In previous work, Ohno [Ohn97] conjectured, and Nakagawa [Nak98] proved, relations between the counting functions of certain cubic fields. These relations may be viewed as complements to the Scholz reflection principle, and Ohno and Nakagawa deduced them as consequences of 'extra functional equations' involving the Shintani zeta functions associated to the prehomogeneous vector space of binary cubic forms.In the present paper we generalize their result by proving a similar identity relating certain degree ℓ fields with Galois groups D ℓ and F ℓ respectively, for any odd prime ℓ, and in particular we give another proof of the Ohno-Nakagawa relation without appealing to binary cubic forms.

We introduce n(n−1)/2 natural involutions ("toggles") on the set S of noncrossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π) exhibits the homomesy phenomenon: the average of f over the elements of a T -orbit is the same for all T -orbits. We can apply our method of proof more broadly to toggle operations back on the collection of independent sets of certain graphs. We utilize this generalization to prove a theorem about toggling on a family of graphs called "2-cliquish." More generally, the philosophy of this "toggle-action," proposed by Striker, is a popular topic of current and future research in dynamic algebraic combinatorics.

Given a finite endomorphism ϕ of a variety X defined over the field of fractions K of a Dedekind domain, we study the extension K(ϕ −∞ (α)) := n≥1 K(ϕ −n (α)) generated by the preimages of α under all iterates of ϕ. In particular when ϕ is post-critically finite, i.e. there exists a non-empty, Zariski-open W ⊆ X such that ϕ −1 (W ) ⊆ W and ϕ : W → X isétale, we prove that K(ϕ −∞ (α)) is ramified over only finitely many primes of K. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire [AHM05] in the case X = A 1 and Cullinan-Hajir, Jones-Manes [CH12, JM14] in the case X = P 1 . Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for X = P 1 . The proof relies on Faltings' theorem and a local argument.

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