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Kevin Keating

Publications and source records attributed to Kevin Keating.

26 records · Page 2Linked to original sources
arXiv2013

Indices of inseparability for elementary abelian p-extensions

Let K be a local field whose residue field is a finite field of characteristic p, and let L/K be a finite totally ramified Galois extension. Fried and Heiermann defined the "indices of inseparability" of L/K, a refinement of the ramification data of L/K. We give a method for computing the indices of inseparability of the extension L/K in terms of the norm group N_{L/K}(L^*) in the case where K has characteristic p and Gal(L/K) is an elementary abelian p-group with a single ramification break. In some cases our methods lead to simple formulas for the indices of inseparability.

math.NT↗
arXiv2008

Wintenberger's Functor for Abelian Extensions

Let $k$ be a finite field. Wintenberger used the field of norms to give an equivalence between a category whose objects are totally ramified abelian $p$-adic Lie extensions $E/F$, where $F$ is a local field with residue field $k$, and a category whose objects are pairs $(K,A)$, where $K\cong k((T))$ and $A$ is an abelian $p$-adic Lie subgroup of $\Aut_k(K)$. In this paper we extend this equivalence to allow $\Gal(E/F)$ and $A$ to be arbitrary abelian pro-$p$ groups.

math.NT↗
arXiv2006

Intersection numbers of Heegner divisors on Shimura curves

We compute the arithmetic intersection numbers of certain Heegner divisors on integral models of Shimura curves over Q. Our formulas generalize the formulas of Gross-Kohnen-Zagier for intersection numbers of Heegner divisors on integral models of modular curves.

math.NT↗
arXiv2005

Extensions of local fields and truncated power series

Let $K$ be a finite tamely ramified extension of $\Q_p$ and let $L/K$ be a totally ramified $(\Z/p^n\Z)$-extension. Let $π_L$ be a uniformizer for $L$, let $σ$ be a generator for $\Gal(L/K)$, and let $f(X)$ be an element of $Ø_K[X]$ such that $σ(π_L)=f(π_L)$. We show that the reduction of $f(X)$ modulo the maximal ideal of $Ø_K$ determines a certain subextension of $L/K$ up to isomorphism. We use this result to study the field extensions generated by periodic points of a $p$-adic dynamical system.

math.NT↗
arXiv2005

How close are pth powers in the Nottingham group?

Let F be a field of characteristic p > 0 and let g, h be elements of the Nottingham group N(F) such that g has depth k and gh^{-1} has depth n >= k. We find the best possible lower bound for the depth of g^ph^{-p}.

math.GR↗
arXiv2001

Enumeration of Isomorphism Classes of Extensions of p-adic Fields

Let $Ω$ be an algebraic closure of ${\mathbb Q}_p$ and let $F$ be a finite extension of ${\mathbb Q}_p$ contained in $Ω$. Given positive integers $f$ and $e$, the number of extensions $K/F$ contained in $Ω$ with residue degree $f$ and ramification index $e$ was computed by Krasner. This paper is concerned with the number ${\mathfrak I}(F,f,e)$ of $F$-isomorphism classes of such extensions. We determine ${\mathfrak I}(F,f,e)$ completely when $p^2\nmid e$ and get partial results when $p^2\parallel e$. When $s$ is large, ${\mathfrak I}({\mathbb Q}_p,f,e)$ is equal to the number of isomorphism classes of finite commutative chain rings with residue field ${\mathbb F}_{p^f}$, ramification index $e$, and length $s$.

math.NT↗
arXiv1999

Signed shape tilings of squares

Let T be a tile in the Cartesian plane made up of finitely many rectangles whose corners have rational coordinates and whose sides are parallel to the coordinate axes. This paper gives necessary and sufficient conditions for a square to be tilable by finitely many \Q-weighted tiles with the same shape as T, and necessary and sufficient conditions for a square to be tilable by finitely many \Z-weighted tiles with the same shape as T. The main tool we use is a variant of F. W. Barnes's algebraic theory of brick packing, which converts tiling problems into problems in commutative algebra.

math.CO↗