Geometric structure of class two nilpotent groups and subgroup growth
aa r X i v : . [ m a t h . G R ] F e b The geometri stru ture of lass two nilpotentgroups and subgroup growthPirita PaajanenS hool of Mathemati sSouthampton UniversityUniversity RoadSouthampton SO17 1BJ, UKp.m.paajanensoton.a .ukO tober 26, 2018Abstra tIn this paper we derive an expli it expression for the normal zetafun tion of lass two nilpotent groups whose asso iated Pfa(cid:30)an hyper-surfa e is smooth. In parti ular, we show how the lo al zeta fun tiondepends on ounting F p -rational points on related varieties, and wedes ribe the varieties that an appear in su h a de omposition. Asa orollary, we also establish expli it results on the degree of polyno-mial subgroup growth in these groups, and we study the behaviour ofpoles of this zeta fun tion. Under ertain geometri onditions, we also on(cid:28)rm that these fun tions satisfy a fun tional equation.1 Introdu tionZeta fun tions of a (cid:28)nitely generated group G were introdu ed in [12℄ asa non- ommutative analogue to the Dedekind zeta fun tion of a number(cid:28)eld. They are used to study the arithmeti and asymptoti properties ofthe sequen e of numbers a n ( G ) = |{ H ≤ f G : | G : H | = n }| . The fa t that G is (cid:28)nitely generated ensures that a n ( G ) is (cid:28)nite for all n .The term subgroup growth is used to des ribe the study of the sequen es a n ( G ) and s n ( G ) = P ni =1 a i ( G ) .We an also onsider di(cid:27)erent variants of subgroup growth, for example,we may only ount the normal subgroups of (cid:28)nite index. To do this we de(cid:28)nethe sequen e a ⊳n ( G ) = |{ H ⊳ f G : | G : H | = n }| .
1r we an de(cid:28)ne a sequen e a ∧ n ( G ) = |{ H ≤ f G : | G : H | = n, ˆ H ∼ = ˆ G }| , to ount those subgroups whose pro(cid:28)nite ompletion is isomorphi to thepro(cid:28)nite ompletion of the group itself. We de(cid:28)ne the zeta fun tion of G tobe the formal Diri hlet series ζ ∗ G ( s ) = ∞ X n =1 a ∗ n ( G ) n − s , where ∗ ∈ {≤ , ⊳, ∧} , and s is a omplex variable. The abs issa of onver-gen e of ζ ∗ G ( s ) , whi h we denote by α ∗ G , determines the degree of polynomialsubgroup growth sin e α ∗ G := inf { α ≥ c > s ∗ n ( G ) < cn α for all n } . When G is a (cid:28)nitely generated torsion-free nilpotent group, alled hen e-forth a T -group, the fun tion a ∗ n ( G ) grows polynomially and is multipli ative,in the sense that a ∗ n ( G ) = Q i a ∗ p kii ( G ) where n = Q i p k i i . Therefore, like thezeta fun tions in number theory, we an write the Diri hlet series as an Eulerprodu t de omposition of the lo al zeta fun tions, i.e., ζ ∗ G ( s ) = Y p prime ζ ∗ G,p ( s ) , where ζ ∗ G,p ( s ) = ∞ X k =0 a ∗ p k ( G ) p − ks ounts only the subgroups with p -power index.A fundamental theorem of Grunewald, Segal and Smith [12℄ states thatthese lo al zeta fun tions of T -groups are rational fun tions in p − s . This ra-tionality implies that the oe(cid:30) ients a ∗ n ( G ) behave smoothly; in parti ular,they satisfy a linear re urren e relation. This is proved by expressing thezeta fun tion as a p -adi integral and then using a model theoreti bla k boxto dedu e the rationality. Results of Denef on p -adi integrals arising from alo al Igusa zeta fun tion have been generalised by Grunewald and du Sautoyin [9℄ and by Voll [19℄. This has lead to some new interesting problems. Forexample it transpires that the Hasse-Weil zeta fun tion of smooth proje tivevarieties is possibly a better analogue for the zeta fun tions in the groupsetting than the Dedekind zeta fun tion of number (cid:28)elds, as previously men-tioned. This initiated an arithmeti -geometri approa h to zeta fun tionsof groups. Theorem 1.6 in [9℄ gives an expli it de omposition for the zetafun tion as a sum of rational fun tions P i ( p, p − s ) , with oe(cid:30) ients omingfrom ounting points mod p on various varieties. In a subsequent paper [8℄2u Sautoy presents a group whose zeta fun tion depends