Character sums and abelian Ramanujan graphs

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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, "Cohomologie étale (SGA 4 1 2)," Lecture Notes in Mathemmatics, Vol. 569, Springer-Verlag, Berlin/Heidelberg/New York) for the case n = 2: |Σx ∈ N2 χ(x) ψ(x)| ≤ 2q 1 2 for all nontrivial characters (χ, ψ) of N2 × F2. Here F2 is a quadratic extension of F and N2 consists of norm 1 (to F) elements in F2. 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.

Original languageEnglish (US)
Pages (from-to)199-217
Number of pages19
JournalJournal of Number Theory
Issue number2
StatePublished - Jun 1992

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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