Number Theory Seminar: Jonathan Tilling Niemann

Speaker: Jonathan Tilling Niemann (DTU)

Title: Non-isomorphic maximal curves of the same genus.

Abstract: A curve over a finite field is called maximal if it attains the (upper) Hasse-Weil bound. Maximal curves have been studied extensively during the last decades, both for intrinsic mathematical reasons and because they are well suited for creating good error-correcting codes. More precisely, they are a key ingredient in so-called algebraic geometry (AG) codes, which generalize Reed-Solomon codes in a way that allows for arbitrarily long codes. Based on joint work with Peter Beelen, Maria Montanucci, and Luciane Quoos, we will discuss a particular family of maximal curves. Some of these curves appear extremely similar, in the sense that several of the invariants typically used to distinguish curves are identical, but they are, in fact, non-isomorphic. This indicates that even after finding a maximal curve, the problem of determining whether it is isomorphic to a known curve might be very difficult.

 
In this talk, we will give a brief introduction to maximal curves over finite fields, outline the way they are used for constructing codes with good parameters, and finally discuss the challenging task of finding and classifying such curves.