AI 中文总结
本文给出Q(t)上Mordell-Weil秩17的椭圆K3曲面X的公式，证明其二次基变换可得到秩18的椭圆曲面，一对复合后得秩19的椭圆曲面，对应无穷多条Q上高秩椭圆曲线。
AI 中文摘要
在[Elkies 2006, Elkies 2007]中，我们宣布存在K3曲面X/Q，其Néron–Severi群NS(X)=NS_Q(X)的秩为19，且具有Mordell-Weil秩17的椭圆纤维化。这是Q(t)上椭圆K3曲面所能达到的最大Mordell-Weil秩。该曲面的一条纤维是2006至2024年间已知的最高秩椭圆曲线，其秩至少为28。我们进一步宣布，该纤维化存在二次基变换，可得到Q(t)上Mordell-Weil秩18的椭圆曲面；且存在这样的一对二次基变换，它们的复合是椭圆曲线E₀/Q上Mordell-Weil秩19的椭圆曲面，其中#(E₀(Q))=∞，由此可知存在无穷多条Q上秩至少为19的椭圆曲线。我们展示了这样一对基变换。后续论文将说明如何利用[Elkies 2007]中也宣布的技术计算X及其纤维化。
英文摘要
In [Elkies 2006, Elkies 2007] we announced a K3 surface X/Q with Néron--Severi group NS(X) = NS_Q(X) of rank 19 and an elliptic fibration of Mordell--Weil rank 17. This is the largest possible Mordell--Weil rank over Q(t) for an elliptic K3 surface. One of the fibers of this surface is the elliptic curve of rank at least 28 that was the elliptic curve of highest rank known during the years 2006--2024. We further announced that there are quadratic base changes of this fibration to elliptic surfaces of Mordell--Weil rank 18 over Q(t), and that there are pairs of such quadratic base changes whose compositum is an elliptic surface of Mordell--Weil rank 19 over an elliptic curve E_0/Q with #(E_0(Q)) = \infty, whence there are infinitely many elliptic curves of rank at least 19 over Q. We exhibit one such pair. A subsequent paper will show (using a technique also announced in [Elkies 2007] how we computed X and its fibration.
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