Seminar des SFB/TRR 326 GAUS The kernel of the adjoint exponential in Anderson 𝑡-modules

Given an algebraically closed complete valued field K over 𝔽_q, an Anderson t-module of dimension d is given by the topological 𝔽_q-vector space K^d, endowed with an 𝔽_q-linear action ϕ_t = ∑_i≥0T_iτⁱ ∈ M_d×d(K)[τ], where τ : K^d → K^d sends (v₁, …, v_d) to (v₁^q, …, v_d^q). In analogy with complex abelian varieties, there is an analytic map exp = ∑_i≥0E_iτⁱ : K^d → K^d—which is not necessarily surjective—such that ϕ_texp = expT₀. The adjoint exponential, defined as the series exp^* := ∑_i≥0τ⁻ⁱE_i^T, determines a (non-analytic) continuous map K^d → K^d. Using the factorization properties of K[[x]], Poonen proved that there is a perfect duality of topological 𝔽_q-vector spaces ker(exp) × ker(exp^*) → 𝔽_q under the condition d = 1. 
In this talk, I explain that for an arbitrary abelian Anderson t-module, we have a collection of perfect pairings ker(ϕ_tⁿ) × ker(ϕ^*_tⁿ) → 𝔽_q, and that we can use them to obtain a canonical generating series (F_ϕ)_c ∈ M_d×d(K)[[τ⁻¹,τ]] for all c ∈ 𝔽_q((t⁻¹))/𝔽_q(t). The study of the properties of F_ϕ allows us to prove that, if exp is surjective, ker(exp^*) is compact and isomorphic to the Pontryagin dual of ker(exp). Moreover, we deduce an alternative explicit description of the Hartl–Juschka pairing, obtained by Gazda and Maurischat in a recent preprint.