论文标题
椭圆曲线的较高模块性超过功能场
Higher modularity of elliptic curves over function fields
论文作者
论文摘要
我们研究了功能场上椭圆曲线的“较高模块化”的概念。鉴于这样的椭圆曲线$ e $和一个整数$ r \ geq 1 $，我们说$ e $是$ r $ - 二型时，当$ r $ $ $ legged shtukas和$ r $ r $ $ fold的产品之间存在代数的对应关系时，$ e $ e $ e $被认为是Elliptic Surfictic表面。 （已知的）情况$ r = 1 $类似于椭圆曲线的模块化概念，而Mathbf {q} $。我们的主要定理是，如果$ e/\ mathbf {f} _q（t）$是一种非异常椭圆曲线，其导体具有4度，则$ e $是2型。最终，证明使用K3表面的属性。在途中，我们证明了独立兴趣的结果：K3表面在且仅当其PICARD晶格与此类Kummer表面的PICARD晶格上合理地等距时，在椭圆曲线的产物上接收有限的形态。
We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of $r$-legged shtukas, and the $r$-fold product of $E$ considered as an elliptic surface. The (known) case $r=1$ is analogous to the notion of modularity for elliptic curves over $\mathbf{Q}$. Our main theorem is that if $E/\mathbf{F}_q(t)$ is a nonisotrivial elliptic curve whose conductor has degree 4, then $E$ is 2-modular. Ultimately, the proof uses properties of K3 surfaces. Along the way we prove a result of independent interest: A K3 surface admits a finite morphism to a Kummer surface attached to a product of elliptic curves if and only if its Picard lattice is rationally isometric to the Picard lattice of such a Kummer surface.
