Kummer classes are isotropic for the Weil cup product #
Let E be an elliptic curve over a field K and m an integer invertible in K. The Kummer map
δ : E(K) → H¹(K, E[m]) and the cup product along the Weil pairing
e_m : E[m] × E[m] → μ_m combine into a pairing E(K) × E(K) → H²(K, μ_m). This file proves
that it vanishes: the image of the Kummer map is isotropic for the Weil cup product,
δ P ∪ δ Q = 0 for all P, Q ∈ E(K)
(WeierstrassCurve.torsionH1WeilPairing_kummerMap_kummerMap_eq_zero). Over a nonarchimedean local
field it says that the local condition δ(E(K)) of the m-Selmer group is isotropic for local
Tate duality (WeierstrassCurve.localTateWeilPairing_kummerMap_kummerMap_eq_zero); there it is
half of the statement that this local condition is its own annihilator.
Main results #
WeierstrassCurve.torsionH1WeilPairing_kummerMap_kummerMap_eq_zero: the cup product of two Kummer classes along the Weil pairing vanishes inH²(K, μ_m).
Implementation notes #
Write Kˢ for the separable closure of K, G_K for its Galois group, [m]^* for pullback of
divisors along multiplication by m, and τ_S^* for translation by S on functions. The Weil
pairing is e_m(S, T) = τ_S^* g / g for any g with divisor [m]^* (T) - [m]^* (O)
(TauCeti.Isogeny.weilPairing).
For a point R of E(Kˢ) with m² R ∈ E(K), fix a function f with divisor
[m]^* (-m R) - [m]^* (O) - ((-m² R) - (O)). Then g(σ) = τ_{-R}^* (σ f / f) has divisor
[m]^* (-m a(σ)) - [m]^* (O), where a(σ) = σ R - R, and
g(σ τ) = g(σ) · τ_{a(σ)}^* σ(g(τ)). Read in the group of pairs (S, g) with
div g = [m]^* (-m S) - [m]^* (O), multiplied by (S, g) (S', g') = (S + S', g · τ_S^* g'), this
says that σ ↦ (a(σ), g(σ)) is a cocycle lifting a; and the commutator of (S, c) and
(S', g') for S ∈ E[m] is the Weil pairing value τ_S^* g' / g'.
Let P' ∈ E(Kˢ) with m P' = P, let Q' ∈ E(Kˢ) with m Q' = Q, and let R ∈ E(Kˢ) with
m R = -Q'. The lifts g₁, g₂, g₃ attached to P', R and P' + R lift the Kummer
cocycles of m P, -Q and m P - Q for multiplication by m². The lift g₁ is constant, and
g₁ g₂ = c · g₃ for a cochain c : G_K → (Kˢ)ˣ; comparing the cocycle identities shows that the
coboundary of c is the cup cocycle (σ, τ) ↦ e_m(σ P' - P', σ (τ Q' - Q')). The cochain c is
continuous, being constant on the cosets of the open subgroup fixing finitely many elements of
Kˢ. So the cup product vanishes in H²(K, (Kˢ)ˣ), hence in H²(K, μ_m), which injects into it
by Hilbert's theorem 90.
This is the theta-group argument: the obstruction to lifting a class of H¹(K, E[m]) to a Mumford
theta group is a quadratic refinement of the Weil cup product, and it vanishes on Kummer classes.
References #
- Yu. G. Zarhin, Noncommutative cohomology and Mumford groups, Math. Notes 15 (1974), 241–244.
- B. Poonen and E. Rains, Self cup products and the theta characteristic torsor, Math. Res. Lett. 18 (2011), 1305–1318, §4.
- J. S. Milne, Arithmetic Duality Theorems, 2nd ed., I §3.
- J. Silverman, The Arithmetic of Elliptic Curves, III.8 and X.1.
Kummer classes are isotropic for the Weil cup product: for all points P and Q of
E(K), the cup product of their Kummer classes along the Weil pairing vanishes in H²(K, μ_m),
for every m invertible in K.