Zeta Functions of Lie Rings Archive

Welcome to the archive of zeta functions of Lie rings. Click on any of the links below to bring up a PDF file detailing the zeta function calculated.

Each PDF file contains the local zeta functions, the functional equation they satisfy (if there is one), the abscissa of convergence of the global zeta function and the order of the pole, the ghost zeta function, and the natural boundary if it has one. Ghost zeta functions are introduced in [3], and natural boundaries in [5].

Most of the names of the Lie rings are (hopefully) standard. Others that may not be familiar are as follows:

• Q₅ was the name given to the only Lie algebra of dimension 5 that I had not previously encountered. The 5 signifies the dimension of it, but there is no significance in the letter Q.
• L_W was the example I came up with whose ideal zeta function has no functional equation. L_W2 is a similar, larger example, whose ideal zeta function also has no functional equation.
• L_3,3 is another Lie ring I came up with, and predates L_W. By removing y₂ from the presentation, you obtain L_W.
• Fil₄ is the unique filiform Lie ring of class 4 not isomorphic to the maximal class Lie ring M₄.
• L(E) is du Sautoy's elliptic curve example.
• H(O_K2) and H(O_K3) are the Heisenberg Lie ring over a quadratic and cubic number field respectively.

We also use a couple of classifications of nilpotent Lie algebras over R to obtain presentations of Lie rings. g_6,n arises from the nth Lie algebra in the list of nilpotent Lie algebras over R of dimension 6 in [9], and names such as g_37B correspond to 7-dimensional Lie algebras as listed in [6].

The calculations of the zeta functions of the plane crystallographic groups (see also [4]) are also included. These are not the only zeta functions of groups recorded here; via the Mal'cev correspondence, calculations for nilpotent Lie rings translate into calculations of zeta functions of torsion-free finitely-generated nilpotent groups, for all but finitely many primes. Calculating these zeta functions however is often easier in the Lie-ring setting (the notable exception being the aforementiond crystallographic groups), which is where almost all calculations to date have been done.

I believe all these calculations are correct but offer no absolute guarantee. If anyone has any comments, queries or suggestions, please let me know.

Class 2 nilpotent

Ring	Rank	Presentation	Ideals	All subrings
H	3	⟨ x, y, z | [x, y] = z ⟩	PDF [8]	PDF [8]
H × Z	4	⟨ x, y, z, w | [x, y] = z ⟩	PDF [8]	PDF [12]
G3	5	⟨ z, x1, x2, y1, y2 | [z, x1] = y1, [z, x2] = y2 ⟩	PDF [8]	PDF [7]
H × Z2	5	⟨ x, y, z, a, b | [x, y] = z ⟩	PDF [8]	PDF [12]
H ×Z H	5	⟨ x1, x2, x3, x4, y | [x1, x3] = y, [x2, x4] = y ⟩	PDF [8]	PDF [12]
F2,3	6	⟨ x1, x2, x3, y1, y2, y3 | [x1, x2] = y1, [x1, x3] = y2, [x2, x3] = y3 ⟩	PDF [8]	PDF [10]
g6,4	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x5, [x1, x3] = x6, [x2, x4] = x6 ⟩	PDF [8]	PDF [12]
H2	6	⟨ x1, x2, y1, y2, z1, z2 | [x1, y1] = z1, [x2, y2] = z2 ⟩	PDF [8]	PDF [10]
H(OK2)	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x4] = x6, [x1, x3] = x5, [x2, x3] = x6, [x2, x4] = αx5 + βx6 ⟩†	PDF [8]
H × Z3	6	⟨ x, y, a, b, c, z | [x, y] = z ⟩	PDF [8]	PDF [12]
G4	7	⟨ z, x1, x2, x3, y1, y2, y3 | [z, x1] = y1, [z, x2] = y2, [z, x3] = y3 ⟩	PDF [7]	PDF [7]
g17	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x7, [x3, x4] = x7, [x5, x6] = x7 ⟩	PDF [8]
g27A	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x6, [x1, x4] = x7, [x3, x5] = x7 ⟩	PDF [5]
g27B	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x6, [x1, x5] = x7, [x2, x3] = x7, [x3, x4] = x6 ⟩	PDF [5]
g37C	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x2, x3] = x6, [x2, x4] = x7, [x3, x4] = x5 ⟩	PDF [5]
g37D	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x1, x3] = x7, [x2, x4] = x7, [x3, x4] = x6 ⟩	PDF [5]
T4	7	⟨ x1, x2, x3, x4, y1, y2, y3 | [x1, x2] = y1, [x2, x3] = y2, [x3, x4] = y3 ⟩	PDF [10]	PDF [12]
G5	9	⟨ z, x1, x2, x3, x4, y1, y2, y3, y4 | [z, x1] = y1, [z, x2] = y2, [z, x3] = y3, [z, x4] = y4 ⟩	PDF [7]	PDF [7]
H3	9	⟨ x1, x2, x3, y1, y2, y3, z1, z2, z3 | [x1, y1] = z1, [x2, y2] = z2, [x3, y3] = z3 ⟩	PDF [10]
H(OK3)	9	⟨ x, y, z | [x, y] = z ⟩‡	PDF [8, 10]
L(E)	9	⟨ x1, x2, x3, x4, x5, x6, y1, y2, y3 | [x1, x4] = y3, [x1, x5] = y1, [x1, x6] = y2, [x2, x4] = y2, [x2, x6] = x1, [x3, x4] = x1, [x3, x5] = x1 ⟩	PDF [11]
H4	12	⟨ x1, x2, x3, x4, y1, y2, y3, y4, z1, z2, z3, z4 | [x1, y1] = z1, [x2, y2] = z2, [x3, y3] = z3, [x4, y4] = z4 ⟩	PDF [12]

