# 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.
• LW was the example I came up with whose ideal zeta function has no functional equation. LW2 is a similar, larger example, whose ideal zeta function also has no functional equation.
• L3,3 is another Lie ring I came up with, and predates LW. By removing y2 from the presentation, you obtain LW.
• 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(OK2) and H(OK3) 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. g6,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 g37B 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

RingRankPresentationIdealsAll subrings
H3⟨ xyz | [xy] = z ⟩PDF [8]PDF [8]
H × Z4⟨ xyzw | [xy] = z ⟩PDF [8]PDF [12]
G35⟨ zx1x2y1y2 | [zx1] = y1, [zx2] = y2 ⟩PDF [8]PDF [7]
H × Z25⟨ xyzab | [xy] = z ⟩PDF [8]PDF [12]
H ×Z H5⟨ x1x2x3x4y | [x1x3] = y, [x2x4] = y ⟩PDF [8]PDF [12]
F2,36⟨ x1x2x3y1y2y3 | [x1x2] = y1, [x1x3] = y2, [x2x3] = y3 ⟩PDF [8]PDF [10]
g6,46⟨ x1x2x3x4x5x6 | [x1x2] = x5, [x1x3] = x6, [x2x4] = x6 ⟩PDF [8]PDF [12]
H26⟨ x1x2y1y2z1z2 | [x1y1] = z1, [x2y2] = z2 ⟩PDF [8]PDF [10]
H(OK2)6⟨ x1x2x3x4x5x6 | [x1x4] = x6, [x1x3] = x5, [x2x3] = x6, [x2x4] = αx5 + βx6 ⟩PDF [8]
H × Z36⟨ xyabcz | [xy] = z ⟩PDF [8]PDF [12]
G47⟨ zx1x2x3y1y2y3 | [zx1] = y1, [zx2] = y2, [zx3] = y3 ⟩PDF [7]PDF [7]
g177⟨ x1x2x3x4x5x6x7 | [x1x2] = x7, [x3x4] = x7, [x5x6] = x7 ⟩PDF [8]
g27A7⟨ x1x2x3x4x5x6x7 | [x1x2] = x6, [x1x4] = x7, [x3x5] = x7 ⟩PDF [5]
g27B7⟨ x1x2x3x4x5x6x7 | [x1x2] = x6, [x1x5] = x7, [x2x3] = x7, [x3x4] = x6 ⟩PDF [5]
g37C7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x2x3] = x6, [x2x4] = x7, [x3x4] = x5 ⟩PDF [5]
g37D7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x1x3] = x7, [x2x4] = x7, [x3x4] = x6 ⟩PDF [5]
T47⟨ x1x2x3x4y1y2y3 | [x1x2] = y1, [x2x3] = y2, [x3x4] = y3 ⟩PDF [10]PDF [12]
G59⟨ zx1x2x3x4y1y2y3y4 | [zx1] = y1, [zx2] = y2, [zx3] = y3, [zx4] = y4 ⟩PDF [7]PDF [7]
H39⟨ x1x2x3y1y2y3z1z2z3 | [x1y1] = z1, [x2y2] = z2, [x3y3] = z3 ⟩PDF [10]
H(OK3)9⟨ xyz | [xy] = z ⟩PDF [8, 10]
L(E)9⟨ x1x2x3x4x5x6y1y2y3 | [x1x4] = y3, [x1x5] = y1, [x1x6] = y2, [x2x4] = y2, [x2x6] = x1, [x3x4] = x1, [x3x5] = x1 ⟩PDF [11]
H412⟨ x1x2x3x4y1y2y3y4z1z2z3z4 | [x1y1] = z1, [x2y2] = z2, [x3y3] = z3, [x4y4] = z4 ⟩PDF [12]

