# Counting points on hyperelliptic curves of type $y^2=x^{2g+1} + ax^{g+1} + bx$

## TL;DR

This paper develops efficient algorithms for counting points on hyperelliptic curves of specific types over finite fields, providing both computational methods and characteristic polynomial classifications for various genera.

## Contribution

It introduces new algorithms with sub-polynomial complexity for point counting on certain hyperelliptic curves and classifies Frobenius polynomials for genus 2 to 7.

## Key findings

- Algorithms for genus 3 and 4 with complexity $ ilde{O}( ext{log}^4 p)$ and $ ilde{O}( ext{log}^8 p)$
- Complete list of Frobenius characteristic polynomials for genus 2-7 curves
- Enhanced understanding of point counting on hyperelliptic curves of specific forms

## Abstract

In this work, we investigate hyperelliptic curves of type $C: y^2 = x^{2g+1} + ax^{g+1} + bx$ over the finite field $\mathbb{F}_q, q = p^n, p > 2$. For the case of $g = 3$ and $4$ we propose algorithms to compute the number of points on the Jacobian of the curve with complexity $\tilde{O}(\log^4{p})$ and $\tilde{O}(\log^8{p})$. For curves of genus $2-7$ we give a complete list of the characteristic polynomials of Frobenius endomorphism modulo $p$.