Indeed, due to the Poincar´e uniformisation theorem any Riemann surface can be represented as a quotient of the hyperbolic plane by a discrete group.
Moduli spaces of convex projective structures on surfaces
Let π ∈ V be a uniformising element, and let l = V /(π) be the residue ﬁeld of V .
Moduli schemes of generically simple Azumaya modules
We give a complete list of orbifolds uniformised by discrete nonelementary two-generator subgroups of PSL(2, C) without invariant plane whose generators and their commutator have real traces.
Two-generator Kleinian orbifolds
A discrete rank 1 valuation ν of k(X ) is geometric (sometimes called divisorial in the valuation theory literature) if it has a uniformisation, that is, a pair E ⊂ Y of a normal variety Y with k(Y ) = k(X ), and a prime divisor E ⊂ Y such that ν = multE measures multiplicity along E .
Hypergeometric Equations and Weighted Projective Spaces
Let ν be a geometric valuation with centre on X and uniformisation f : E ⊂ Y → X .
Hypergeometric Equations and Weighted Projective Spaces
It is easy to see that the discrepancy does not depend on the choice of uniformisation and the differential ω .
Hypergeometric Equations and Weighted Projective Spaces
If ν is a geometric valuation of X with uniformisation f : (E ⊂ Y ) → X , and KX + B is Q-Cartier, we deﬁne the discrepancy of ν with respect to the pair (X, B ) as: a(ν, B ) = multE (cid:0)KY − f ∗ (KX + B )(cid:1).
Hypergeometric Equations and Weighted Projective Spaces
It is easy to see that X has canonical singularities if a(ν ) ≥ 0 for all ν which are uniformised by a ﬁxed resolution f : Y → X .
Hypergeometric Equations and Weighted Projective Spaces
Proof: The ﬁrst part is well-known and follows easily from using nonarchimedean uniformisations of the abelian varieties A and A′ that have purely toric reduction at all ℓ ∈ Q and ℓ|N (ρ)pδ (see Chapter III of [Ri3] and Section 3 of [GS] for instance).
On isomorphisms between deformation rings and Hecke rings
This follows from the fact that pro jective determinacy implies that pro jective sets are completely Ramsey and, moreover, that they can be uniformised by pro jective sets (see ).
Infinite asymptotic games
In (2) we may therefore assume that π ∗x = hm for h a local uniformiser at a.
A Theory of Divisors for Algebraic Curves
As a is a non-singular, we can ﬁnd a uniformising element t in the local ring Oa,D of D .
A Theory of Divisors for Algebraic Curves
Clearly, F r is a bijection on points, hence it is Zariski unramiﬁed. F r has algebraic multiplicity p everywhere, as, for any point x ∈ F ′ , we can choose a local uniformiser t at x such that F r∗(t) = tp .
A Theory of Divisors for Algebraic Curves
These isomorphisms give us an identiﬁcation of GL2 (A∞F ) with (D ⊗Q A∞ )× ; • A uniformiser ̟x of OF,x for each ﬁnite place x.
On the modularity of supersingular elliptic curves over certain totally real number fields
As every manifold is locally compact and Hausdorff, hence completely regular, it follows that every manifold is uniformisable ([38, Proposition 11.5]).
Metrisability of Manifolds
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