The Tate–Shafarevich group of a Galois module #
This file defines a general notion of a Tate–Shafarevich group of a Galois module (i.e. an abelian
group A equipped with a continuous action of the absolute Galois group G_K of a field K), as
the intersection of the kernels of the maps Hⁿ(G_K, A) → Hⁿ(G_Kᵥ, A), induced by the map
G_Kᵥ → G_K, where Kᵥ is a field extension of K for each v in an arbitrary indexing set V.
Main definitions #
ContinuousCohomology.tateShafarevich: the Tate–Shafarevich group of a Galois module.
TODOs #
- Add a notation for
ContinuousCohomology.tateShafarevich. - Prove that
ContinuousCohomology.tateShafarevichof a discrete Galois module is torsion. - Define the classical Tate–Shafarevich group in
Mathlib/AlgebraicGeometry/EllipticCurve.
Implementation notes #
Mathematicians seem to use "Tate–Shafarevich group" in at least three distinct contexts. They have
in common the set-up that K is a global field, A is a discrete G_K-module, V is the set of
places of K, and Kᵥ is the completion of K at a place v in V.
The first context is when A is finite. This is used, for example, in the statement of global
Poitou-Tate duality. In this theorem, the Tate–Shafarevich group is defined exactly as in this file.
It agrees with Definition 8.6.2 of [Neukirch–Schmidt–Wingberg].
The second context is when A is the Kˢ-valued points of a group scheme J such as an abelian
variety. Then the Tate–Shafarevich group is usually defined as the intersection of the kernels of
the maps H¹(G_K, J(Kˢ)) → H¹(G_Kᵥ, J(Kᵥˢ)). Note that in particular the module changes as well as
the group, so technically this is not quite what is happening in this definition. However the
inclusion H¹(G_Kᵥ, J(Kᵥˢ)) → H¹(G_Kᵥ, J(Kˢ)) is an isomorphism by the Greenberg approximation
theorem, so the definition in this file is still mathematically correct.
See Remark I.3.10 of [Milne].
The third context is when A is the quotient of a p-adic G_K-module by a G_K-invariant
lattice, which is different from the definition in this PR. See Section 1 of [Flach].
References #
The Tate–Shafarevich group of a Galois module.
Equations
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