A Carlitz module analogue of the Grunwald-Wang Theorem

The Theorem of Grunwald -

The theorem of Grunwald-Wang in the setting of valuation theory

Given a field K with finitely many valuations; does there exist an extension of K which at these valuations has a prescribed local behavior? The Grunwald-Wang theorem answers this question in the case of abelian field extensions. Originally developed for algebraic number fields in the context of class field theory, it has turned out that it is valid quite generally, for arbitrary multi-valued f...

Grunwald–wang for Global Character Groups

Let F denote C or Q` for a prime ` 6= char(k), equipped with its natural topology, and let Gk = Gal(ks/k) for a choice of separable closure ks/k. For each place v of k choose a separable closure kv,s/kv and an embedding ks ↪→ kv,s over k ↪→ kv, thereby identifying Gkv with a decomposition group at v in Gk. It is surprising (but confirmed by consultation with experts) that the following natural ...

A Carlitz Module Analogue of a Conjecture of Erdős and Pomerance

Let A = Fq [T ] be the ring of polynomials over the finite field Fq and 0 = a ∈ A. Let C be the A-Carlitz module. For a monic polynomial m ∈ A, let C(A/mA) and ā be the reductions of C and a modulo mA respectively. Let fa(m) be the monic generator of the ideal {f ∈ A,Cf (ā) = 0̄} on C(A/mA). We denote by ω(fa(m)) the number of distinct monic irreducible factors of fa(m). If q = 2 or q = 2 and a ...

A Carlitz–von Staudt Type Theorem for Finite Rings

We compute the kth power-sum polynomials (for k > 0) over an arbitrary finite ring R, obtained by summing the kth powers of (T + r) for r ∈ R. For R non-commutative, this extends the work of Brawley–Carlitz–Levine [Duke Math. J. 41], and resolves a conjecture by Fortuny Ayuso, Grau, Oller-Marcén, and Rúa (2015). For R commutative, our results bring together two classical programs in the literat...