Counting points on elliptic curves over finite fields

Elliptic curves play an important role in number theory and cryptography. This report explores essential aspects of elliptic curves, such as their group structure and their torsion subgroup and isogenies - with particular emphasis on the Frobenius map. Special focus is given to Hasse's bound and division polynomials - both are an essential foundation for the study of René Schoof's algorithm described in [Sch85]. This algorithm, published in 1985, allows the computation of the number of points on an elliptic curve defined over a finite field with a significant time saving to previous approaches. This work provides a detailed analysis of this algorithm: we expand key steps which were only briefly mentioned, and even correct minor mistakes in the original document. To enhance understanding, we complement our report with detailed examples and SageMath-generated illustrations for many of the concepts covered.