We will mainly interest with curves over finite fields which are supersingular. They have many applications in cryptography, coding theory and they appear on some number theoretic problems in finite fields.
A quadratic form over a finite field can be seen as a supersingular curve. These curves are in a special type of Artin-Schreier curves and they create Reed-Muller codes. Counting their number of rational points plays a huge role here.
Finding the number of rational points of a curve over a finite field is related to the roots of its L-polynomial. Moreover, roots of L-polynomial of a supersingular curve over finite fields are roots of unity. The reciprocal sum of these roots is in the quadratic field related to the characteristics of the finite field. Because of this reason, we will analyze the relation of quadratic and cyclotomic fields. This relation reduces to time-consuming to count the number of rational points.
We will also find the number of irreducible polynomials with prescribed coefficients by using supersingular curves. We will analyze the supersingular curves here in more detail.
