Rational nodal curves with no smooth Weierstrass points

Rational Nodal Curves with No Smooth Weierstrass Points

LetX denote the rational curve with n+1 nodes obtained from the Riemann sphere by identifying 0 with∞ and ζj with −ζj for j = 0, 1, . . . , n−1, where ζ is a primitive (2n)th root of unity. We show that if n is even, then X has no smooth Weierstrass points, while if n is odd, then X has 2n smooth Weierstrass points. C. Widland [14] showed that the rational curve with three nodes obtained from P...

Elliptic Curves with No Rational Points

The existence of infinitely many elliptic curves with no rational points except the origin oo is proved by refining a theorem of DavenportHeilbronn. The existence of infinitely many quadratic fields with the Iwasawa invariant A3 = 0 is proved at the same time.

On Weierstrass Points and Optimal Curves

We use Weierstrass Point Theory and Frobenius orders to prove the uniqueness (up to isomorphism) of some optimal curves. This paper continues the study, begun in [FT] and [FGT], of curves over finite fields with many rational points, based on Stöhr-Voloch’s approach [SV] to the Hasse-Weil bound by way of Weierstrass Point Theory and Frobenius orders. Some of the results were announced in [T]. A...

Weierstrass Gap Sequence at Total Inflection Points of Nodal Plane Curves

C be the normalization of Γ . Let g = (d− 1)(d− 2) 2 − δ; the genus of C. We identify smooth points of Γ with the corresponding points on C. In particular, if P is a smooth point on Γ then the Weierstrass gap sequence at P is considered with respect to C. A smooth point P ∈ Γ is called an (e − 2)-inflection point if i(Γ, T ;P ) = e ≥ 3 where T is the tangent line to Γ at P (cf. Brieskorn–Knörre...

Rational points on curves