# A Two-Witness Criterion for Strong BSD in Positive-Density Non-Semistable Quadratic-Twist Families **Neo.K** **Derived theorem note — Version 0.1, 13 August 2026** > **Status.** This paper formulates and proves a reusable sufficient criterion obtained by > combining existing \(2\)-primary, ordinary/multiplicative, additive-twist, and > supersingular \(p\)-part results. It does not assert novelty or priority. ## Abstract We isolate a finite certificate on a non-semistable elliptic curve \(E/\mathbf Q\) that produces a positive-density set of quadratic twists satisfying the strong Birch--Swinnerton--Dyer conjecture. The criterion is designed for curves whose only additive reduction occurs at \(2\). Two odd multiplicative primes play complementary roles: the prime \(3\) and a second nonsplit multiplicative prime \(\lambda\), both with minimal-discriminant valuation one. These primes provide all ramification witnesses required by the ordinary, multiplicative, additive-twist, and supersingular Iwasawa theorems. The twist support is defined by requiring primes to split completely in a finite abelian \(2\)-extension controlling all bad-prime splitting, while having Frobenius a \(3\)-cycle in the Galois closure of the \(2\)-division cubic. The support set has explicit positive Chebotarev density \[ \frac{[L_E\cap K_E:\mathbf Q]}{3[K_E:\mathbf Q]}. \] The curve \(696.e1\) is recovered as a concrete corollary, with density \(1/24\). --- ## 1. Setup Let \(E/\mathbf Q\) be an elliptic curve with conductor \(N_E\), minimal discriminant \(\Delta_E\), and no rational \(2\)-torsion. Let \[ f_E(x) \] denote the cubic whose roots are the \(x\)-coordinates of the nonzero points of \(E[2]\), and let \(L_E\) be its Galois closure. For an odd prime \(\ell\), put \[ \ell^*=(-1)^{(\ell-1)/2}\ell. \] Define the finite abelian extension \[ K_E = \mathbf Q\left( \zeta_8,\, \sqrt{\ell^*}\ :\ \ell\mid N_E,\ \ell\text{ odd} \right). \] We write \(E^{(q)}\) for the quadratic twist by a positive prime \(q\). --- ## 2. The finite two-witness certificate ### Definition 2.1 We say that \(E\) satisfies the **two-witness BSD certificate** if the following conditions hold. **(T1) \(2\)-primary anchor.** The curve \(E\) is optimal of analytic rank \(0\), its Manin constant is odd, \[ E(\mathbf Q)[2]=0,\qquad \Delta_E<0,\qquad \operatorname{ord}_2L^{(\mathrm{alg})}(E,1)=0, \] and \[ \operatorname{BSD}(E,2) \] is known. **(T2) \(S_3\) two-division field.** The polynomial \(f_E\) is irreducible and \[ \operatorname{Gal}(L_E/\mathbf Q)\simeq S_3. \] **(T3) controlled non-semistability.** The only prime at which \(E\) has additive reduction is \(2\). Every odd bad prime is multiplicative. **(T4) first ramification witness.** The prime \(3\) is multiplicative and \[ v_3(\Delta_E)=1. \] **(T5) nonsplit Fouquet--Wan witness.** There exists an odd prime \[ \lambda\neq3 \] such that \(E\) has nonsplit multiplicative reduction at \(\lambda\) and \[ v_\lambda(\Delta_E)=1. \] **(T6) residual irreducibility.** For every odd prime \(p\), \[ E[p] \] is absolutely irreducible. **(T7) optimality after twisting.