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Register a new userThe number of reducible space curves over a finite field
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Most hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible.
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Let F be the cubic field of discriminant -23 and O its ring of integers. Let Gamma be the arithmetic group GL_2 (O), and for any ideal n subset O let Gamma_0 (n) be the congruence subgroup of level n. In a previous paper, two of us (PG and DY) computed the cohomology of various Gamma_0 (n), along with the action of the Hecke operators. The goal of that paper was to test the modularity of elliptic curves over F. In the present paper, we complement and extend this prior work in two ways. First, we tabulate more elliptic curves than were found in our prior work by using various heuristics (old and new cohomology classes, dimensions of Eisenstein subspaces) to predict the existence of elliptic curves of various conductors, and then by using more sophisticated search techniques (for instance, torsion subgroups, twisting, and the Cremona-Lingham algorithm) to find them. We then compute further invariants of these curves, such as their rank and representatives of all isogeny classes. Our enumeration includes conjecturally the first elliptic curves of ranks 1 and 2 over this field, which occur at levels of norm 719 and 9173 respectively.
We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time--space complexity is roughly quadratic in the logarithm of the cardinality of the field and a geometric invariant of the input system (called its degree), which is always bounded by the Bezout number of the system. Our algorithm works for fields of any characteristic, but requires the cardinality of the field to be greater than a quantity which is roughly the fourth power of the degree of the input variety.
Let $Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $lambda$ of the field of algebraic numbers which is prime to p, consider the $lambda$-adic pro-semisimple completion of $Pi$ as an object of the groupoid whose objects are pro-semisimple groups and whose morphisms are isomorphisms up to conjugation by elements of the neutral connected component. We prove that this object does not depend on $lambda$. If dim X=1 we also prove a crystalline generalization of this fact. We deduce this from the Langlands conjecture for function fields (proved by L. Lafforgue) and its crystalline analog (proved by T. Abe) using a reconstruction theorem in the spirit of Kazhdan-Larsen-Varshavsky. We also formulate two related conjectures, each of which is a reciprocity law involving a sum over all $l$-adic cohomology theories (including the crystalline theory for $l=p$).
We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
Elkies proposed a procedure for constructing explicit towers of curves, and gave two towers of Shimura curves as relevant examples. In this paper, we present a new explicit tower of Shimura curves constructed by using this procedure.
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