透过Eisenstein理想的视角
Through the lens of the Eisenstein ideal
AI总结：
本文通过Eisenstein理想建立模曲线几何与分圆域算术的联系，定义两个猜想互逆的映射，并给出其等变构造及Euler系统的应用。
AI中文摘要：
本文通过Eisenstein理想阐述了模曲线几何与分圆域算术之间的关系。对于整数$N \ge 5$和奇素数$p$，我们定义了两个猜想互逆的映射，分别连接$X_1(N)$的同调$P$的Eisenstein约化的实部与第$N$个分圆整数环的第二$K$-群的$p$-部分$Y$的实部。我们回顾了从$P$到$Y$的映射的一个动机性构造，并描述了最近关于模符号上底层映射的Eisenstein性质的证明。我们利用$X_1(N)$的第一étale上同调的Eisenstein约化，提供了从$Y$到$P$的第二个映射的完全等变构造。然后，我们解释了利用模曲线的相对上同调群中的扩张类对分圆单位的Euler系统进行等变构造，并描述了它在我们的纲领中的地位。
英文摘要：
This paper exposits relationships between the geometry of modular curves and the arithmetic of cyclotomic fields through the Eisenstein ideal. For an integer $N \ge 5$ and an odd prime $p$, we define two conjecturally inverse maps between the real part $P$ of the Eisenstein reduction of the homology $P$ of $X_1(N)$ and the real part of the $p$-part $Y$ of the second $K$-group of the $N$th cyclotomic integer ring. We recall a motivic construction of the map from $P$ to $Y$ and describe the recent proof of the Eisenstein property of the underlying map on modular symbols. We provide a fully equivariant construction of the second map from $Y$ to $P$ using the Eisenstein reduction of the first étale cohomology of $X_1(N)$. We then explain an equivariant construction of the Euler system of cyclotomic units using extension classes in relative cohomology groups of modular curves and describe its place in our program.
