We prove Berhuy-Reichstein’s conjecture on the canonical dimension of orthogonal groups showing that for any integer n ≥ 1, the canonical dimension of SO2n+1 and of SO2n+2 is equal to n(n + 1)/2. More precisely, for a given (2n + 1)-dimensional quadratic form φ defined over an arbitrary field F of characteristic 6= 2, we establish certain property of the correspondences on the orthogonal grassm...

We introduce a valuation-theoretic approach to the problem of semistable reduction (i.e., existence of logarithmic extensions on suitable covers) of overconvergent isocrystals with Frobenius structure. The key tool is the quasicompactness of the Riemann-Zariski space associated to the function field of a variety. We also make some initial reductions, which allow attention to be focused on valua...

If q is an algebraic Lie algebra and Q is an algebraic group with Lie algebra q, then ind q equals the transcendence degree of the field of Q-invariant rational functions on q. If q is reductive, then q and q are isomorphic as q-modules and hence ind q = rk q. It is an important invariant-theoretic problem to study index and, more generally, the coadjoint representation for non-reductive Lie al...

We resolve the local semistable reduction problem for overconvergent F -isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree 0). We first introduce a higher-dimensional analogue of the generic radius of convergence for a p-adic differential module, which obeys a convexity property. We then combine this convexity property with a form of the p-adic...

In this paper we discuss the question ”When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same R-place?”. We use this to show that if R contains a dense real closed subfield R′, then the spaces of R-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X) we prove that its space M(K) of R-places is metrizible if and only...

For n = 1, the space of R-places of the rational function field R(x1, . . . , xn) is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of R-places of the rational function field R(x, y) is not metrizable. We explain how the proof generalizes to show that the space of R-places of any finitely generated formally real field extensi...

Let K be a field and let A be a finitely generated prime K-algebra. We generalize a result of Smith and Zhang, showing that if A is not PI and does not have a locally nilpotent ideal, then the extended centre of A has transcendence degree at most GKdim(A) − 2 over K. As a consequence, we are able to show that if A is a prime K-algebra of quadratic growth, then either the extended centre is a fi...

For n = 1, the space of R-places of the rational function field R(x1, . . . , xn) is homeomorphic to the real projective line. For n ≥ 2, the structure is much more complicated. We prove that the space of R-places of the rational function field R(x, y) is not metrizable. We explain how the proof generalizes to show that the space of R-places of any finitely generated formally real field extensi...

We prove a pro-p Hom-form of the birational anabelian conjecture for function fields over sub-p-adic fields. Our starting point is the corresponding Theorem of Mochizuki in the case of transcendence degree 1.

Let ξ1, . . . , ξr be complex numbers with K = Q(ξ1, . . . , ξr) having transcendence degree r − 1 over Q. Consider the equation a1x1 + · · ·+ akxk = 1, (1) in which the ai’s are fixed elements of K ×, no proper subsum ai1xi1 + · · ·+ aijxij vanishes, and we seek solutions xi ∈ Γ = ξ1, . . . , ξr . It is well–known that (1) has only finitely many solutions; we present here an elementary proof o...