Zeta functions of curves with no rational points

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.

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...

Rational points on curves

Rational points on curves

2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to `-adic representations 21 2.6 The isogen...

Rational Points on Curves

This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve C over Q. The focus is on practical aspects of this problem in the case that the genus of C is at least 2, and therefore the set of rat...