### Class 3 nilpotent

RingRankPresentationIdealsAll subrings
M34⟨ zx1x2x3 | [zx1] = x2, [zx2] = x3 ⟩PDF [10]PDF [10]
F3,25x1x2y1z1z2 | [x1x2] = y1, [x1y1] = z1, [x2y1] = z2 ⟩PDF [12]PDF [12]
M3 × Z5⟨ zx1x2ax3 | [zx1] = x2, [zx2] = x3 ⟩PDF [12]PDF [12]
Q55⟨ x1x2x3x4x5 | [x1x2] = x3, [x1x3] = x5, [x2x4] = x5 ⟩PDF [12]PDF [12]
F3,2 × Z6⟨ x1x2yaz1z2 | [x1x2] = y, [x1y] = z1, [x2y] = z2 ⟩PDF [12]
g6,66⟨ x1x2x3x4x5x6 | [x1x2] = x6, [x1x3] = x4, [x1x4] = x5, [x2x3] = x5 ⟩PDF [12]
g6,76⟨ x1x2x3x4x5x6 | [x1x3] = x4, [x1x4] = x5, [x2x3] = x6 ⟩PDF [12]
g6,86⟨ x1x2x3x4x5x6 | [x1x2] = x3 + x5, [x1x3] = x4, [x2x5] = x6 ⟩PDF [12]
g6,96⟨ x1x2x3x4x5x6 | [x1x2] = x3, [x1x3] = x4, [x1x5] = x6, [x2x3] = x6 ⟩PDF [12]
g6,10(γ)6⟨ x1x2x3x4x5x6 | [x1x2] = x4, [x1x4] = x6, [x1x3] = x5, [x2x3] = x6, [x2x4] = αx5 + βx6 ⟩PDF [12]
g6,126⟨ x1x2x3x4x5x6 | [x1x3] = x5, [x1x5] = x6, [x2x4] = x6 ⟩PDF [12]PDF [12]
g6,136⟨ x1x2x3x4x5x6 | [x1x2] = x5, [x1x3] = x4, [x1x4] = x6, [x2x5] = x6 ⟩PDF [12]
g6,14(γ)6⟨ x1x2x3x4x5x6 | [x1x3] = x4, [x1x4] = x6, [x2x3] = x5, [x2x5] = γx6 ⟩PDF [12]
LW6⟨ zw1w2x1x2y1 | [zw1] = x1, [zw2] = x2, [zx1] = y1 ⟩PDF [12]PDF [12]
M3 × Z26⟨ zx1x2a1a2x3 | [zx1] = x2, [zx2] = x3 ⟩PDF [12]PDF [12]
Q5 × Z6⟨ x1x2x3ax4x5 | [x1x2] = x4, [x1x4] = x5, [x2x3] = x5 ⟩PDF [12]PDF [5]
g137B7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x1x5] = x7, [x2x4] = x7, [x3x4] = x6, [x3x6] = x7 ⟩PDF [5]
g137C7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x1x4] = x6, [x1x6] = x7, [x2x3] = x6, [x3x5] = −x7 ⟩PDF [5]
g137D7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x1x4] = x6, [x1x6] = x7, [x2x3] = x6, [x2x4] = x7, [x3x5] = −x7 ⟩PDF [5]
g147A7⟨ x1x2x3x4x5x6x7 | [x1x2] = x4, [x1x3] = x5, [x1x6] = x7, [x2x5] = x7, [x3x4] = x7 ⟩PDF [5]
g147B7⟨ x1x2x3x4x5x6x7 | [x1x2] = x4, [x1x3] = x5, [x1x4] = x7, [x2x6] = x7, [x3x5] = x7 ⟩PDF [5]
g1577⟨ x1x2x3x4x5x6x7 | [x1x2] = x3, [x1x3] = x7, [x2x4] = x7, [x5x6] = x7 ⟩PDF [5]
g247B7⟨ x1x2x3x4x5x6x7 | [x1x2] = x4, [x1x3] = x5, [x1x4] = x6, [x3x5] = x7 ⟩PDF [13]
g257A7⟨ x1x2x3x4x5x6x7 | [x1x2] = x3, [x1x3] = x6, [x1x5] = x7, [x2x4] = x6 ⟩PDF [5]
g257B7⟨ x1x2x3x4x5x6x7 | [x1x2] = x3, [x1x3] = x6, [x1x4] = x7, [x2x5] = x7 ⟩PDF [5]
g257C7⟨ x1x2x3x4x5x6x7 | [x1x2] = x3, [x1x3] = x6, [x2x4] = x6, [x2x5] = x7 ⟩PDF [5]
g257K7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x1x5] = x6, [x2x5] = x7, [x3x4] = x6 ⟩PDF [5]
H × M37⟨ tzux1vx2x3 | [tu] = v, [zx1] = x2, [zx2] = x3 ⟩PDF [12]
L3,37⟨ zw1w2x1x2y1y2 | [zw1] = x1, [zw2] = x2, [zx1] = y1, [zx2] = y2 ⟩PDF [12]
M3 ×Z M37⟨ z1z2w1w2x1x2y | [z1w1] = x1, [z2w2] = x2, [z1x1] = y, [z2x2] = y ⟩PDF [5]
H × Q58⟨ tux1x2vx3x4x5 | [tu] = v, [x1x2] = x3, [x1x3] = x5, [x2x4] = x5 ⟩PDF [12]
LW28⟨ zw1w2w3x1x2x3y | [zw1] = x1, [zw2] = x2, [zw3] = x3, [zx1] = y ⟩PDF [12]
M3 × M38⟨ z1z2w1w2x1x2y1y2 | [z1w1] = x1, [z1x1] = y1, [z2w2] = x2, [z2x2] = y2 ⟩PDF [13]
H × g6,129⟨ tux1x2x3x4vx5x6 | [tu] = v, [x1x3] = x5, [x1x5] = x6, [x2x4] = x6 ⟩PDF [12]
H × LW9⟨ tzuw1w2vx1x2y | [tu] = v, [zw1] = x1, [zw2] = x2, [zx1] = y ⟩PDF [13]
G3 × Q510⟨ cx1x2x4a1a2x3x5b1b2 | [x1x2] = x3, [x1x3] = x5, [x2x4] = x5, [ca1] = b1, [ca2] = b2 ⟩PDF [5]
H × H × M310⟨ zx1t1t2u1u2x2v1v2x3 | [zx1] = x2, [zx2] = x3, [t1u1] = v1, [t2u2] = v2 ⟩PDF [13]

