论文标题
有限场上的德林菲尔德模块的同构类别
Isomorphism classes of Drinfeld modules over finite fields
论文作者
论文摘要
我们研究了有限田$ k $的Drinfeld $ a $模型，并具有交换性内态代数$ d $，以描述固定同学类别中的同构类别。我们研究$ d $的最小订单$ a [π] $ a [π] $ a [π] $ a [π] $ a [π] $ a [π]通过证明何时在$π$处是本地最大的情况，并证明这是当且仅当同等基类是普通或$ k $的情况下才会发生这种情况。然后，我们描述了Drinfeld Module $ nodule $ d $ d $ d $ lineare等于$ d $ lineareare $ nlodalism ring $ \ mathcal $ \ mathcal {e} $的基本理想的单体如何作用于Hayes的精神上的同构$ ϕ $。我们表明，当限于内核理想时，该动作是免费的，我们给出了三个等效的定义，并确定何时该动作是传递的。特别是，该动作是在同构类别中的同构类别的自由和传递性的，该类别是普通的，或在黄金场上定义的，在这些情况下产生了完整而明确的描述。
We study isogeny classes of Drinfeld $A$-modules over finite fields $k$ with commutative endomorphism algebra $D$, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order $A[π]$ of $D$ occurs as an endomorphism ring by proving when it is locally maximal at $π$, and show that this happens if and only if the isogeny class is ordinary or $k$ is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring $\mathcal{E}$ of a Drinfeld module $ϕ$ up to $D$-linear equivalence acts on the isomorphism classes in the isogeny class of $ϕ$, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases.
