Rational points on curves over finite fields

We have also sketched families of higher genus hyperelliptic curves whose deterministic algebraic parameterization is based on solvable polynomials of higher degree arising from

The minimum and maximum number of rational points on jacobian surfaces over finite fields..

— We détermine ail groups that can occur as thé group of rational points of an elliptic curve over a given finite field.. Let Fq dénote thé finite field of

The first two exceptional cases N=40, 48 are the values of N for which Xo(N) is hyperelliptic, but the hyperelliptic involution is not of Atkin-Lehner type (this is also the case

familiar modular curves, give a modular interpretation of their Fq-defined points, and use Weil’s estimate to approximate the number of their Fq-defined points...

In Section 3, we give some background concerning curves in characteristic p and prove facts for regular nonsmooth curves that hold in any absolute genus.. The latter reduction was

If it is the case, the corresponding isogeny class contains the product of elliptic curves of trace −m and these curves have q+1+m ≥ 3+1+3 = 7 rational points, thus at least

Once one has good upper bounds, one would like to show that these bounds are met by some curve, therefore one needs good methods of constructing curves; these methods include