Permanence criteria for semi-free profinite groups

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual permanence properties of free groups carry over to semi-free groups. Using this, we conclude that if k is a separably closed field, then many field extensions of k((x,y)) have free absolute Galois groups.Comment: 24 page

Applications of patching to quadratic forms and central simple algebras

This paper provides applications of patching to quadratic forms and central simple algebras over function fields of curves over henselian valued fields. In particular, we use a patching approach to reprove and generalize a recent result of Parimala and Suresh on the u-invariant of p-adic function fields, for p odd. The strategy relies on a local-global principle for homogeneous spaces for rational algebraic groups, combined with local computations.Comment: 48 pages; connectivity now required in the definition of rational group; beginning of Section 4 reorganized; other minor change

Analytic curves in algebraic varieties over number fields

We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and $p$-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces, of these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200

Number Fields Ramified at One Prime

Abstract. For G a finite group and p a prime, a G-p field is a Galois number field K with Gal(K/Q) ∼ = G and disc(K) = ±pa for some a. We study the existence of G-p fields for fixed G and varying p. For G a finite group and p a prime, we define a G-p field to be a Galois number field K ⊂ C satisfying Gal(K/Q) ∼ = G and disc(K) = ±pa for some a. Let KG,p denote the finite, and often empty, set of G-p fields. The sets KG,p have been studied mainly from the point of view of fixing p and varying G; see [Har94], for example. We take the opposite point of view, as we fix G and let p vary. Given a finite group G, we let PG be the sequence of primes where each prime p is listed |KG,p | times. We determine, for various groups G, the first few primes in PG and their corresponding fields. Only the primes p dividing |G | can be wildly ramified in a G-p field, and so the sequences PG which are infinite are dominated by tamely ramified fields. In Sections 1, 2, and 3, we consider the cases when G is solvable with length 1, 2, and ≥ 3 respectively, using mainly class field theory. Section 4 deals wit

Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v