on the F p -rationalpoints on an ellipti urve. The expli it zeta fun tion ζ ⊳G,p ( s ) of this ellipti urve example is al ulated by Voll in [21℄, and he also proves that it satis-(cid:28)es a fun tional equation. Sin e the zeta fun tion depends on ounting thenumber of F p -points on the ellipti urve, an expression that is more om-pli ated than just a polynomial in p , the proof that it satis(cid:28)es a fun tionalequation uses the fa t that the Hasse-Weil zeta fun tion of smooth proje tivevarieties also satis(cid:28)es a fun tional equation. In [22℄, he extends this to allsmooth hypersurfa es, with the restri tion that these hypersurfa es shouldnot ontain any lines.In the urrent paper, we generalise Voll's work by removing the onditionon linear subspa es. We still require that the hypersurfa e asso iated withthe groups is smooth. The main ontribution of this generalisation is that itsheds some light on whi h varieties an arise in the ontext of zeta fun tionsof groups. In parti ular, we an des ribe the varieties and the numeri aldata asso iated to the poles of the zeta fun tion quite expli itly.Let us now de(cid:28)ne the parti ular varieties that will be of interest to us.De(cid:28)nition 1.1. Let Γ be a lass 2 T -group with Z (Γ) = [Γ , Γ] , and apresentation of the form Γ := h x , . . . , x d , y , . . . , y d ′ : [ x i , x j ] = M ( y ) ij i , where d is even and M ( y ) is the matrix of relations; an antisymmetri d × d matrix with entries that are linear forms in the y i . We all the variety de(cid:28)nedby the equation P Γ : det M ( y ) = 0 the Pfa(cid:30)an hypersurfa e asso iated to Γ . We ex lude from this de(cid:28)nitionthe ase when the determinant is identi ally zero.We shall extend this de(cid:28)nition for a general lass 2 T -group G by writing G = Γ × Z m , where Γ is as above and m ≥ . Then P G := P Γ .Note that the ondition that d is an even number is ne essary to ensurethat the orresponding Pfa(cid:30)an hypersurfa e is non-trivial. Similarly the ex-tended de(cid:28)nition, we want to ex lude the Z m , be ause otherwise the Pfa(cid:30)anwould be identi ally zero.De(cid:28)nition 1.2. The set of ( k − -planes in P d ′ − forms a variety alled theGrassmannian, denoted by G ( k − , d ′ − . The a(cid:30)ne analogue is the set of k -planes in a(cid:30)ne d ′ -dimensional spa e, whi h we denote by G ( k, d ′ ) .It is well-known that G ( k − , d ′ − an be embedded in P N via thePlü ker embedding, where N = (cid:0) nk (cid:1) . 3e(cid:28)nition 1.3. The Fano variety asso iated to a hypersurfa e X ⊂ P d ′ − is the variety F k − ( X ) = { Π ∈ G ( k − , d ′ −
1) : Π ⊂ X } of ( k − -planes ontained in X .By abuse of notation we will identify ( k − -dimensional linear spa es Π ∈ P d ′ − with the orresponding point Π ∈ G ( k − , d ′ − .We shall show in this paper that we an derive a de omposition of thenormal lo al zeta fun tion of G , whi h is similar to that of Theorem 1.6 in[9℄, and also des ribe the varieties that appear in our de omposition exa tlyas the Pfa(cid:30)an hypersurfa e, and omponents of the Fano varieties of thelinear subspa es ontained on the Pfa(cid:30)an. Here omponents may arise be- ause the Fano variety may be redu ible and, moreover, sin e we onsiderdeterminantal varieties, the rank of the de(cid:28)ning matrix needs to be takeninto a ount, and di(cid:27)erent omponents may give di(cid:27)erent ranks.The lo al zeta fun tions of (cid:28)nitely generated torsion-free nilpotent groups al ulated over the years, see [7℄ for a olle tion of these, exhibit urioussymmetries whi h suggest the possibility all zeta fun tions of groups mayhave a similar lo al fun tional equation. In a re ent paper [19℄, Voll hasshown that a fun tional equation is satis(cid:28)ed for lo al zeta fun tions of T -groups ounting all subgroups, onjuga y lasses of subgroups and represen-tations. However, for the lo al normal zeta fun tion Voll's proof works onlyin nilpoten y lass two. Expli itly, let G = Γ × Z m be