lemmaroutine

Two-descent criterion for twist E_l

Matrix conditions give 2-Selmer rank 3 for twist ElE_l, with extra class (β,1)(\beta,1)
controls the Selmer group so a point can be lifted
Lemma·par:descent·§3.3 A criterion for one twist

Statement (paper)

Let l>0l>0 be squarefree, prime to 66, with l≡3(mod4)l\equiv3\pmod4 and l≡2(mod3)l\equiv2\pmod3. Suppose that its support QQ contains the nonempty set BB of Lemma  and a prime outside BB, and suppose that ker⁡K=⟨1,1B⟩,λ∉im⁡K. \ker K=\langle\mathbf1,\mathbf1_B\rangle, \qquad \lambda\notin\operatorname{im}K. Then dim⁡F2Sel⁡2(El/Q)=3, \dim_{\F_2}\Sel_2(E_l/\Q)=3, and (β,1)(\beta,1) is a Selmer class outside the subgroup of rational two-torsion classes, in Kummer coordinates (x+3l,x−l)(x+3l,x-l).

In plain English

For squarefree l>0l>0 prime to 6 with l≡3(mod4)l\equiv3\pmod4, l≡2(mod3)l\equiv2\pmod3, whose support QQ contains the set BB and a further prime, if ker⁡K=⟨1,1B⟩\ker K=\langle\mathbf1,\mathbf1_B\rangle and λ∉im⁡K\lambda\notin\operatorname{im}K, then dim⁡F2Sel⁡2(El/Q)=3\dim_{\F_2}\Sel_2(E_l/\Q)=3. Moreover (β,1)(\beta,1) is a Selmer class not coming from rational two-torsion, in Kummer coordinates (x+3l,x−l)(x+3l,x-l).

Why it's here

This turns two matrix conditions on the Rédei-type matrix KK and vector λ\lambda into exact control of the 2-Selmer group of ElE_l. The upper bound comes from reducing the local conditions to linear equations and solving them via the kernel hypotheses. The lower bound comes from exhibiting (β,1)(\beta,1) beyond the two-torsion classes. The extra class (β,1)(\beta,1) is what Lemma 3.4 uses to look for a rational point with x∈βQ×2x\in\beta\Q^{\times2}.
Classified routine:
This is a full 2-descent on the twist E_l, with the local conditions at primes of Q encoded as linear algebra over F_2 via a Rédei-type matrix. That is the standard technique in the style of Heath-Brown and Swinnerton-Dyer for 2-Selmer groups of quadratic twists, specialized here to the paper's matrix K and vector λ.

Uses

Nothing else in the paper

Used by

lemSelmer class lifts to rational point

Inside the proof

claimTwo-torsion Kummer classes are independentclaimLocal Kummer images at odd primes and 3claimNormalized Selmer pairs with positive coordinatesclaimSelmer conditions as Rédei matrix equationsclaimKernel computation bounds Selmer group by eightclaimThe class (beta,1) is a new Selmer class

Sources

PaperOpen
Section
§3.3 A criterion for one twist
Page
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Label
par:descent
Line
sections/03-parity.tex:300
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