# The $a$-number of Certain Hyperelliptic Curves

## TL;DR

This paper derives formulas for the $a$-number of specific hyperelliptic curves, expanding understanding of their algebraic properties for infinitely many cases, including curves defined by $y^2= x^m+1$ and $y^2= x^m+x$.

## Contribution

It provides explicit formulas for the $a$-number of hyperelliptic curves of the form $y^2= x^m+1$ and $y^2= x^m+x$, for infinitely many values of $m$, which was previously unknown.

## Key findings

- Formulas for the $a$-number of $y^2= x^m+1$ curves.
- Formulas for the $a$-number of $y^2= x^m+x$ curves.
- Extension of $a$-number computations to infinitely many cases.

## Abstract

In this paper, we compute a formula for the $a$-number of certain hyperelliptic curves given by the equation $y^2= x^m+1$ for infinitely many values of $m$. The same question is studied for the curve corresponding to $y^2= x^m+x$.