Homology of Projective Space over Finite Fields

Homology of Projective Space over Finite Fields

Extending Self-maps to Projective Space over Finite Fields

Using the closed point sieve, we extend to finite fields the following theorem proved by A. Bhatnagar and L. Szpiro over infinite fields: if X is a closed subscheme of P over a field, and φ : X → X satisfies φOX(1) ' OX(d) for some d ≥ 2, then there exists r ≥ 1 such that φ extends to a morphism P → P.

On Computing Homology Gradients over Finite Fields

Recently the so-called Atiyah conjecture about l-Betti numbers has been disproved. The counterexamples were found using a specific method of computing the spectral measure of a matrix over a complex group ring. We show that in many situations the same method allows to compute homology gradients, i.e. generalizations of l-Betti numbers to fields of arbitrary characteristic. As an application we ...

Computing in Picard groups of projective curves over finite fields

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that various desirable operations can be performed efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite mor...

Space Filling Curves over Finite Fields

In this note, we construct curves over finite fields which have, in a certain sense, a “lot” of points, and give some applications to the zeta functions of curves and abelian varieties over finite fields. In fact, we found the basic construction, given in Lemma 1, of curves in A which go through every rational point, as part of an unsuccessful attempt to find curves of growing genus over a fixe...