### Class 4 nilpotent

RingRankPresentationIdealsAll subrings
Fil45⟨ zx1x2x3x4 | [zx1] = x2, [zx2] = x3, [zx3] = x4, [x1x2] = x4 ⟩PDF [12]
M45⟨ zx1x2x3x4 | [zx1] = x2, [zx2] = x3, [zx3] = x4 ⟩PDF [10]PDF [10]
Fil4 × Z6⟨ zx1x2ax3x4 | [zx1] = x2, [zx2] = x3, [zx3] = x4, [x1x2] = x4 ⟩PDF [12]
g6,156⟨ x1x2x3x4x5x6 | [x1x2] = x3 + x5, [x1x3] = x4, [x1x4] = x6, [x2x5] = x6 ⟩PDF [12]
g6,166⟨ x1x2x3x4x5x6 | [x1x3] = x4, [x1x4] = x5, [x1x5] = x6, [x2x3] = x5, [x2x4] = x6 ⟩PDF [12]
g6,176⟨ x1x2x3x4x5x6 | [x1x2] = x3, [x1x3] = x4, [x1x4] = x5, [x2x5] = x6 ⟩PDF [12]
M4 × Z6⟨ zx1x2ax3x4 | [zx1] = x2, [zx2] = x3, [zx3] = x4 ⟩PDF [12]
g1357A7⟨ x1x2x3x4x5x6x7 | [x1x2] = x4, [x1x4] = x5, [x1x5] = x7, [x2x3] = x5, [x2x6] = x7, [x3x4] = −x7 ⟩PDF [5]
g1357B7⟨ x1x2x3x4x5x6x7 | [x1x2] = x4, [x1x4] = x5, [x1x5] = x7, [x2x3] = x5, [x3x4] = −x7, [x3x6] = x7 ⟩PDF [5]
g1357C7⟨ x1x2x3x4x5x6x7 | [x1x2] = x4, [x1x4] = x5, [x1x5] = x7, [x2x3] = x5, [x2x4] = x7, [x3x4] = −x7, [x3x6] = x7 ⟩PDF [5]
g1357G7⟨ x1x2x3x4x5x6x7 | [x1x2] = x3, [x1x4] = x6, [x1x6] = x7, [x2x3] = x5, [x2x5] = x7 ⟩PDF [5]
g1357H7⟨ x1x2x3x4x5x6x7 | [x1x2] = x3, [x1x4] = x6, [x1x6] = x7, [x2x3] = x5, [x2x5] = x7, [x2x6] = x7, [x3x4] = −x7 ⟩PDF [5]
g1457A7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x1x5] = x6, [x1x6] = x7, [x3x4] = x7 ⟩PDF [5]
g1457B7⟨ x1x2x3x4x5x6x7 | [x1x2] = x5, [x1x5] = x6, [x1x6] = x7, [x2x5] = x7, [x3x4] = x7 ⟩PDF [5]

### Soluble

RingRankPresentationIdealsAll 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

RingRankPresentationIdealsAll subrings
sl2(Z)3⟨ feh | [he] = 2e, [hf] = −2f, [ef] = h ⟩PDF [2, 12]PDF [1, 2]

### Plane crystallographic groups

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

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, g6,4 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 LW2 had the wrong presentation. The presentation of LW had been given instead.
• The zeta function counting ideals in G3 was given incorrectly in its PDF. An incorrect extra factor of ζp(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 g257K, M3 × M3, tr5(Z) and tr6(Z) was reported as being at 'Re(s) = Re(s) = ...'. The duplicate Re(s) has been removed.
• g37B is isomorphic to T4, so the former has been removed.