ConjCoord: a class number congruence modulo 16 for p = 5 (mod 16) and Conjecture coord of Paper V
Source: github.com/jkfarland-bot/ConjCoord/tree/c956f4808ba6c6f15a1…
For primes p = 5 (mod 16), p = a^2 + b^2 with b even, the companion note proves h(-8p) = h(-4p) + 8 [b = +-6 (mod 16)] (mod 16), which settles Conjecture 5.3 of Paper V of the K Project series. The proof writes 4(h(-8p) - h(-4p)) as a combination of representation numbers of x^2 + 2y^2 + 4z^2 and x^2 + 2y^2 + 2z^2. It shows that a weight-2 form Phi built from sieved theta series vanishes modulo 256, by Sturm's theorem and a finite check below the bound 512, and reads off the coefficient at p. This development formalizes the proof with the modular input (Serre-Stark modularity and Sturm's…
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