arXiv subjects

James M. Borger

Publications and source records attributed to James M. Borger.

3 recordsLinked to original sources

arXiv2015

Witt vectors, semirings, and total positivity

We extend the big and $p$-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the $p$-typical case, it uses positivity with respect to an apparently new basis of the $p$-typical symmetric functions. We also give explicit descriptions of the big Witt vectors of the natural numbers and of the nonnegative reals, the second of which is a restatement of Edrei's theorem on totally positive power series. Finally we give some negative results on the relationship between truncated Witt vectors and $k$-Schur positivity, and we give ten open questions.

math.CO↗

arXiv2002

Conductors and the moduli of residual perfection

Let A be a complete discrete valuation ring with possibly imperfect residue field. The purpose of this paper is to give a notion of conductor for Galois representations over A that generalizes the classical Artin conductor. The definition rests on two general results: there is a moduli space that parametrizes the ways of modifying A so that its residue field is perfect, and any information about a Galois-theoretic object over A can be recovered from its pullback to the (residually perfect) discrete valuation ring corresponding to the generic point of this moduli space.

math.NT↗

arXiv2002

Kato's conductor and generic residual perfection

Let A be a complete discrete valuation ring with possibly imperfect residue field, and let $χ$ be a one-dimensional Galois representation over A. I show that the non-logarithmic variant of Kato's Swan conductor is the same for $χ$ and the pullback of $χ$ to the generic residual perfection of A. This implies the conductor from "Conductors and the moduli of residual perfection" (math.NT/0112305) extends the non-logarithmic variant of Kato's.

math.NT↗