论文标题
在有限场上的德林菲尔德模块的同构和同构类别的明确描述
Explicit description of isogeny and isomorphism classes of Drinfeld modules over finite field
论文作者
论文摘要
从数字理论到函数字段理论时，人们不能错过等级1德林菲尔德模块与统一根群之间的深层类比，以及等级2 Drinfeld模块和椭圆曲线之间的类比。但是到目前为止，在数量字段理论中尚无类似于较高等级r> 2的德林菲尔德模块的结构。 Yu对同学类别的分类（本田在Abelian品种中的类似物）。 Actually Yu has also explicitly did that work for r = 2. To complete the classification, we define the new notion of fine isomorphy invariants for any rank r Drinfeld module and we prove that the fine isomorphy invariants together with J-invariants completely determine the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.
When travelling from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r > 2. In this paper we investigate the classes of those Drinfeld modules of higher rank r > 2. We describe explicitly the Weil polynomials defining the isogeny classes of rank r Drinfeld modules for any rank r > 2. our explicit description of the Weil polynomials depends heavily on Yu's classification of isogeny classes (analogue of Honda-Tate at abelian varieties). Actually Yu has also explicitly did that work for r = 2. To complete the classification, we define the new notion of fine isomorphy invariants for any rank r Drinfeld module and we prove that the fine isomorphy invariants together with J-invariants completely determine the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.
