Finitely ramified iterated extensions

Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold iterate of f, is absolutely irreducible over F; we compute a recursion for its discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the iterated monodromy group of f. The iterated extension L/F is finitely ramified if and only if f is post-critically finite (pcf). We show that, moreover, for pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely ramified over K, pointing to the possibility of studying Galois groups with restricted ramification via tree representations associated to iterated monodromy groups of pcf polynomials. We discuss the wildness of ramification in some of these representations, describe prime decomposition in terms of certain finite graphs, and also give some examples of monogene number fields.Comment: 19 page

Transitive and self-dual codes attaining the Tsfasman-Vladut-Zink bound

A major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good. In this correspondence-as a generalization of cyclic codes-the notion of transitive codes is introduced (see Definition 1.4 in Section I), and it is shown that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over F-q, for all squares q = l(2). It is also shown that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E-0 subset of E-1 subset of E-2 subset of center dot center dot center dot of function fields over F-q (with q = l(2)), where all extensions E-n/E-0 are Galois

Category forcings, MM+++MM^{+++}, and generic absoluteness for the theory of strong forcing axioms

We introduce a category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms with a stationary set preserving quotient. We show that the cut of this category at a rank initial segment of the universe of height a super compact which is a limit of super compact cardinals is a stationary set preserving partial order which forces $MM^{++}$ and collapses its size to become the second uncountable cardinal. Next we argue that any of the known methods to produce a model of $MM^{++}$ collapsing a superhuge cardinal to become the second uncountable cardinal produces a model in which the cutoff of the category of stationary set preserving forcings at any rank initial segment of the universe of large enough height is forcing equivalent to a presaturated tower of normal filters. We let $MM^{+++}$ denote this statement and we prove that the theory of $L(Ord^{\omega_1})$ with parameters in $P(\omega_1)$ is generically invariant for stationary set preserving forcings that preserve $MM^{+++}$. Finally we argue that the work of Larson and Asper\'o shows that this is a next to optimal generalization to the Chang model $L(Ord^{\omega_1})$ of Woodin's generic absoluteness results for the Chang model $L(Ord^{\omega})$. It remains open whether $MM^{+++}$ and $MM^{++}$ are equivalent axioms modulo large cardinals and whether $MM^{++}$ suffices to prove the same generic absoluteness results for the Chang model $L(Ord^{\omega_1})$.Comment: - to appear on the Journal of the American Mathemtical Societ