Let $p$ be a prime and $f$ a positive integer, greater than $1$ if $p=2$. We construct liftings of the Artin-Schreier curve $C(p,f)$ in characteristic $p$ defined by the equation $y^e=x-x^p$ (where $e=p^f-1$) to a curve $ilde{C}(p,f)$ over a certain polynomial ring $R'$ in characteristic $0$ which shares the following property with $C(p,f)$. Over a certain quotient of $R'$, the formal completion of the Jacobian $J(ilde{C}(p,f))$ has a $1$-dimensional formal summand of height $(p-1)f$. Along the way we show how Honda’s theory of commutative formal group laws can be extended to more general rings and prove a conjecture of his about the Fermat curve.
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机译：假设$ p $是质数，$ f $是正整数，如果$ p = 2 $，则大于$ 1 $。我们在由等式$ y ^ e = xx ^ p $（其中$ e = p ^ f-1 $）定义的特征$ p $中构造Artin-Schreier曲线$ C（p，f）$的提升$ tilde {C}（p，f）$在特征为$ 0 $的某个多项式环$ R'$上，与$ C（p，f）$共享以下属性。在$ R'$的一个商中，雅可比行列式$ J（ tilde {C}（p，f））$的形式完成具有高度为$（p-1）f $的维度为$ 1 $的形式加数。一路上，我们展示了本田的可交换形式群定律理论可以扩展到更一般的环，并证明了他对费马曲线的猜想。
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