We show that the moduli space of elliptic curves of minimal degree in a general Fano variety of lines of a cubic fourfold is a non-singular curve of genus $631$. The curve admits a natural involution with connected quotient. We find that the general Fano contains precisely $3780$ elliptic curves of minimal degree with fixed (general) $j$-invariant. More generally, we express (modulo a transversality result) the enumerative count of elliptic curves of minimal degree in hyper-Kahler varieties with fixed $j$-invariant in terms of Gromov--Witten invariants. In $K3[2]$-type this leads to explicit formulas of these counts in terms of modular forms.

中文翻译：

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Hyper-Kähler 品种中的椭圆曲线

我们表明，在一般的 Fano 各种四重三次直线中，最小阶椭圆曲线的模空间是 $631$ 属的非奇异曲线。该曲线允许具有连通商的自然对合。我们发现一般 Fano 包含精确的 $3780$ 具有固定（一般）$j$-invariant 的最小度数的椭圆曲线。更一般地，我们用 Gromov--Witten 不变量表示（以横向结果为模）具有固定 $j$-不变量的超 Kahler 变体中最小度数的椭圆曲线的枚举计数。在 $K3[2]$-type 中，这导致了这些计数在模形式方面的明确公式。