Peculiarities in low dimensions or low order or etc

Question

I have been pondering about certain conjectures and theorems viewed as either low vs high dimensions, or smaller vs larger primes, or anything of the sort "low vs high order". Let me mention a couple of such mathematical phenomena that might be more familiar.

Poincaré's conjecture (now a theorem) in dimension $3$ persisted much longer than in higher dimensions.

Congruence modulo primes for the partition function $p(n)$ lingers for primes $p=2, 3$ while a recent work on Maass forms settles such for higher primes.

Hoping that these citations shed light, I like to ask:

QUESTION. Do you know of conjectures (problems) which manifested to be either notoriously harder or unsolved for "lower dimensions/orders/primes" compared to their "higher dimensional/order/prime" cousins?

Answer 1

In general, Galois representations $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\operatorname{GL}_d(\mathbb{Z}_\ell)$ are tamely ramified at primes $p\ge d+1$, which makes it much easier to analyze what's happening at large primes. To give a concrete example, consider Ogg's formula that relates the conductor and discriminant of an elliptic curve $E/\mathbb{Q}$: $$ \operatorname{ord}_p(N_E) = \operatorname{ord}_p(\Delta_E)+1-m_p, $$ where $m_p$ is the number of irreducible components on the fiber of the Neron model at $p$. This is quite easy to prove for $p\ge5$, Ogg proved it for $p=3$ in 1967, and Saito finally proved it for $p=2$ in 1988. (Actually, they proved the analogous formula over all number fields.)

Answer 2

In probability theory, critical percolation on the integer lattice is known to hold in dimensions $2$ and $\ge 19$, but as far as I know, it remains open in dimensions 3 through 18. (I recall hearing that the techniques used for high dimensions could perhaps, with sufficient hard work, be extended down to dimension 16 or so, but not further.)

See this nice discussion by Louigi Addario-Berry.

Answer 3

The whole "chaos" program would fit into this category. When statistical physics (i.e. dynamics of large number of particles aka N-body problem for large N) and erogdic theory were developed in early 20th century, there was a wide belief that dynamical systems are in some sense generically ergodic, called the "Ergodic hypothesis".

Almost half a century later KAM (Kolmogorov-Arnold-Moser) theorem resolved the issue in negative, showing (loosely) that for low dimensional systems, such as the three-body problem, generic perturbations of integrable systems would not lead to ergodicity. Rather, the phase space remains a mix of chaotic and ordered zones.

Answer 4

Finding an $n\times n$ magic square with entries consecutive primes is not hard for $n>3$, compared to $n=3$.

Answer 5

All irreducible spherical buildings (i.e. with finite Weyl group) of rank greater than 2 are associated to simple algebraic or classical groups. This is not the case for rank $\leq$ 2.

See https://en.wikipedia.org/wiki/Building_(mathematics)