# A proof of the Hasse-Weil inequality for genus 2 \`a la Manin

## TL;DR

This paper proves the Hasse-Weil inequality for genus 2 curves of a specific form, employing elementary methods inspired by Manin's 1956 proof for genus 1, advancing understanding of algebraic curves.

## Contribution

It provides a new elementary proof of the Hasse-Weil inequality for genus 2 curves, extending Manin's approach from genus 1 to genus 2.

## Key findings

- Established the Hasse-Weil inequality for genus 2 curves of the form y^2=f(x)
- Demonstrated the effectiveness of elementary methods in higher genus cases
- Extended classical proof techniques to more complex algebraic curves

## Abstract

We prove the Hasse-Weil inequality for genus 2 curves given by an equation of the form y^2 = f(x) with f a polynomial of degree 5, using arguments that mimic the elementary proof of the genus 1 case obtained by Yu. I. Manin in 1956.