TY - JOUR
T1 - Character sums and abelian Ramanujan graphs
AU - Li, Wen Ch ing Winnie
AU - Feng, Keqin
N1 - Funding Information:
in part by NSA Grant MDA904-90-H-1021.
PY - 1992/6
Y1 - 1992/6
N2 - 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.
AB - 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.
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U2 - 10.1016/0022-314X(92)90120-E
DO - 10.1016/0022-314X(92)90120-E
M3 - Article
AN - SCOPUS:0000989231
SN - 0022-314X
VL - 41
SP - 199
EP - 217
JO - Journal of Number Theory
JF - Journal of Number Theory
IS - 2
ER -