Class 3 nilpotent

Ring	Rank	Presentation	Ideals	All subrings
M3	4	⟨ z, x1, x2, x3 | [z, x1] = x2, [z, x2] = x3 ⟩	PDF [10]	PDF [10]
F3,2	5	⟨x1, x2, y1, z1, z2 | [x1, x2] = y1, [x1, y1] = z1, [x2, y1] = z2 ⟩	PDF [12]	PDF [12]
M3 × Z	5	⟨ z, x1, x2, a, x3 | [z, x1] = x2, [z, x2] = x3 ⟩	PDF [12]	PDF [12]
Q5	5	⟨ x1, x2, x3, x4, x5 | [x1, x2] = x3, [x1, x3] = x5, [x2, x4] = x5 ⟩	PDF [12]	PDF [12]
F3,2 × Z	6	⟨ x1, x2, y, a, z1, z2 | [x1, x2] = y, [x1, y] = z1, [x2, y] = z2 ⟩	PDF [12]
g6,6	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x6, [x1, x3] = x4, [x1, x4] = x5, [x2, x3] = x5 ⟩	PDF [12]
g6,7	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x3] = x4, [x1, x4] = x5, [x2, x3] = x6 ⟩	PDF [12]
g6,8	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x3 + x5, [x1, x3] = x4, [x2, x5] = x6 ⟩	PDF [12]
g6,9	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x3, [x1, x3] = x4, [x1, x5] = x6, [x2, x3] = x6 ⟩	PDF [12]
g6,10(γ)	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x4, [x1, x4] = x6, [x1, x3] = x5, [x2, x3] = x6, [x2, x4] = αx5 + βx6 ⟩†	PDF [12]
g6,12	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x3] = x5, [x1, x5] = x6, [x2, x4] = x6 ⟩	PDF [12]	PDF [12]
g6,13	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x5, [x1, x3] = x4, [x1, x4] = x6, [x2, x5] = x6 ⟩	PDF [12]
g6,14(γ)	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x3] = x4, [x1, x4] = x6, [x2, x3] = x5, [x2, x5] = γx6 ⟩	PDF [12]
LW	6	⟨ z, w1, w2, x1, x2, y1 | [z, w1] = x1, [z, w2] = x2, [z, x1] = y1 ⟩	PDF [12]	PDF [12]
M3 × Z2	6	⟨ z, x1, x2, a1, a2, x3 | [z, x1] = x2, [z, x2] = x3 ⟩	PDF [12]	PDF [12]
Q5 × Z	6	⟨ x1, x2, x3, a, x4, x5 | [x1, x2] = x4, [x1, x4] = x5, [x2, x3] = x5 ⟩	PDF [12]	PDF [5]
g137B	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x1, x5] = x7, [x2, x4] = x7, [x3, x4] = x6, [x3, x6] = x7 ⟩	PDF [5]
g137C	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x1, x4] = x6, [x1, x6] = x7, [x2, x3] = x6, [x3, x5] = −x7 ⟩	PDF [5]
g137D	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x1, x4] = x6, [x1, x6] = x7, [x2, x3] = x6, [x2, x4] = x7, [x3, x5] = −x7 ⟩	PDF [5]
g147A	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x4, [x1, x3] = x5, [x1, x6] = x7, [x2, x5] = x7, [x3, x4] = x7 ⟩	PDF [5]
g147B	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x4, [x1, x3] = x5, [x1, x4] = x7, [x2, x6] = x7, [x3, x5] = x7 ⟩	PDF [5]
g157	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x3, [x1, x3] = x7, [x2, x4] = x7, [x5, x6] = x7 ⟩	PDF [5]
g247B	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x4, [x1, x3] = x5, [x1, x4] = x6, [x3, x5] = x7 ⟩	PDF [13]
g257A	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x3, [x1, x3] = x6, [x1, x5] = x7, [x2, x4] = x6 ⟩	PDF [5]
g257B	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x3, [x1, x3] = x6, [x1, x4] = x7, [x2, x5] = x7 ⟩	PDF [5]
g257C	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x3, [x1, x3] = x6, [x2, x4] = x6, [x2, x5] = x7 ⟩	PDF [5]
g257K	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x1, x5] = x6, [x2, x5] = x7, [x3, x4] = x6 ⟩	PDF [5]
H × M3	7	⟨ t, z, u, x1, v, x2, x3 | [t, u] = v, [z, x1] = x2, [z, x2] = x3 ⟩	PDF [12]
L3,3	7	⟨ z, w1, w2, x1, x2, y1, y2 | [z, w1] = x1, [z, w2] = x2, [z, x1] = y1, [z, x2] = y2 ⟩	PDF [12]
M3 ×Z M3	7	⟨ z1, z2, w1, w2, x1, x2, y | [z1, w1] = x1, [z2, w2] = x2, [z1, x1] = y, [z2, x2] = y ⟩	PDF [5]
H × Q5	8	⟨ t, u, x1, x2, v, x3, x4, x5 | [t, u] = v, [x1, x2] = x3, [x1, x3] = x5, [x2, x4] = x5 ⟩	PDF [12]
LW2	8	⟨ z, w1, w2, w3, x1, x2, x3, y | [z, w1] = x1, [z, w2] = x2, [z, w3] = x3, [z, x1] = y ⟩	PDF [12]
M3 × M3	8	⟨ z1, z2, w1, w2, x1, x2, y1, y2 | [z1, w1] = x1, [z1, x1] = y1, [z2, w2] = x2, [z2, x2] = y2 ⟩	PDF [13]
H × g6,12	9	⟨ t, u, x1, x2, x3, x4, v, x5, x6 | [t, u] = v, [x1, x3] = x5, [x1, x5] = x6, [x2, x4] = x6 ⟩	PDF [12]
H × LW	9	⟨ t, z, u, w1, w2, v, x1, x2, y | [t, u] = v, [z, w1] = x1, [z, w2] = x2, [z, x1] = y ⟩	PDF [13]
G3 × Q5	10	⟨ c, x1, x2, x4, a1, a2, x3, x5, b1, b2 | [x1, x2] = x3, [x1, x3] = x5, [x2, x4] = x5, [c, a1] = b1, [c, a2] = b2 ⟩	PDF [5]
H × H × M3	10	⟨ z, x1, t1, t2, u1, u2, x2, v1, v2, x3 | [z, x1] = x2, [z, x2] = x3, [t1, u1] = v1, [t2, u2] = v2 ⟩	PDF [13]

