We introduce the use of $p$-descent techniques for elliptic surfaces over a perfect field of characteristic not $2$ or $3$. Under mild hypotheses, we obtain an upper bound for the rank of a nonconstant elliptic surface. When $p=2$, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa’s inequality. This answers a question raised by Ulmer. We give some applications to rank bounds for elliptic surfaces over the rational numbers.

中文翻译：

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Mordell-Weil 秩的椭圆曲面下降和算术界

我们介绍使用$p$- 椭圆表面的下降技术在一个完美的特征场上不是$2$或者$3$. 在温和的假设下，我们获得了非常数椭圆曲面秩的上限。什么时候$p=2$，这个界限是对从 Igusa 不等式推导出来的等级的众所周知的几何界限的算术改进。这回答了 Ulmer 提出的问题。我们给出了一些应用程序来对有理数上的椭圆曲面的边界进行排序。