A hidden clock

0:00A hidden clock

Take quaternions, numbers like a + bi + cj + dk, with rational coordinates and length one: a² + b² + c² + d² = 1. Multiply two of them, and you get another. They form a group. Here is a strange rule. Write the quaternion over a common denominator, in lowest terms. If the denominator is odd, give it color 0. If it’s even, multiply the four numerators. A remainder of 3, mod 4, gives color 1, and a remainder of 1 gives color 2. Now multiply. (1 + i + j + k)/2 has color 2. (1 + i + j − k)/2 has color 1. Their product is j, with color 0. 2 + 1 = 0, mod 3. That’s no accident: colors always add, mod 3. So the color-0 quaternions form a normal subgroup. The group hides a clock with three hours. Where does the clock come from? From the prime 2. The Margulis–Platonov conjecture says that clocks like this can only come from a few special primes. Result 018 in OpenAI’s math repository claims to prove it, over every global field.

1:13Where clocks come from

First, what is a normal subgroup? It’s a subgroup that conjugation can’t move, exactly what a homomorphism sends to the identity, like the color-0 quaternions. A group with no normal subgroups, except the obvious ones, is called simple. Many groups of rational matrices are simple, up to their centers. Take 2-by-2 rational matrices with determinant 1, modulo ±1. That group contains unipotent elements, like (1 1; 0 1), and they generate everything. Groups with elements like these are called isotropic, and for them, simplicity over global fields was fully proved by 2009. Our quaternion group has no unipotent elements. It’s anisotropic. So look at it one prime at a time, over the p-adic numbers. At every odd prime, Hamilton’s quaternions become 2-by-2 matrices, and the group is isotropic there. But at the prime 2, they stay a division algebra, and the length-one quaternions form a compact group. Compact p-adic groups have many finite quotients. Reducing them at the prime 2 lands in a field with four elements. Its three nonzero elements are the clock. In the real picture, the group is a three-dimensional sphere. It’s compact too, but connected, and the three colors are mixed together everywhere on it. So geometry can’t see the clock. Only the prime 2 can. The conjecture says this is the only source. Take an absolutely almost simple, simply connected group over a global field, such as the rationals. It is compact at only finitely many primes. Every normal subgroup that isn’t central is pulled back from an open normal subgroup at those primes. The subgroup can be completely arbitrary, with no topology assumed, and it is still detected there. If there are no such primes, the group is simple, up to its center.

3:07What was known

The question goes back to Martin Kneser, who asked about normal subgroups of these quaternion groups. Vladimir Platonov proposed a general local-to-global version, and Gregory Margulis formulated the precise statement in 1979. He also showed that such normal subgroups have finite index. Then the cases fell, one family at a time. The isotropic groups, finished by 2009. Most anisotropic types, in the 1980s. Quaternions, by Margulis in 1980. Then division algebras in general, completed in 2002 by Yoav Segev and Gary Seitz, building on a reduction by Rapinchuk and Potapchik. That step needed the classification of finite simple groups. Three families remained over number fields: unitary groups of division algebras, called outer type A; triality forms of D₄; and forms of E₆. Over function fields, only the unitary family of the three can be anisotropic, and it was open in every characteristic, including 2.

4:13The claim

Two preprints claim to finish the job. The first, 78 pages, dated September 23, covers number fields. The second, 81 pages, dated October 5, covers function fields, including characteristic 2. Together, they claim the Margulis–Platonov conjecture over every global field. There’s a consequence over number fields. A recent theorem of Andrei Rapinchuk, which assumed the conjecture, would now hold unconditionally. When the primes allowed in the denominators avoid the special primes and have positive density, in a precise sense, every finite-index normal subgroup of these arithmetic groups contains a congruence subgroup.

4:56How the proofs go

The proof starts from what’s already known. A noncentral normal subgroup has finite index, so it contains every product of e-th powers, for some exponent e. The compact groups at the special primes predict what that power subgroup should be. Any gap between the two is a finite group, and the job is to show it’s trivial. For unitary groups of odd-degree division algebras, a nontrivial gap would produce one of two things: an abelian character, like a clock, or a map onto a nonabelian finite simple group. The abelian case is ruled out with an algebraic identity and the known inner type A case. For the nonabelian case, color each rational element by its image in the finite simple group. The paper builds a random, translation-invariant pattern of colors, in which every color is equally likely, using Eisenstein series and mixing in p-adic groups. A recurrence theorem of Furstenberg and Katznelson then forces colors to repeat in a rigid way. A lemma about finite simple groups, checked family by family with the classification, says that can’t happen. Over function fields, that recurrence argument breaks down, so the second paper uses finite Fourier analysis instead. Unitary groups of every degree then follow by induction. Back over number fields, triality D₄ is handled through products of two involutions, and E₆ is reduced to D₄ and type A pieces.

6:23Where it stands

There is no Lean proof for this result. These are preprints, not peer reviewed, 159 pages in all. They rely on many deep theorems, including the classification of finite simple groups, and the second paper builds on the first. We found no public critique, and nothing has been withdrawn. The rational quaternions hide a clock with three hours, set by the prime 2. If the proofs hold, every clock of this kind, in every such group, is set by a prime like that.