** The rational isogeny class of \(E\) is a singleton. The last condition is included explicitly to keep the period comparison independent of any auxiliary isogeny argument. --- ## 3. The Chebotarev support set Let \(C_3\subset S_3\) be the conjugacy class of \(3\)-cycles. ### Definition 3.1 Let \(\mathcal P_E\) be the set of rational primes \(q\), unramified in \(L_EK_E\), such that \[ \operatorname{Frob}_q|_{K_E}=1 \] and \[ \operatorname{Frob}_q|_{L_E}\in C_3. \] ### Lemma 3.2 If \(q\in\mathcal P_E\), then: 1. \(q\equiv1\pmod8\); 2. every prime \(\ell\mid N_E\) splits in \(\mathbf Q(\sqrt q)\); 3. \(f_E\bmod q\) is irreducible; 4. \(q\) is a prime of good ordinary reduction for \(E\). #### Proof The trivial Frobenius condition on \(\mathbf Q(\zeta_8)\subset K_E\) gives \(q\equiv1\pmod8\). For an odd \(\ell\mid N_E\), trivial Frobenius on \(\mathbf Q(\sqrt{\ell^*})\) gives \[ \left(\frac{\ell^*}{q}\right)=1. \] Since \(q\equiv1\pmod4\), quadratic reciprocity rewrites this as \[ \left(\frac q\ell\right)=1, \] which is exactly the splitting condition for \(\ell\) in \(\mathbf Q(\sqrt q)\). The condition at \(2\) follows from \(q\equiv1\pmod8\). The \(3\)-cycle condition is equivalent to irreducibility of the \(2\)-division cubic modulo \(q\). On \(E[2]\), \[ \operatorname{GL}_2(\mathbf F_2)\simeq S_3, \] and a \(3\)-cycle has trace \(1\). Thus \[ a_q(E)\equiv1\pmod2. \] Hence \(a_q(E)\) is odd. Because \(q\equiv1\pmod8\), we have \(q\ge17\). If \(E\) were supersingular at \(q\), then \(q\mid a_q(E)\), and the Hasse bound would force \(a_q(E)=0\), contradiction. Thus \(q\) is ordinary. \(\square\) ### Proposition 3.3 — positive density Let \[ e_E=[L_E\cap K_E:\mathbf Q]. \] Then \[ e_E\in\{1,2\} \] and \[ \boxed{ \delta(\mathcal P_E) = \frac{e_E}{3[K_E:\mathbf Q]}. } \] In particular \(\mathcal P_E\) has positive natural prime density. #### Proof Since \(K_E/\mathbf Q\) is abelian, \(L_E\cap K_E\) is an abelian Galois subextension of the \(S_3\)-extension \(L_E/\mathbf Q\). The only possibilities are \(\mathbf Q\) and the unique quadratic resolvent of \(L_E\). Hence \(e_E\in\{1,2\}\). A \(3\)-cycle acts trivially on the quadratic resolvent. Therefore the condition “identity on \(K_E\)” is compatible with either of the two \(3\)-cycles in \(L_E\). The compositum has degree \[ [L_EK_E:\mathbf Q] = \frac{6[K_E:\mathbf Q]}{e_E}. \] Exactly two elements of its Galois group have identity \(K_E\)-component and \(3\)-cycle \(L_E\)-component. Chebotarev gives \[ \delta(\mathcal P_E) = \frac{2}{6[K_E:\mathbf Q]/e_E} = \frac{e_E}{3[K_E:\mathbf Q]}. \] \(\square\) --- ## 4. Main criterion ### Theorem 4.1 — Two-Witness Non-Semistable Criterion Assume that \(E/\mathbf Q\) satisfies the two-witness BSD certificate (T1)--(T7). Then for every \[ q\in\mathcal P_E \] the strong Birch--Swinnerton--Dyer conjecture holds for the quadratic twist \(E^{(q)}\): \[ \boxed{ \operatorname{BSD}(E^{(q)}). } \] Moreover the set of such primes has natural density \[ \boxed{ \delta(\mathcal P_E) = \frac{[L_E\cap K_E:\mathbf Q]} {3[K_E:\mathbf Q]} >0. } \] #### Proof Fix \(q\in\mathcal P_E\) and write \[ A=E^{(q)}. \] ### Step 1: the \(2\)-part By Lemma 3.2, \[ q\equiv1\pmod4, \] every prime dividing \(N_E\) splits in \(\mathbf Q(\sqrt q)\), and the unique prime \(q\mid d=q\) is inert in the cubic \(2\)-division field. Thus the no-rational-\(2\)-torsion branch of Banwait--Huang's \(2\)-primary twist theorem applies using (T1), and