# Modular hyperbolas and bilinear forms of Kloosterman sums

## TL;DR

This paper explores incidence geometry of hyperbolas over finite fields and applies linear sum-product techniques to derive new bounds for bilinear Kloosterman sum forms.

## Contribution

It introduces a combinatorial approach to bounding bilinear Kloosterman sums using incidence results for hyperbolas in finite fields.

## Key findings

- Established incidence bounds for hyperbolas in finite fields
- Derived a new upper bound for bilinear Kloosterman sums
- Demonstrated the effectiveness of sum-product methods on hyperbolic curves

## Abstract

In this paper we study incidences for hyperbolas in $\mathbf{F}_p$ and show how linear sum--product methods work for such curves. As an application we give a purely combinatorial proof of a nontrivial upper bound for bilinear forms of Kloosterman sums.