The A-module Structure Induced by a Drinfeld A-module over a Finite Field

2004/12/18 by saadbouh, Mohamed Ahmed Mohamed
#14J32 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0412368

Abstract

Let Φ be a Drinfeld F_q[T]-module of rank 2, over a finite field L, a finite extension of n degrees for a finite field with q elements % F_q. Let P_Φ(X)= X²-cX+μP^m (c an element of % F_q[T] and μ a no null element of F_q, m the degree of the extension L over the field F_q[T]/P, P is a F%_q[T]-characteristic of L and d the degree of the polynomial P) the characteristic polynomial, of the Frobenius F of L. We will interested to the structure of finite F_q[T]-module L^Φ deduct by Φ over L and will proof our main result, the analogue of Deuring theorem for the elliptic curves : Let M=\fracF%_q[T]I₁⊕ \fracF_q[T]I₂, where I₁=(i₁),% I₂=(i₂) (i₁, i₂ two polynomials of F_q[T]%) and such that : i₂| (c-2). Then there exists an ordinary Drinfeld F_q[T]% -module Φ over L of rank 2, such that : % L^Φ ≃ M. We finish by a statistic about the cyclicity of such structure L^Φ, and we prove that is cyclic only for the trivial extensions of F_q.

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