On Mordell's conjecture for algebraic curves over function fields
On Mordell's conjecture for algebraic curves over function fields
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关于函数域上代数曲线的莫德尔猜想
DOI：
10.2969/jmsj/01820182
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发表时间：
1966
影响因子：
0.7
通讯作者：
Megumu Miwa
中科院分区：
文献类型：
--
作者：
Megumu Miwa
In this paper, we are concerned with Mordell's conjecture on the set of rational points on algebraic curves in " relative case " (cf. [2] p. 139). Let k be any field and K be a function field with k as constant field, i, e. a regular extension of finite type of k. Let C be a complete non-singular curve defined over K. We say that C is trivially defined, if there is a curve Co defined over k which is birationally equivalent to C over K. Then our main Theorem reads: If the genus g of C is >_2, then the set of all rational points of C over K is a finite set or C is trivially defined*). This was proved by Grauert [3] in the case where the characteristic of k is 0 and k is algebraically closed. Manin [4] obtained the same result with a transcendental method. We shall prove the above Theorem for the field k of any characteristic p (which may be == 0 or * 0), without supposing k to be algebraically closed. The proof is given in two cases (1) p = 0 (~ 1), (2) p : 0 (~ 2)1'. We shall use the results of [3] as formulated at the beginnings of § 1 and § 2, and the theory of abelian varieties (cf. [1], [6]). As to the terminology we follow generally the usage in [1]. More specifically, the method we shall use is that of descent. To explain
