claimroutineinferred_dep

Local Kummer images at odd primes and 3

Describes local Kummer images of ElE_l at odd primes and at 3
local conditions cutting out the Selmer group
Claim·§3.3 A criterion for one twist·inside lemTwo-descent criterion for twist E_l

Statement (paper)

For every odd prime qq, the group El(Qq)/2El(Qq)E_l(\Q_q)/2E_l(\Q_q) has order four. For an odd prime q∤3lq\nmid 3l, the local Kummer image in coordinates (x+3l,x−l)(x+3l,x-l) is exactly the unramified subgroup of pairs with even coordinate valuations. For each prime qi∣lq_i\mid l, the classes τ1,τ2\tau_1,\tau_2 generate the entire local image. At 33, the local image is (Q3×/Q3×2)×{1}(\Q_3^\times/\Q_3^{\times2})\times\{1\}.

In plain English

For each odd prime qq, El(Qq)/2El(Qq)E_l(\Q_q)/2E_l(\Q_q) has order four. For q∤3lq\nmid 3l its Kummer image in coordinates (x+3l,x−l)(x+3l,x-l) is the unramified pairs (even valuations); for q∣lq\mid l it is generated by τ1,τ2\tau_1,\tau_2; and at 33 it is (Q3×/Q3×2)×{1}(\Q_3^\times/\Q_3^{\times2})\times\{1\}.

Why it's here

This gives the explicit local conditions at all odd primes and at 3 that cut out the Selmer group, so the later steps can translate Selmer membership into conditions on pairs (a,b)(a,b) and the matrix KK. At primes dividing ll, the independent torsion classes τ1,τ2\tau_1,\tau_2 from the previous claim fill the order-four local group, so no further local condition arises there.}</br>
Classified routine:
Standard local computation in a full 2-descent: ∣E(Qq)/2E(Qq)∣=∣E(Qq)[2]∣=4|E(\Q_q)/2E(\Q_q)|=|E(\Q_q)[2]|=4 for odd qq, the image at good primes is the unramified subgroup, and at primes of multiplicative/additive reduction it is filled out by the torsion classes (as in Silverman, AEC Ch. X). The paper applies this to the specific twist ElE_l, with the independence of τ1,τ2\tau_1,\tau_2 taken from the previous claim.

Uses

claimTwo-torsion Kummer classes are independent

Used by

claimNormalized Selmer pairs with positive coordinates

Sources

PaperOpen
Section
§3.3 A criterion for one twist
Page
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sections/03-parity.tex:328
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Local Kummer images at odd primes and at 3" rests on claim-par-descent-1