BACKGROUND
Percutaneous aortic valve replacement represents an endovascular alternative to conventional open heart surgery without the need for sternotomy, aortotomy, or cardiopulmonary bypass.
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OBJECTIVES
Transcatheter aortic valve replacement (AVR) is a promising approach to aortic valve disease. The implications of this new therapy are not entirely known. We describe the potential for the… (More)

Suppose that f (resp. g) is a modular form of integral (resp. half-integral) weight with coefficients in the ring of integers OK of a number field K. For any ideal m ⊂ OK , we bound the first prime p… (More)

v det([1− ρ(Frobv)q v ]|W Iv )−1, W Iv = subspace of W fixed by inertia at v. Grothendieck gave a related description of ζ∗ X using continuous p-adic representations GF → GL(H ét,c(XF ,Qp)) =: GL(W… (More)

S h → G R Ad → GL(g) Concretely, C× acts on gC via the characters z/z, 1, z/z. (2) ad(h(±i)) is a Cartan involution of G(R). (3) There is no Q-simple factor Gi of G such that S h → G R → G i,R is… (More)

§1.1. Motivation. The purpose of these notes is to explain the definition and basic properties of the Néron model A of an abelian variety A over a global or local field K. We also give some idea of… (More)

1.1. Shimura varieties are like a brightly colored toy. The motivation for their study comes from considering moduli spaces of Hodge structures. LetX be a smooth, complex, projective variety of… (More)

Fix ρ : GQ,S → GL2(k) absolutely irreducible, and let ρ : GQ,S → GL2(R) be the universal deformation. We’re interested in the map R → Tm for some Hecke algebra defined in terms of ρ. Note that the… (More)

Remark. Apropos of reduction mod p: If V is a Qp-vector space and G ⊂ GL(V ) is a compact subgroup, then there exists a G-fixed lattice in V for the following reason. Pick any lattice L ⊂ V . Then… (More)