纯不可分的Richelot同源
Purely inseparable Richelot isogenies
AI总结：
本文证明特征2代数闭域上超奇异亏格2曲线有无穷多条Richelot同源，并给出任意两条此类曲线间同源数量的精确计数及显式构造。
摘要：
我们证明，若$C$是特征为$2$的代数闭域上的超奇异亏格$2$曲线，则从$C$出发存在无穷多条Richelot同源。这与特征$2$中的非超奇异曲线或特征不为$2$的任意曲线情形形成对比：在这些情形下，从给定亏格$2$曲线出发至多有十五条Richelot同源。更具体地，我们证明若$C_1$和$C_2$是特征$2$的代数闭域上的任意两条超奇异亏格$2$曲线，则从$C_1$到$C_2$恰好有六十条Richelot同源，除非$C_1$或$C_2$同构于曲线$y^2 + y = x^5$。在该情形下，从$C_1$到$C_2$的Richelot同源有十二条或四条，取决于$C_1$是否同构于$C_2$。（这里我们按同构类计数Richelot同源。）我们给出显式构造，产生两条超奇异曲线之间的所有Richelot同源。
英文摘要：
We show that if $C$ is a supersingular genus-$2$ curve over an algebraically-closed field of characteristic $2$, then there are infinitely many Richelot isogenies starting from $C$. This is in contrast to what happens with non-supersingular curves in characteristic $2$, or to arbitrary curves in characteristic not $2$: In these situations, there are at most fifteen Richelot isogenies starting from a given genus-$2$ curve. More specifically, we show that if $C_1$ and $C_2$ are two arbitrary supersingular genus-$2$ curves over an algebraically-closed field of characteristic $2$, then there are exactly sixty Richelot isogenies from $C_1$ to $C_2$, unless either $C_1$ or $C_2$ is isomorphic to the curve $y^2 + y = x^5$. In that case, there are either twelve or four Richelot isogenies from $C_1$ to $C_2$, depending on whether $C_1$ is isomorphic to $C_2$. (Here we count Richelot isogenies up to isomorphism.) We give explicit constructions that produce all of the Richelot isogenies between two supersingular curves.
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