论文标题
椭圆形曲线的平均分析等级
Average analytic ranks of elliptic curves over number fields
论文作者
论文摘要
在任意数字字段上，给出了椭圆曲线平均分析等级的条件结合。 In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $(9\ \text{deg}(K)+1)/2$, when ordered by naive height.证明中的一个关键成分是在具有规定的局部条件的任意数字字段上为椭圆曲线的数量提供渐近学；这些结果是通过证明具有规定的局部条件的加权投影堆栈上有界高度的计数点的一般结果来获得的，这可能是独立的。
A conditional bound is given for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field $K$ are modular and have $L$-functions which satisfy the Generalized Riemann Hypothesis, it is shown that the average analytic rank of isomorphism classes of elliptic curves over $K$ is bounded above by $(9\ \text{deg}(K)+1)/2$, when ordered by naive height. A key ingredient in the proof is giving asymptotics for the number of elliptic curves over an arbitrary number field with a prescribed local condition; these results are obtained by proving general results for counting points of bounded height on weighted projective stacks with a prescribed local condition, which may be of independent interest.
