On a Theorem of Coleman
نویسنده
چکیده
A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all the rational points on certain curves of genus 2. 0 Introduction The following classical result of Chabauty [3] is a curiosity of the literature in that there has been a 50 year period during which applications have been virtually impossible. Proposition 0.1. Let C be a curve of genus g defined over a number field K, whose Jacobian has Mordell-Weil rank g − 1. Then C has only finitely many K-rational points. This result is now superceded by Falting's work. However it has been shown by Coleman [4] that Chabauty's method can be used in many situations to give good bounds for the number of points on a curve. In particular, there are two potential genus 2 applications [4], [8]. Proposition 0.2. Let C be a curve of genus 2 defined over Q, and p 5 be a prime of good reduction. If the Jacobian of C has rank at most 1 and C is the reduction of C mod p then #C(Q) # C(F p) + 2. Proposition 0.3. Let C be a curve of genus 2 defined over Q with 4 rational branch points and good reduction at 3, whose Jacobian has rank at most 1. Then #C(Q) 6. If the the rational branch points of the curve in Proposition 0.3 are mapped to (0, 0), (1, 0), (−1, 0), (1/λ, 0), then there is the following situation for which Coleman's method is guaranteed to determine C(Q) completely.
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