The Weil pairing and the dual isogeny #
Let φ : W₁ → W₂ be a separable isogeny of elliptic curves over a separably closed field F, and
N a positive integer invertible in F. The dual φ̂ : W₂ → W₁ is adjoint to φ for the Weil
pairing (Silverman III.8.2):
e_N(φ S, T) = e_N(S, φ̂ T) for `S ∈ W₁[N]` and `T ∈ W₂[N]`,
and consequently e_N(φ S, φ T) = e_N(S, T) ^ deg φ.
The proof is Silverman's. Let g be a function on W₂ with divisor [N]^* (T) - [N]^* (O), the
function from which e_N(·, T) is built. Over a separably closed field the pullback φ^* (T) is
the fibre ∑_{φ P = T} (P), a translate of the kernel, so φ^* ((T) - (O)) is a degree-zero
divisor whose sum is deg φ • P₀ = φ̂ (φ P₀) = φ̂ T for any P₀ over T. Hence
φ^* ((T) - (O)) - ((φ̂ T) - (O)) is the divisor of a function h. Since φ commutes with [N],
the function φ^* g / [N]^* h has divisor [N]^* (φ̂ T) - [N]^* (O), so it computes
e_N(·, φ̂ T). Translation by an N-torsion point fixes [N]^* h, and moves φ^* g to
φ^* (τ_{φ S}^* g), so e_N(S, φ̂ T) = φ^* (τ_{φ S}^* g / g) = e_N(φ S, T).
Main results #
TauCeti.Isogeny.exists_principal_eq_divisorPullback_sub:φ^* ((T) - (O))and(φ̂ T) - (O)differ by a principal divisor.TauCeti.Isogeny.weilPairing_eq_weilPairing_dual:e_N(φ S, T) = e_N(S, φ̂ T).TauCeti.Isogeny.weilPairing_eq_degree_nsmul_weilPairing:e_N(φ S, φ T) = e_N(S, T) ^ deg φ.
References #
- J. Silverman, The Arithmetic of Elliptic Curves, III.6.1 and III.8.2.
φ^* ((T) - (O)) is linearly equivalent to (φ̂ T) - (O): their difference is principal,
over a separably closed field (Silverman III.6.1). Its sum is deg φ • P₀ - φ̂ T for any P₀ over
T, and φ̂ T = φ̂ (φ P₀) = deg φ • P₀.
Adjointness of the dual for the Weil pairing #
The dual isogeny is adjoint to φ for the Weil pairing: e_N(φ S, T) = e_N(S, φ̂ T) for
S ∈ W₁[N] and T ∈ W₂[N], where φ is a separable isogeny over a separably closed field in
which N is invertible (Silverman III.8.2). The images φ S and φ̂ T are given as the torsion
points S' and T'.
A separable isogeny scales the Weil pairing by its degree:
e_N(φ S, φ T) = e_N(S, T) ^ deg φ for S, T ∈ W₁[N], written additively, since
φ̂ (φ T) = deg φ • T (Silverman III.8.2). The images φ S and φ T are given as the torsion
points S' and T'.