The standardness conjecture for prime ideals in solvable uniform Iwasawa algebras

Let p>2p>2 be a prime, let KK be a finite extension of Qp\mathbb{Q}_p with ring of integers O\mathcal{O}, and let GG be a compact pp-adic Lie group. Its completed group algebra is

OG:=limNO[G/N],\mathcal{O}G:=\varprojlim_N \mathcal{O}[G/N],

where NN ranges over the open normal subgroups of GG. A prime ideal PP of OG\mathcal{O}G is standard if there is a closed normal subgroup HH of GG such that, writing G0:=G/HG_0:=G/H, the group G0G_0 is torsionfree, H1PH-1\subseteq P, and the image of PP in OG0\mathcal{O}G_0 is centrally generated. It is virtually standard if POUP\cap\mathcal{O}U is a finite intersection of standard prime ideals of OU\mathcal{O}U for some open normal subgroup UU of GG.

The standardness conjecture. Let GG be a solvable, uniform pro-pp group, and let PP be a prime ideal in OG\mathcal{O}G. Then PP is virtually standard, and moreover if pPp\in P then PP is standard.

This conjecture seeks to reduce all prime ideals in solvable uniform Iwasawa algebras to augmentation ideals and centrally generated ideals. The supplied text gives no evidence that the conjecture has been resolved.

Sources & referencesView supporting material

Primary source

Adam Jones, “Primitive ideals in rational, nilpotent Iwasawa algebras”, arXiv:2102.04165 (2021).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.