Proposition

Proposition 43 ([Odd coefficient detection]). The forms in Lemma 42 ([Explicit weighted theta construction]), after a fixed adjustment of precision, satisfy v(a_M(D)),j(P_D)-C, v(a_M(D))<M j(P_D)≤v(a_M(D))+C. A visible coefficient detects …

Proposition 43 ([Odd coefficient detection]). The forms in Lemma 42 ([Explicit weighted theta construction]), after a fixed adjustment of precision, satisfy v(aM(D)),j(PD)-C, v(aM(D))<M j(PD)≤v(aM(D))+C. A visible coefficient detects a(E^ (D) )=1 , and a fixed positive witness has bounded coefficient valuation as M→∞ . All constants are independent of M,ℓ,D .

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Lemma 41 ([Kummer detection, including CM]). Let C act on T=T_2E as complex…Lemma 42 ([Explicit weighted theta construction]). For ℓ as in Lemma 41…Proposition
Gross--Zagier identifies the height of the genus sum with the product derivative . Since L(E^ (h_*) ,1)≠0 , vanishing of the derivative makes P_D torsion, and a nontorsion P_D is equivalent to a simple zero of E^ (D) . The genus sum lies in a biquadratic field, so a fixed integer kills all possible torsion: reduce prime-to-residue-characteristic torsion at two fixed good primes of different residue characteristic; the residue degrees are at most four. Include that integer in L . On the free twist line, divisibility by 2^ j(P_D) remains divisibility after the integral maps of reduction and λ_M . Restriction and the two quadratic projections have composite multiplication by 2 , so their integral indices have a fixed bound. Equation therefore proves both valuation assertions, with fixed losses only. Lemma 41 ([Kummer detection, including CM]) gives the witness.
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