### Abstract

For the finite field F_{q} of q elements (q odd) and a quadratic non-residue (that is, a non-square) α ∈ F_{q}, 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 H_{q} = {x + y sqrt(α) | x ∈ F_{q}, y ∈ F_{q}
^{*}} ⊆ F_{q2}. For two sets E, F ⊂ H_{q} with # E = E, # F = F and a non-trivial additive character ψ on F_{q}, we give the following estimate| under(∑, w ∈ E, z ∈ F) ψ (δ (w, z)) | ≤ min {q + sqrt(2 q E), sqrt(3) q^{5 / 4}} sqrt(E F), which (assuming that F ≥ E) is non-trivial if E F / q^{2} → ∞ as q → ∞.

Original language | English |
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Pages (from-to) | 738-747 |

Number of pages | 10 |

Journal | Finite Fields and their Applications |

Volume | 15 |

Issue number | 6 |

DOIs | |

Publication status | Published - Dec 2009 |

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## Cite this

Katz, N. M., Shparlinski, I. E., & Xiong, M. (2009). On character sums with distances on the upper half plane over a finite field.

*Finite Fields and their Applications*,*15*(6), 738-747. https://doi.org/10.1016/j.ffa.2009.08.002