A pointwise 2-converse for elliptic curves with rational two-torsion

1. The even construction has two outputs

Arrange quadratic twists as $h_x=h_0\prod_q(q^*)^{\lambda_q(x)}$ for $x\in\mathbb F_2^b$, with linear forms $\lambda_q$. Here $q^*=(-1/q)q$, the new primes avoid a fixed finite support, and all vertices have the same sign and fixed local squareclasses. Binary tests at minimum normalized coefficient depth, together with finite prime networks, provide nonvanishing and a uniform valuation upper bound at every $x\ne0$. Theorem 3.2 returns them to the base. Proposition 9.7 handles $h_0=1$ with $s_2(E)=0$; Proposition 9.6 supplies a uniform estimate for twists having an appropriate partition certificate, needed later on the odd side.

Propositions 9.6–9.7 · p. 78

2. The order of the odd-side companion choices matters

The fixed $h_*$ defines the Heegner-point detector $P_h$. The varying $k'(h)$, with uniformly bounded prime count, is used only to prove a divisibility lower bound for $P_h$. Afterwards, a separate $k$ is fixed once and for all for the final interpolation over $K=\mathbb Q(\sqrt k)$. Treating $h_*$, $k'(h)$, and $k$ as interchangeable auxiliary twists would use the estimate needed for odd normalization circularly. See the sections on Proposition 9.10 and Proposition 9.14.

Proposition 9.10 · pp. 80–81

3. Rank equality and finiteness of the entire Shafarevich–Tate group

The two interpolation arguments first give $a(E)=s_2(E)\in\{0,1\}$. The established Gross–Zagier–Kolyvagin forward implication then yields $r(E)=a(E)$ and finiteness of the whole group $\operatorname{Sha}(E/\mathbb Q)$. The forward statement recorded in Skinner’s introduction adds no reduction restriction here. The semistability and other conditions in that paper’s converse theorem are not hypotheses of this forward step.

[Sk20, §1, Gross–Zagier–Kolyvagin implication, print p. 330 / PDF p. 2]

Proof of Theorem 1.1; Remark 9.15 · pp. 87–88

1. Separate rational divisibility from uniform integrality

Put $\Lambda=\mathbb Z_2[[t]]$, $\mathcal O=\widehat{\Lambda_{(2)}}$, and $G=(\mathbb Z/2)^b$. The manuscript writes the wedge of the real-place class and a Kato class as a scalar $u$ in the determinant of the positive global complex. Tracing into the lattice of one fixed weight-two eigenform constructs $2^{C_{\rm int}}u\in\mathcal O[G]$ with an exponent independent of the number of auxiliary primes. This does not uniformly identify the integral lattices of varying twisted eigenforms.

Kato’s Theorem 12.5(3) supplies height-one divisibility after inverting 2; Burungale–Tian’s Theorem 2.6 supplies the CM case. Including corrections at the fixed bad primes gives $u_x\in\Lambda[1/2]$ for every character constituent. Kato’s stronger integral statement 12.5(4), which requires $p\ne2$ and an image condition, is not used. Finally Lemma 3.3 applies $\mathcal O\cap\Lambda[1/2]=\Lambda$ to each group-basis coefficient, with no additional $|G|$ loss, to obtain

Central evaluation is performed on $\Lambda[1/2]$ first; there is no evaluation at $t=0$ on the whole completed localization $\mathcal O$.

[Kat04, §§8.9–8.12, print pp. 185–186 / PDF pp. 70–71]

[Kat04, Theorem 12.5(3)–(4), print p. 222 / PDF p. 107]

[BT25, Theorem 2.6; Remark 2.7, p. 5]

§§3.3–3.5; Proposition 3.7 · pp. 10–15

2. A sufficiently large cube has another vertex with the base’s finite digits

At a corank-zero constituent, Proposition 3.7 identifies the vanishing of $u_x(0)$ with that of $L^*(h_x)$ and gives $|v(u_x(0))-D(h_x)|\le C_{\rm loc}$. The nonzero vertices have corank zero by the known analytic-rank-zero theorem; the base has it by hypothesis. Fix $M-C_{\rm int}>B+C_{\rm loc}$ and apply Lemma 3.4 to the constant coefficient of $2^{C_{\rm int}}u$. For $b>2^M-1$, some $x\ne0$ satisfies

If $u_0(0)=0$, or if its valuation exceeds the upper bound, this contradicts $v(u_x(0))\le B+C_{\rm loc}$. Thus the base is nonvanishing too. Uniformity concerns $C_{\rm int},C_{\rm loc},B$, not the prime sizes or the precision needed to realize a family.

Lemmas 3.3–3.4 · pp. 7–8

§3.6 · p. 16

3. Supplying the upper bound at all nonzero vertices

Proposition 7.1 compares central values with squares of weight-$3/2$ coefficients; Theorem 3.1 supplies the lower bound that fixes their minimum normalized depth. A unit test gives $L^*(h)\ne0$ and $v(L^*(h))\le2w(h)+C$. The finite networks of Proposition 7.9 and Theorem 8.3 turn differences in quadratic-residue edge bits into contractions of prime labels and make the test equal to one at every nonzero address. Subtracting $s(h_x)\ge0$ gives the required upper bound for $D(h_x)$. No prescribed curve is discarded into a density-zero exceptional set.

