PhD ceremony Mr. M.A. Soomro: Algebraic curves over finite fields

PhD ceremony: Mr. M.A. Soomro, 9.00 uur, Academiegebouw, Broerstraat 5, Groningen

Dissertation: Algebraic curves over finite fields

Promotor(s): prof. J. Top

Faculty: Mathematics and Natural Sciences

Algebraic curves over finite field is a fascinating topic in number theory and in algebraic geometry. Moreover, it has applications in different fields such as cryptography, coding theory. Namely, such curves can be used to construct error-correcting codes, and also for devising secure protocols for data communication.

Triggered by these applications, mathematicians such as Ihara, Serre, Vladut, Drin'feld studied the maximal number of points on such curves, for a fixed finite fields, and the maximum taken over all curves in a certain class determined by a geometric invariant called the ‘genus’. Upper bounds for this maximum in terms of the genus and the finite field were obtained by Weil and Serre.

In this thesis we propose constructions of curves with many points. For genus 1 and 2 we find explicit curves reaching the maximum. For higher genus, we defined a curve to be ‘good’ if its number of points is within 10% of the upper bound obtained by Weil and Serre.

We are able to construct many ‘good’ curves. This has provided a lot of data for the website www.manypoints.org. Finally, for genus 1 we have extended an elementary proof due to Yuri I. Manin (1956) of a classical result of Helmut Hasse (1933) which gives a good upper bound for the maximum in this case.