Effective Chabauty

Abstract

0. Introduction. By the Mordell-Weil rank of a curve over a number field we mean the rank of the group of points on its Jacobian. In his paper Sur les points rationels des courbes algebriques de genre supbrieur ?t l'unitb, Chabauty showed that a curve of genus g over a number field with Mordell-Weil rank at most g has finitely many points. This result is now superceded by Falting's work. However, as we shall show in this note, Chabauty's method can be used to give good effective bounds for the number of points on a curve as above. For example, we will show: (i) A curve of genus two over Q with good reduction at two or three and Mordell-Weil rank at most one has at most twelve rational points.More generally, (ii) Suppose C is a curve of genus g over a number field K with Mordell-Weil rank at most g-1. Suppose is an unramified prime of K at which C has good reduction, of residue characteristic strictly greater than 2g. Then # C(K)< N + 2g(N + 1)-1.Turning the method around we can show:

The motivic fundamental group of P1∖{0,1,∞} and the theorem of Siegel

“…The reader familiar with the method of Chabauty ([4], [6]) will immediately recognize our proof to be a 'non-abelian lift' of his. As such, we believe that many generalizations and refinements should be possible and hope to discuss them in the near future.…”

Low-degree points on Hurwitz-Klein curves

“…The proof of Case 1 above goes through for k = 1. For k = 2, part of the proof goes through, up to the point where we show that there exists a rational function on F p,r whose divisor equals 2 . By Riemann-Roch, E is special.…”

“…a factor of the form J p,r ) with finite Mordell-Weil group (the predictions of the Birch and Swinnerton-Dyer conjecture regarding Mordell-Weil ranks of Fermat Jacobians are discussed in [8]). The proof of Theorem 1.2 uses the geometry of J 13,3 (studied by Lim in [14]), Coleman's effective Chabauty bound ( [2]) and a technical device (Theorem 1.3 below) which is valid only under a strong (and difficult to settle in general) Mordell-Weil rank condition. Following Lim ([14]), let C 0 denote the quotient of F p,r by the group of automorphisms generated by ρ.…”

Rational points on curves

“…This is what Chabauty proves. Later, the method was taken up by Coleman [15] who used it to deduce upper bounds on the number of rational points on the curve. The method can also be used to determine the set of rational points in certain cases, see [23,26,57] for early examples of this.…”