Proposition 7.1 · pp. 53–54; Lemma 7.5 · pp. 60–61

Theorem 8.3 · pp. 68–69

1. A varying companion supplies an already nonvanishing central value

For each $h$ already known to have analytic rank one, Lemma 9.8 chooses a negative odd fundamental discriminant $k'=k'(h)$. It satisfies $(k',2Nh)=1$, is a square at all places dividing $2Nh$, has $\omega(k')\le A$, and defines neither an extra-unit field nor the CM field of $E$. Bounded-factor completion in Proposition 6.2 and finite label preparations control this prime count. Apply Proposition 9.6 to the nonvanishing parameter $hk'$ to obtain

The actual prime sizes may depend on $h$. No odd minimum-depth normalization has been used at this point.

Lemma 9.8 · pp. 79–80; Proposition 6.2 · p. 48

2. Height comparison cancels the additional Sha term

Write $J_h(k')$ for the index of the conductor-$|h|$ ring-class sum over $\mathbb Q(\sqrt{k'})$. Corollary 4.1 and Lemma 9.9 give

The second identity applies Cai–Shu–Tian’s Theorem 1.1 to the two sums. The level primes split and the conductor is coprime to the level. The original formula’s $c\sqrt{|D|}$ factor differs precisely between conductor $|h|$ and conductor one. Cancelling the derivative and common rank-one regulator leaves the variable factor already included in the normalization of $L^*(hk')$. Excluding extra-unit fields fixes the unit factor as well.

Combine the three inequalities using $w(hk')=w(h)+w(k')$ and $w(k')\le A$:

Only now apply Lemma 7.5. An odd unit simultaneously detects analytic rank one and the upper bound $j(P_h)\le w(h)+C$.

Corollary 4.1 · p. 17

Lemma 9.9 · p. 80

[CST14, Theorem 1.1 and Heegner hypotheses, print pp. 2524–2525 / PDF pp. 3–4]

Proposition 7.4; Lemma 7.5 · pp. 59–61

1. Fix the final companion before the cube dimension

If the prepared odd detector detects the base, the argument is finished. Otherwise take its nonempty minimum seed. A separate even test selects a negative discriminant $k$ with $L(E^{(k)},1)\ne0$. It is a square at $2N$, avoids the old support and required torsion ramification, and excludes the exceptional imaginary fields. Keep $K=\mathbb Q(\sqrt k)$ fixed thereafter. Require every new prime in $h$ to split in $K$. The forward theorem gives $s_2(E^{(k)})=0$, so the quadratic corank identity yields

This is not an integral direct-sum decomposition of Selmer groups.

Lemma 2.2 · p. 5

2. Make separately nonzero polynomials units at the same vertex

The even test reads $hk$, while the odd test reads $h$. Separate nonzero terminal polynomials at an address need not admit the same nonzero assignment of residue symbols. Theorem 8.3 adds primary prime networks and makes both tests equal to one on the same finite graph. No new prime is active at $x=0$.

When the residual fields are quadratic, the two sides of a rational 2-isogeny supply good simple primes and donors. Full rational two-torsion needs a separate argument: Lemma 9.12 ensures a nonempty witness for the even test restricted to $K$-split primes, and Lemma 9.13 corrects the oriented total. These are substantive steps beyond generic prime selection.

After choosing $b$ and the networks, select an actual stage of the odd coefficients realizing the required test tables and precision on that finite weight range. The moving detecting prime $\ell$ and the precision may change; the minimum normalized depth, fixed $k$, and valuation bound do not. A single stage for all cube dimensions is unnecessary.

Theorem 8.3 · pp. 68–69

Lemmas 9.12–9.13 · pp. 82–85

§§9.5–9.7 · pp. 81–87

3. Feed the index upper bound into ring-class interpolation

At every nonzero vertex, $a(E^{(h)})=1$ and $a(E^{(hk)})=0$. The two unit bounds and the height comparison for fixed $k$ give

The finite lengths $s(h),s(hk)$ are nonnegative, so $2J_h(k)-4w(h)-s(h)-s(hk)\le C$. These are the inputs of Theorem 4.2: for a fixed nonexceptional $K$ satisfying the split Heegner condition at $2N$, full Selmer corank one at the base, and arbitrarily large cubes whose nonzero vertices have a simple product zero and this uniform upper bound, the base product $L(E,s)L(E^{(k)},s)$ has a simple zero.

The interpolation uses complexes retaining full Kummer conditions and actual maps to Galois cochains to produce an integral family with a bounded clearing factor, $f_xu_x=d_x$. Proposition 5.14 uniformly controls the valuation of $f_0(0)\ne0$ and the negative Laurent degrees at finite precision. Lemma 5.17 matches finitely many coefficient digits with the base. A vanishing base derivative would give $u_0(0)=0$, contradicting the known valuation bound at a nonzero vertex. As the manuscript emphasizes, general Selmer duality or the Gross–Zagier formula alone does not supply this simultaneous integral family.

Finally $L(E^{(k)},1)\ne0$ gives $a(E)=1$. The passage to finiteness of all of Sha is the last step of the main theorem.

Theorem 4.2 · p. 17

Propositions 5.14–5.15; Lemmas 5.16–5.17; proof of Theorem 4.2 · pp. 45–47