The universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of \pi_1(E-{0},x). This was written down independently by Calaque, Enriquez and Etingof (arXiv:math/0702670), and by Levin and Racinet (arXiv:math/0703237). It generalizes the KZ-equation in genus 0. These notes are in four parts. The first two parts provide a detailed exposition of this connection (following Levin-Racinet); the third is a leisurely exploration of the connection in which, for example, we compute the limit mixed Hodge structure on the unipotent fundamental group of the Tate curve minus its identity. In the fourth part we elaborate on ideas of Levin and Racinet and explicitly compute the connection over the moduli space of elliptic curves with a non-zero abelian differential, showing that it is defined over Q.

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通用椭圆KZB连接注意事项

万能椭圆 KZB 方程是 M_{1,2} 上的 pro-vector 丛上的可积连接，其在对应于椭圆曲线 E 的点和 E 的非零点 x 上的纤维是 \pi_1( E-{0},x)。这是由 Calaque、Enriquez 和 Etingof (arXiv:math/0702670) 以及 Levin 和 Racinet (arXiv:math/0703237) 独立编写的。它概括了属 0 中的 KZ 方程。这些注释分为四个部分。前两部分详细阐述了这种联系（紧随 Levin-Racinet 之后）；第三个是对连接的悠闲探索，例如，我们在泰特曲线的单能基本群上计算极限混合霍奇结构减去其身份。