Order of torsion for reduction of linearly independent points for a\n family of Drinfeld modules

2021/02/28 by Dragos Ghioca, Ghioca, Dragos, Igor E. Shparlinski +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2103.00641

openalex publication_date 2021/02/28 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let q be a power of the prime number p, let K= mathbb F_q(t), and let\nr\≥ 2 be an integer. For points mathbf a, mathbf b\∈ K which are\n mathbbF_q-linearly independent, we show that there exist positive\nconstants N₀ and c₀ such that for each integer \ℓ\≥ N₀ and for each\ngenerator \τ of mathbb F_q^\ℓ/ mathbb F_q, we have that for all\nexcept N₀ values \λ\∈\ mathbbF_q, the corresponding\nspecializations mathbf a, mathbf b(\τ) and mathbf b(\τ) cannot\nhave orders of degrees less than c₀\log\log\ℓ as torsion points for the\nDrinfeld module \Φ^(\τ,\λ): mathbbF_q[T] longrightarrow\n\End_\ mathbbF_q( mathbb G_a) (where mathbb G_a\nis the additive group scheme), given by \Φ^(\τ,\λ)_T(x)=\τ\nx+\λ x^q + x^qr.\n

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