and abelian Ramanujan graphs

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
41
Issue
2
Identifiers
DOI: 10.1016/0022-314x(92)90120-e

Abstract

Abstract Let F be a finite field of q elements. In this paper we obtain several estimates on character sums derived from the Riemann hypothesis for curves over F. In particular, we establish an estimate on twisted generalized Kloosterman sums as conjectured by P. Deligne (1977, “Cohomologieétale (SGA 4½),” Lecture Notes in Mathemmatics, Vol. 569, Springer-Verlag, Berlin/Heidelberg/New York) for the case n = 2: |Σ x ∈ N 2 χ( x) ψ( x)| ≤ 2 q 1/2 for all nontrivial characters (χ, ψ) of N 2 × F 2. Here F 2 is a quadratic extension of F and N 2 consists of norm 1 (to F) elements in F 2. We also present new constructions of Ramanujan graphs based on abelian groups. The character sum estimates are used to prove that these are indeed Ramanujan graphs.

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