椭圆曲线与实域和p-adic域上代数函数域的Hilbert第十问题
Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p-adic fields
摘要：
设k是特征为零的域，V是k上的光滑正维拟投射簇，D是V上的非空有效因子，K是V的函数域，A是D在K中的半局部环.本文证明了：（1）A，在所有情况下;（2）K，当k是（形式上）实数，V有一个实点;（3）K，当k是p-adic域的子域时，对于某个奇素数p。为了实现这一点，我们使用Denef的方法：从Q上的椭圆曲线E，不需要复乘，可以构造E在Q（t）上的二次扭曲E '，其具有Mordell-Weil秩1。大部分的文件是致力于证明（使用定理R。Noot），可以在K中选择f，在D处消失，使得从域扩张K/Q（f）= Q（t）导出的群E '（K）等于E'（Q（t））。然后，我们模仿Denef（对于真实情况）和Kim和Bauh（对于p-adic情况）的论点。
英文摘要：
Let k be a field of characteristic zero, V a smooth, positive-dimensional, quasiprojective variety over k, and D a nonempty effective divisor on V. Let K be the function field of V, and A the semilocal ring of D in K. In this paper, we prove the Diophantine undecidability of: (1) A, in all cases; (2) K, when k is (formally) real and V has a real point; (3) K, when k is a subfield of a p-adic field, for some odd prime p. To achieve this, we use Denef's method: from an elliptic curve E over Q, without complex multiplication, one constructs a quadratic twist E' of E over Q(t), which has Mordell-Weil rank one. Most of the paper is devoted to proving (using a theorem of R. Noot) that one can choose f in K, vanishing at D, such that the group E'(K) deduced from the field extension K/Q(f)=Q(t) is equal to E'(Q(t)). Then we mimic the arguments of Denef (for the real case) and of Kim and Roush (for the p-adic case).
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