From 7848802dea2c5e9d3aca43840c6a8a10aaa5b1df Mon Sep 17 00:00:00 2001 From: Taksh Date: Sun, 23 Aug 2026 20:44:02 +0530 Subject: [PATCH] Attach the right witness to the 857.567 record in 87a The [HMR2019] row quoted 3^4*5^4*7^4*13^2*29^4*53^2*109^2 as yielding rd <= 857.5662. That product is a 24-digit integer, so it cannot be a root discriminant, and it is not the one behind this record: it is the discriminant of Martin's degree-8 field in HMR2019 section 3.3.1, whose 8th root is 913.4927 -- the value on the row above. 857.5662 comes from section 3.3.3, a different, degree-12 totally real field with rd_K < 770.6432, cut at one prime above 13 of norm 13: rd_K * 13^(1/24) = 857.56620..., which the paper calls a new record with savings by a factor of 13^(1/24). The "8-th root ... rank 8" description belonged to 3.3.1 as well. The record value itself is unchanged. Reported in issue 150. --- constants/87a.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/constants/87a.md b/constants/87a.md index 1952990..66e10f1 100644 --- a/constants/87a.md +++ b/constants/87a.md @@ -16,8 +16,8 @@ Upper bounds on $C_{87}$ come from explicit constructions of infinite tamely ram | ----- | --------- | -------- | | $1058.565$ | [Mar1978] | Martinet's original construction of infinite $2$-class field towers of totally real number fields. | | $954.293$ | [HM2002] | Hajir–Maire, refined Golod–Shafarevich with tame ramification. | -| $913.493$ | [Mar2006] | Martin, further refinement of the [HM2002] construction. | -| $857.567$ | [HMR2019] | Hajir–Maire–Ramakrishna, "cutting towers" via the refined Golod–Shafarevich criterion; explicit example is an $8$-th root class-field tower over the totally real field of [HM2002] with $2$-class group of rank $8$, yielding $\mathrm{rd} \le 3^4\cdot 5^4\cdot 7^4\cdot 13^2\cdot 29^4\cdot 53^2\cdot 109^2 \le 857.5662\dots$ Current record. | +| $913.493$ | [Mar2006] | Martin, further refinement of the [HM2002] construction; the degree-$8$ field of [HMR2019, §3.3.1] has discriminant $3^4\cdot 5^4\cdot 7^4\cdot 13^2\cdot 29^4\cdot 53^2\cdot 109^2$ and root discriminant $< 913.4927$. | +| $857.567$ | [HMR2019] | Hajir–Maire–Ramakrishna, "cutting towers" via the refined Golod–Shafarevich criterion. The totally real example [HMR2019, §3.3.3] is a *degree-$12$* field $\mathrm{K}$ with $\mathrm{rd}_{\mathrm{K}} < 770.6432$, cut at a single prime above $13$ of norm $13$, giving $\mathrm{rd}_{\mathrm{K}_S^{[1]}} = \mathrm{rd}_{\mathrm{K}}\cdot 13^{\frac{1}{12}(1-\frac{1}{2})} < 857.5662\dots$ — a saving of a factor $13^{1/24}$. Current record. | ## Known lower bounds