Class 4 nilpotent

Ring	Rank	Presentation	Ideals	All subrings
Fil4	5	⟨ z, x1, x2, x3, x4 | [z, x1] = x2, [z, x2] = x3, [z, x3] = x4, [x1, x2] = x4 ⟩	PDF [12]
M4	5	⟨ z, x1, x2, x3, x4 | [z, x1] = x2, [z, x2] = x3, [z, x3] = x4 ⟩	PDF [10]	PDF [10]
Fil4 × Z	6	⟨ z, x1, x2, a, x3, x4 | [z, x1] = x2, [z, x2] = x3, [z, x3] = x4, [x1, x2] = x4 ⟩	PDF [12]
g6,15	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x3 + x5, [x1, x3] = x4, [x1, x4] = x6, [x2, x5] = x6 ⟩	PDF [12]
g6,16	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x3] = x4, [x1, x4] = x5, [x1, x5] = x6, [x2, x3] = x5, [x2, x4] = x6 ⟩	PDF [12]
g6,17	6	⟨ x1, x2, x3, x4, x5, x6 | [x1, x2] = x3, [x1, x3] = x4, [x1, x4] = x5, [x2, x5] = x6 ⟩	PDF [12]
M4 × Z	6	⟨ z, x1, x2, a, x3, x4 | [z, x1] = x2, [z, x2] = x3, [z, x3] = x4 ⟩	PDF [12]
g1357A	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x4, [x1, x4] = x5, [x1, x5] = x7, [x2, x3] = x5, [x2, x6] = x7, [x3, x4] = −x7 ⟩	PDF [5]
g1357B	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x4, [x1, x4] = x5, [x1, x5] = x7, [x2, x3] = x5, [x3, x4] = −x7, [x3, x6] = x7 ⟩	PDF [5]
g1357C	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x4, [x1, x4] = x5, [x1, x5] = x7, [x2, x3] = x5, [x2, x4] = x7, [x3, x4] = −x7, [x3, x6] = x7 ⟩	PDF [5]
g1357G	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x3, [x1, x4] = x6, [x1, x6] = x7, [x2, x3] = x5, [x2, x5] = x7 ⟩	PDF [5]
g1357H	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x3, [x1, x4] = x6, [x1, x6] = x7, [x2, x3] = x5, [x2, x5] = x7, [x2, x6] = x7, [x3, x4] = −x7 ⟩	PDF [5]
g1457A	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x1, x5] = x6, [x1, x6] = x7, [x3, x4] = x7 ⟩	PDF [5]
g1457B	7	⟨ x1, x2, x3, x4, x5, x6, x7 | [x1, x2] = x5, [x1, x5] = x6, [x1, x6] = x7, [x2, x5] = x7, [x3, x4] = x7 ⟩	PDF [5]

Soluble

Ring	Rank	Presentation	Ideals	All subrings
tr2(Z)	3	(2 × 2 upper-triangular matrices over Z)	PDF [13]	PDF [13]
