On character sums with distances on the upper half plane over a finite field

Nicholas M. Katz, Igor E. Shparlinski*, Maosheng Xiong

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

For the finite field Fq of q elements (q odd) and a quadratic non-residue (that is, a non-square) α ∈ Fq, we define the distance functionδ (u + v sqrt(α), x + y sqrt(α)) = frac((u - x)2 - α (v - y)2, v y) on the upper half plane Hq = {x + y sqrt(α) | x ∈ Fq, y ∈ Fq *} ⊆ Fq2. For two sets E, F ⊂ Hq with # E = E, # F = F and a non-trivial additive character ψ on Fq, we give the following estimate| under(∑, w ∈ E, z ∈ F) ψ (δ (w, z)) | ≤ min {q + sqrt(2 q E), sqrt(3) q5 / 4} sqrt(E F), which (assuming that F ≥ E) is non-trivial if E F / q2 → ∞ as q → ∞.

Original languageEnglish
Pages (from-to)738-747
Number of pages10
JournalFinite Fields and their Applications
Volume15
Issue number6
DOIs
Publication statusPublished - Dec 2009

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