On the surjectivity of \mathfrakp-adic Galois representations attached to Drinfeld modules of rank 2

2025/02/26 by Kumar, Narasimha, Shit, Dwipanjana
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.19084

Abstract

Let \mathbbF_q be the finite field with q≥ 5 elements and A:=\mathbbF_q[T]. For a class of \mathfrakp ∈ Spec(A) ∖ \(0)\, but fixed, we produce infinitely many Drinfeld A-modules of rank 2, for which the associated \mathfrakp-adic Galois representation is surjective. This result is a variant of the work of~[Ray24] for \mathfrakp=(T). We also show that for a class of \mathfrakl=(l) ∈ Spec(A), where l is a monic polynomial, the \mathfrakp-adic Galois representation, attached to the Drinfeld A-module φ_T=T+g₁τ-l^q-1τ² with g₁ ∈ A ∖ \mathfrakl, is surjective for all \mathfrakp ∈ Spec(A)∖\(0)\. This result generalizes the work of [Zyw11] from \mathfrakl=(T), g₁=1.

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• Surjectivity of the adelic Galois Representation associated to a Drinfeld Module of prime rank
• A class of Drinfeld A-modules of rank 3 with surjective Galois representations
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