gives \[ L(A,1)\neq0 \] and \[ \operatorname{BSD}(A,2). \] ### Step 2: residual irreducibility after twisting For odd \(p\), \[ A[p]\simeq E[p]\otimes\chi_q. \] Tensoring by a one-dimensional character preserves absolute irreducibility. Hence (T6) gives absolute irreducibility of \(A[p]\) for every odd \(p\). ### Step 3: the additive twist prime \(p=q\) Lemma 3.2 shows that \(q\ge5\) is good ordinary for the base curve \(E\). The additive-twist theorem used in Banwait--Huang's proof of their odd-prime router therefore applies once the ramified auxiliary prime is supplied explicitly. Take \[ \ell=\lambda. \] By (T5), \[ v_\lambda(\Delta_E)=1, \] so \(E[q]\) is residually ramified at \(\lambda\). Since \(q\in\mathcal P_E\), the prime \(\lambda\) splits in \(\mathbf Q(\sqrt q)\), and \(q\neq\lambda\). This supplies the required ramification hypothesis and yields \[ \operatorname{BSD}(A,q). \] ### Step 4: odd good ordinary primes Let \(p\) be an odd good ordinary prime for \(A\). Then \(p\neq\lambda\). The curve \(A\) has multiplicative reduction at \(\lambda\), and local splitting of \(q\) at \(\lambda\) means that the twist does not alter that local representation. Since \[ v_\lambda(\Delta_A)=1, \] the representation \(A[p]\) is ramified at \(\lambda\). Skinner's ordinary/multiplicative theorem applies and gives \[ \operatorname{BSD}(A,p). \] ### Step 5: fixed odd multiplicative primes Let \(p\mid N_E\) be odd. By (T3), \(A\) remains multiplicative at \(p\). If \[ p\neq\lambda, \] use \(\lambda\) as the distinct ramified multiplicative witness; the valuation-one condition in (T5) works for every odd \(p\neq\lambda\). If \[ p=\lambda, \] use the prime \(3\). By (T4), \[ v_3(\Delta_A)=1, \] so \(A[\lambda]\) is ramified at \(3\). Skinner's theorem therefore gives \[ \operatorname{BSD}(A,p) \] for every odd multiplicative bad prime \(p\). ### Step 6: odd good supersingular primes Let \(p\) be an odd good supersingular prime for \(A\). Since \(3\) is bad multiplicative by (T4), we have \(p\neq3\); hence \(p\ge5\). Apply Fouquet--Wan. Absolute irreducibility is supplied by Step 2. At a good supersingular prime \(p\ge5\), the local residual representation is irreducible, so its semisimplification cannot be one of the forbidden one-dimensional character sums. For the auxiliary Steinberg prime take again \[ \ell=\lambda. \] By (T5), \(E\), hence also \(A\), is nonsplit multiplicative at \(\lambda\); the support condition makes the local quadratic twist trivial there. The condition \[ v_\lambda(\Delta_A)=1 \] implies residual ramification for every odd good \(p\neq\lambda\). Thus the Fouquet--Wan residual hypotheses hold and their rank-zero \(p\)-part BSD consequence applies. For the period comparison, (T7) and twisting imply that \(A\) is the unique curve in its rational isogeny class and hence optimal. Prime divisors of the Manin constant are supported at additive reduction primes. The additive primes of \(A\) are \(2\) and \(q\), whereas the present \(p\) is good. Thus the modular and Néron periods have the same \(p\)-adic valuation, and \[ \operatorname{BSD}(A,p) \] holds. ### Step 7: exhaustion The curve \(A\) has: - additive reduction at \(2\) and \(q\); - multiplicative reduction at the odd primes dividing \(N_E\); - good reduction