tr3(Z)	6	(3 × 3 upper-triangular matrices over Z)	PDF [13]
tr4(Z)	10	(4 × 4 upper-triangular matrices over Z)	PDF [13]
tr5(Z)	15	(5 × 5 upper-triangular matrices over Z)	PDF [13]
tr6(Z)	21	(6 × 6 upper-triangular matrices over Z)	PDF [13]
tr7(Z)	28	(7 × 7 upper-triangular matrices over Z)	PDF [13]

Insoluble

Ring	Rank	Presentation	Ideals	All subrings
sl2(Z)	3	⟨ f, e, h | [h, e] = 2e, [h, f] = −2f, [e, f] = h ⟩	PDF [2, 12]	PDF [1, 2]

Plane crystallographic groups

Group	Presentation	Normal subgroups	All subgroups
p1	⟨ x, y | [x, y] ⟩	PDF [4]	PDF [4]
p2	⟨ x, y, r | [x, y], r2, xr = x−1, yr = y−1 ⟩	PDF [4]	PDF [4]
pm	⟨ x, y, m | [x, y], m2, xm = x, ym = y−1 ⟩	PDF [4]	PDF [4]
pg	⟨ x, y, t | [x, y], t2 = x, yt = y−1 ⟩	PDF [4]	PDF [4]
p2mm	⟨ x, y, p, q | [x, y], [p, q], p2, q2, xp = x, xq = x−1, yp = y−1, yq = y ⟩	PDF [4]	PDF [4]
p2mg	⟨ x, y, m, t | [x, y], t2, m2 = y, xt = x, xm = x−1, yt = y−1, mt = m−1 ⟩	PDF [4]	PDF [4]
p2gg	⟨ x, y, u, v | [x, y], u2 = x, v2 = y, xv = x−1, yu = y−1, (uv)2 ⟩	PDF [4]	PDF [4]
cm	⟨ x, y, t | [x, y], t2, yt = y−1, xt = xy ⟩	PDF [4]	PDF [4]
c2mm	⟨ x, y, m, r | [x, y], m2, r2, ym = y−1, xm = xy, yr = y−1, xr = x−1, rm = r−1 ⟩	PDF [4]	PDF [4]
p4	⟨ x, y, r | [x, y], r4, yr = x−1, xr = y ⟩	PDF [4]	PDF [4]
p4mm	⟨ x, y, r, m | [x, y], r4, m2, yr = x−1, xr = y, xm = y, rm = r−1 ⟩	PDF [4]	PDF [4]
p4gm	⟨ x, y, r, t | [x, y], r4, t2, yr = x−1, xr = y, xt = y, rt = r−1x−1 ⟩	PDF [4]	PDF [4]
p3	⟨ x, y, r | [x, y], r3, xr = x−1y, yr = x−1 ⟩	PDF [4]	PDF [4]
p31m	⟨ x, y, r, t | [x, y], r2, t2, (tr)3, xr = x, yt = y, xt = x−1y, yr = xy−1 ⟩	PDF [4]	PDF [4]
p3m1	⟨ x, y, r, m | [x, y], r3, m2, rm = r−1, xr = x−1y, yr = x−1, xm = x−1, ym = x−1y  ⟩	PDF [4]	PDF [4]
p6	⟨ x, y, r | [x, y], r6, xr = y, yr = x−1y ⟩	PDF [4]	PDF [4]
p6mm	⟨ x, y, r, m | [x, y], r6, m2, yr = x−1y, xr = y, xr = y, xm = x−1, ym = x−1y, rm = r−1y ⟩	PDF [4]	PDF [4]

