Old and new conjectures on the number of points of algebraic sets over finite fields

In the late 1980's, Tsfasman conjectured an explicit formula for the maximum number of $\mathbb{F}_q$-rational points on a (not necessarily irreducible) hypersurface of a given degree in a projective space of a given dimension. This was soon proved by Serre. Subsequently, two generalizations were proposed. One, the so-called Ghorpade-Lachaud Conjecture, that proposed a bound on the number of $\mathbb{F}_q$-rational points of a projective algebraic set when it is a complete intersection. And another, called Tsfasman-Boguslavsky Conjecture (TBC), that proposed an intricate, but explicit, formula for the maximum number of $\mathbb{F}_q$-rational points on a projective algebraic set that is cut out by a given number of linearly independent homogeneous polynomials, all having the same degree. About five years ago, Couvreur showed that the former conjecture holds in the affirmative and in fact, proved a more general inequality for the number of $\mathbb{F}_q$-rational points of a projective algebraic set in terms of the degrees and dimensions of its irreducible components and of course the size q of the finite ground field. On the other hand, it was also shown around the same time that the TBC can be false, in general even though it holds in the affirmative in some cases. Later, a refined conjecture that ameliorates the TBC has been proposed first in a special case, and more recently, in the general case. It is still open, but it has also been shown to hold in the affirmative in a number of special cases. We will outline these developments, with a particular emphasis on the new conjecture.
This talk is based on joint works with Mrinmoy Datta as well as with Peter Beelen and Mrinmoy Datta.