at every other prime. The \(2\)-part is handled in Step 1; \(p=q\) in Step 3; all odd multiplicative primes in Step 5; and all other odd primes are good ordinary or good supersingular, handled in Steps 4 and 6. Thus the BSD formula holds at every finite prime. Since \(L(A,1)\neq0\), the rank-zero Gross--Zagier--Kolyvagin consequences give rank equality and finiteness of \(\Sha(A)\), and the local-to-global formulation of the strong BSD formula yields \[ \operatorname{BSD}(A). \] \(\square\) --- ## 5. The explicit curve \(696.e1\) ### Corollary 5.1 Let \[ E:\quad y^2=x^3+x^2+8x-16 \] be the curve \(696.e1\). Then \(E\) satisfies the two-witness certificate with \[ 3 \quad\text{and}\quad \lambda=29. \] Moreover \[ [K_E:\mathbf Q]=16, \qquad [L_E\cap K_E:\mathbf Q]=2. \] Hence \[ \delta(\mathcal P_E) = \frac{2}{3\cdot16} = \frac1{24}. \] In concrete congruence form, \[ \mathcal P_E = \left\{ q: q\equiv1\pmod{24},\ \left(\frac q{29}\right)=1,\ x^3+x^2+8x-16 \text{ irreducible mod }q \right\}, \] and every \(q\in\mathcal P_E\) satisfies \[ \operatorname{BSD}(E^{(q)}). \] #### Proof The arithmetic certificate is: \[ N_E=2^3\cdot3\cdot29,\qquad \Delta_E=-2^{11}\cdot3\cdot29, \] with additive reduction only at \(2\), split multiplicative reduction at \(3\), nonsplit multiplicative reduction at \(29\), and \[ v_3(\Delta_E)=v_{29}(\Delta_E)=1. \] The remaining conditions (T1), (T2), (T6), and (T7) are exactly the finite data isolated in the explicit \(696.e1\) theorem note. The \(2\)-division Galois closure has quadratic resolvent \[ \mathbf Q(\sqrt{-174}), \] which lies in \[ K_E=\mathbf Q(\zeta_{24},\sqrt{29}), \] so the intersection has degree \(2\), giving density \(1/24\). \(\square\) --- ## 6. What the criterion does and does not say Theorem 4.1 is intentionally a **sufficient criterion**, not a classification. The assumptions \[ v_3(\Delta_E)=v_\lambda(\Delta_E)=1 \] are stronger than necessary. They are chosen because they make the ramification witnesses uniform for every odd prime without an exceptional prime audit. A more general version should replace valuation one by suitable gcd conditions. Likewise, the assumption that the only additive prime is \(2\) avoids the finite but nontrivial Fouquet--Wan local checks at fixed odd additive primes. Removing this hypothesis is the next natural extension. Thus there are two immediate generalizations: 1. **gcd-witness criterion:** replace valuation-one witnesses by a finite set of multiplicative primes whose discriminant valuations have odd-prime-free gcd; 2. **odd-additive extension:** permit finitely many odd additive primes and verify Fouquet--Wan's local residual hypotheses at those fixed primes. These extensions preserve the same global architecture: \[ \boxed{ \text{finite base certificate} \longrightarrow \text{positive-density Chebotarev support} \longrightarrow \text{all-prime BSD routing}. } \] --- ## References - B. S. Banwait and X. Huang, *On the identification of elliptic curves that admit infinitely many twists satisfying the Birch--Swinnerton--Dyer conjecture*, arXiv:2601.16044v3 (2026). - A. A. Burungale, C. Skinner, Y. Tian and X. Wan, *Zeta elements for elliptic curves and applications*, arXiv:2409.01350. - C. 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