Footnotes

^†γ will be a nonzero squarefree integer, and α and β depend on γ:

• If γ ≡ 1 (mod 4), then α = ¼(γ − 1) and β = 1.
• if γ ≡ 2 or 3 (mod 4), then α = γ and β = 0.

^‡This is a presentation as a O_L-Lie ring. I don't know of any way to write down the integral basis of an arbitrary cubic number field, and a presentation as a Z-Lie ring would seem to depend on this.

References

1. Marcus du Sautoy and Gareth Taylor. The zeta function of sl₂(Z) and resolution of singularities. Math. Proc. Camb. Phil. Soc., 132:57-73, 2002.
2. M.P.F. du Sautoy. The zeta function of sl₂(Z). Forum Mathematicum, 12:197-221, 2000.
3. M.P.F. du Sautoy and F.J. Grunewald. Zeta functions of groups: zeros and friendly ghosts. American Journal Of Mathematics, 124:1-48, 2002.
4. M.P.F. du Sautoy, J.J. McDermott and G.C. Smith. Zeta functions of crystallographic groups and analytic continuation. Proc. London Math. Soc. (3), 79:511-534, 1999.
5. M.P.F. du Sautoy, L. Woodward. Zeta Functions of Groups and Rings. Lecture Notes in Mathematics 1925, 2008.
6. M.-P. Gong, Classification Of Nilpotent Lie Algebras Of Dimension 7 (Over Algebraically Closed Fields and R), PhD thesis, University of Waterloo, 1998. Available electronically at http://etd.uwaterloo.ca/etd/mpgong1998.pdf.
7. D. Grenham. Some topics in nilpotent group theory. PhD thesis, University of Oxford, 1988.
8. F.J. Grunewald, D. Segal, and G.C. Smith. Subgroups of finite index in nilpotent groups. Invent. Math., 93:185-223, 1988.
9. L. Magnin. Sur les algèbres de Lie nilpotentes de dimension ≤ 7. J. Geom. Phys., 3:119--144, 1986.
10. Gareth Taylor. Zeta Functions of Algebras and Resolution of Singularities. PhD thesis, University of Cambridge, 2001.
11. Christopher Voll. Zeta functions of groups and enumeration in Bruhat-Tits buildings. PhD thesis, University of Cambridge, 2002.
12. Luke Woodward. Zeta functions of groups: computer calculations and functional equations, DPhil thesis, University of Oxford, 2005.
13. Luke Woodward, not published elsewhere.

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Errata 2013-06-07: a number of errors in the PDFs in this archive have come to light:

• The ghosts of the zeta functions counting all subrings in Q₅, g_6,4 and T₄ were incorrectly reported as friendly when they are in fact unfriendly. In all three cases, the unfriendly factor was of the form −1 ± Y + Y². The ghosts were all correct; it was the determination of whether they were friendly or unfriendly that was wrong in these three cases.
• The PDF containing the zeta function counting ideals in L_W2 had the wrong presentation. The presentation of L_W had been given instead.
• The zeta function counting ideals in G₃ was given incorrectly in its PDF. An incorrect extra factor of ζ_p(3s − 3) had been multipled in.
• The PDF containing the zeta function counting ideals in H × M₃ had the wrong presentation. The relation [t,u] = z should have been [t,u] = v.
• The presentation of the crystallographic group p2mm was missing a comma in the two PDFs.
• The location of the natural boundary in the zeta functions counting ideals in g_257K, M₃ × M₃, tr₅(Z) and tr₆(Z) was reported as being at 'Re(s) = Re(s) = ...'. The duplicate Re(s) has been removed.
• g_37B is isomorphic to T₄, so the former has been removed.

Additionally, a couple of changes were made:

• A number of Lie rings have now had their presentations split over two lines.
• The numerator polynomials of a couple of zeta functions have been shortened by multiplying back in factors that had previously been removed. It happens that multiplying these factors in makes the polynomials shorter.

These corrections have now all been applied.

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