Counting the number of solutions to polynomial equations over finite fields, or equivalently computing the zeta function of algebraic varieties over finite fields, is a very central problem in computational number theory. For example, it is important for investigating famous conjectures like the (generalised) Birch and Swinnerton-Dyer, Sato-Tate, Lang-Trotter conjectures and parts of the Langlands program, but also has applications to public key cryptography and coding theory. Up until recently (apart from trivial cases) Magma could only really compute zeta functions of hyperelliptic curves. Recent work and code of mine (partly joint with Wouter Castryck) extends this to a much larger class of curves. My talk will give a survey of what Magma could do before it included my code, what it can do now and what it might be able to do in the future. The talk will not be very technical and should have something interesting for anyone interested in computer algebra or number theory.

The slides from this talk are on Jan's web page at https://perswww.kuleuven.be/~u0055310/magma.pdf