REU: Rational Curves over Finite Fields
نویسنده
چکیده
Our main object of study is X ⊂ P5F2 , where X is the degree 5 Fermat hypersurface, which we are looking at in P over F2. X is the zero locus {X 0 +X 1 + ...+X 5 = 0}. The question is, “what kind of rational curves lie on X?” Since F2 only has one invertible element, P(F2) = F 2 \ 0, which suggests that the Fermat quintic has the same zero locus as the linear hypersurface X0 + X1 + ... + X5 = 0; however, we are interested in field extensions as well, so X is a “thing,” not just the space of solutions.
منابع مشابه
On the maximum number of rational points on singular curves over finite fields
We give a construction of singular curves with many rational points over finite fields. This construction enables us to prove some results on the maximum number of rational points on an absolutely irreducible projective algebraic curve defined over Fq of geometric genus g and arithmetic genus π.
متن کاملThe family of indefinite binary quadratic forms and elliptic curves over finite fields
In this paper, we consider some properties of the family of indefinite binary quadratic forms and elliptic curves. In the first section, we give some preliminaries from binary quadratic forms and elliptic curves. In the second section, we define a special family of indefinite forms Fi and then we obtain some properties of these forms. In the third section, we consider the number of rational poi...
متن کاملOn Curves over Finite Fields with Many Rational Points
We study arithmetical and geometrical properties of maximal curves, that is, curves defined over the finite field Fq2 whose number of Fq2 -rational points reachs the Hasse-Weil upper bound. Under a hypothesis on non-gaps at rational points we prove that maximal curves are Fq2 -isomorphic to y q + y = x for some m ∈ Z.
متن کاملApprentice Linear Algebra , 3 rd day , 07 / 06 / 05 REU 2005 Instructor : László Babai
Regarding the entire discussion of vector spaces and their properties, we may replace R by any field F , and define vector space “over F” for which F is the domain of scalars. Note that it only makes sense to consider linear maps between vector spaces over the same field. We are already familiar with the field of real numbers R, but also with rational numbers Q, complex numbers C, and the finit...
متن کاملOn the number of rational points on curves over finite fields with many automorphisms
Using Weil descent, we give bounds for the number of rational points on two families of curves over finite fields with a large abelian group of automorphisms: Artin-Schreier curves of the form y−y = f(x) with f ∈ Fqr [x], on which the additive group Fq acts, and Kummer curves of the form y q−1 e = f(x), which have an action of the multiplicative group Fq . In both cases we can remove a √ q fact...
متن کامل