Contributions to: Rational points on curves over finite fields (J.-P. Serre, edited by A. Bassa, E. Lorenzo García, C. Ritzenthaler, and R. Schoof, with contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler, adapted and expanded from notes by Fernando Gouvêa of Serre's 1985 lectures at Harvard University), Société mathématique de France, 2020 (to appear).

Fernando Gouvêa's handwritten notes to Serre's famous 1985 course at Harvard, on the number of points on curves over finite fields, have been passed around “as clandestine literature,” as René Schoof puts it in the introduction to this forthcoming book. Over the past few years, a team of editors and volunteers TeXed the notes and Serre has gone over them, editing them and updating them in places.

Several authors contributed new material for this book. My contributions consist of “postfaces” to three chapters.

For Chapter II (“Refinements of Weil's bound”) I contributed a short section describing a theorem that Kristin Lauter and I proved (with versions in these two papers) that extends a result of Serre, which says that if the real Weil polynomial of an abelian variety over a finite field can be written as the product of two factors whose resultant is a unit, then the abelian variety is not isogenous to a Jacobian.

For Chapter V (“General results”) I contributed a section discussing questions related to curves over finite fields with few points for their genus.

And for Chapter VI (“Optimization in the explicit formulas”) I contributed a section detailing how the Oesterlé bound on the number of points of a genus-g curve over a finite field can be computed using rational arithmetic (and no transcendental functions).