# On certain maximal hyperelliptic curves related to Chebyshev polynomials

## TL;DR

This paper investigates hyperelliptic curves derived from Chebyshev polynomials over finite fields, aiming to identify conditions under which these curves are maximal over specific finite field extensions.

## Contribution

It characterizes pairs (q, d) for which hyperelliptic curves related to Chebyshev polynomials are maximal over finite fields, extending the understanding of their arithmetic properties.

## Key findings

- Identifies (q, d) pairs for maximality of curves from Chebyshev polynomials.
- Analyzes variants involving shifted Chebyshev polynomials and quadratic factors.
- Provides criteria for maximality over finite fields for these classes of hyperelliptic curves.

## Abstract

We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs $(q,d)$ such that the hyperelliptic curve $\cC$ over a finite field $\FF_{q^2}$ corresponding to the equation $y^2 = \varphi_{d}(x)$ is maximal over the finite field $\FF_{q^2}$ of cardinality $q^2$. Here $\varphi_{d}(x)$ denotes the Chebyshev polynomial of degree $d$. The same question is studied for the curves corresponding to $y^2=(x\pm 2) \varphi_{d}(x)$, and also for $y^2=(x^2-4)\varphi_d(x)$.