claimroutineinferred_dep

Selmer conditions as Rédei matrix equations

Rewrites Selmer conditions as Ku+Dλv=δλKu+D_\lambda v=\delta\lambda, Nv=0Nv=0
linear algebra form for the kernel computation
Claim·§3.3 A criterion for one twist·inside lemTwo-descent criterion for twist E_l

Statement (paper)

A pair (3δ∏iqiui, ∏iqivi)\left(3^\delta\prod_iq_i^{u_i},\ \prod_iq_i^{v_i}\right) with δ∈F2\delta\in\F_2 and u,v∈F2Qu,v\in\F_2^Q satisfies the local conditions at the primes qi∈Qq_i\in Q (i.e. agrees with the local torsion class τ1uiτ2vi=(lui3vi,(−l)vi)\tau_1^{u_i}\tau_2^{v_i}=(l^{u_i}3^{v_i},(-l)^{v_i})) if and only if Ku+Dλv=δλKu+D_\lambda v=\delta\lambda and Nv=0Nv=0. The additional requirement that the second coordinate be a square at 33 is equivalent to (λ+s)tv=0(\lambda+s)^{\mathsf t}v=0.

In plain English

A normalized pair (3δ∏qiui,∏qivi)(3^\delta\prod q_i^{u_i},\prod q_i^{v_i}) meets the local conditions at the primes qi∈Qq_i\in Q exactly when Ku+Dλv=δλKu+D_\lambda v=\delta\lambda and Nv=0Nv=0. The extra condition that the second coordinate is a square at 33 is equivalent to the single linear equation (λ+s)tv=0(\lambda+s)^{\mathsf t}v=0.

Why it's here

This turns the local Selmer conditions at odd primes and at 33 into linear algebra over F2\F_2 for the vectors (δ,u,v)(\delta,u,v) from the normalized-pair claim. The matrix equations are what the kernel computation in the next claim uses, together with the hypotheses ker⁡K=⟨1,1B⟩\ker K=\langle\mathbf1,\mathbf1_B\rangle and λ∉im⁡K\lambda\notin\operatorname{im}K, to bound the Selmer group.
Classified routine:
A standard translation of local 2-descent conditions into Legendre-symbol (Rédei-type) matrix equations over F_2, as in Heath-Brown's and Monsky's treatments of congruent-number Selmer groups. It is direct bookkeeping of local Hilbert symbols on top of the normalized-pair claim.

Uses

claimNormalized Selmer pairs with positive coordinates

Used by

claimKernel computation bounds Selmer group by eight

Sources

PaperOpen
Section
§3.3 A criterion for one twist
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sections/03-parity.tex:392
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Selmer conditions as Redei matrix equations" rests on claim-par-descent-3