a lass two nilpotentgroup, where Z (Γ) = [Γ , Γ] with h (Γ ab ) = d and h ( Z (Γ)) = d ′ , whi h yields h ( G ) = d + d ′ + m , h ( Z ( G )) = m + d ′ , h ([ G, G ]) = d ′ , h ( G/ [ G, G ]) = m + d and h ( G/Z ( G )) = d . Here h is the Hirs h length, whi h for a (cid:28)nitely gen-erated group G is the number of in(cid:28)nite y li fa tors in a subnormal seriesof G . Then, as Voll shows, the fun tional equation in the lass two is of theform ζ ⊳G,p ( s ) | p p − = ( − d + d ′ + m p ( d + d ′ + m ) − (2 d + d ′ + m ) s ζ ⊳G,p ( s ) . Examples of lo al normal zeta fun tions of groups of nilpoten y lassthree al ulated by Woodward [7℄ show that a fun tional equation is notalways satis(cid:28)ed. However, [7℄ ontains also a number of examples of lo alnormal zeta fun tions in higher lasses where the fun tional equation holds.It remains mysterious when there is a fun tional equation for the normalzeta fun tion and when not.Other type of results on fun tional equations onne ted to zeta fun -tions of groups in lude work by Lubotzky and du Sautoy [11℄ on zeta fun -tions ounting those subgroups whose pro(cid:28)nite ompletion is isomorphi tothe pro(cid:28)nite ompletion. These results have been partially generalised byBerman [4℄.In the urrent paper we on(cid:28)rm with a di(cid:27)erent method than Voll's thatthe fun tional equation is satis(cid:28)ed in the ase where the Pfa(cid:30)an hypersurfa e4sso iated to the group, and the omponents of its Fano varieties, are smoothand absolutely irredu ible. This is the best possible result using our methods,sin e the (cid:28)nal step uses the fa t that the Hasse-Weil zeta fun tion of pointson a smooth and absolutely irredu ible algebrai variety satis(cid:28)es a fun tionalequation.Our main theorem also has appli ations to subgroup growth. In parti -ular, we provide eviden e that some of the best bounds known so far forsubgroup growth may be exa t. We also dedu e results on the number andnature of poles of the normal zeta fun tion for lass two nilpotent groups.We refer the reader to [10℄ for the most resent survey of the general theoryof these zeta fun tions.1.1 ResultsBefore stating our main results, we begin with a few preliminary de(cid:28)nitions.A lassi al result, e.g. [12℄ states that the zeta fun tion of the free abeliangroup of rank d an be expressed in terms of ζ ( s ) , the Riemann zeta fun tion,as follows: ζ Z d ( s ) = ζ ( s ) ζ ( s − . . . ζ ( s − ( d − . We write ζ p ( s ) = − p − s for its lo al fa tors, where p is a prime.De(cid:28)nition 1.4. A (cid:29)ag of type I in V = F d ′ p , where I = { i , . . . , i l } < ⊆{ , . . . , d ′ } , (where { i , . . . , i l } < is an indexing set ordered su h that i d ′ . In number theory there are mu h (cid:28)ner asymptoti estimates on erningthe asymptoti behaviour of the oe(cid:30) ients of a Diri hlet series; these arethe so- alled Tauberian theorems. To apply these results we need to beable to analyti ally ontinue the zeta fun tion to the left of its abs issa of onvergen e. The analyti ontinuation by some ε > for zeta fun tionsof T -groups is proved in [9℄ using the analyti ontinuation of the Artin L -fun tions.Using the ombination of Theorem 1.7 and the appropriate Tauberiantheorems it is possible to obtain more detailed results on subgroup growth.Corollary 1.12. Let β ⊳G ∈ N be the multipli ity of the pole of the zetafun tion lo ated at the abs issa of onvergen e. Then exists c G ∈ R su h that s ⊳n ( G ) ∼ c G · n α ⊳G (log n ) β ⊳G − . as n → ∞ .Remark 1.13. It is lear from the statement of Theorem 1.7 that all thepoles are simple, ex ept when the numeri al data is su h that it produ es amultiple pole. In parti ular, we have β ⊳G = 1 .Theorem 1.7 indi ates that the existen e of a multiple pole is just a oin iden e of the numeri al data, rather than any stru tural properties ofthe fun tion or the group. It should be noted that these oin iden es existfor smooth Pfa(cid:30)ans. Moreover, it is not di(cid:30) ult to prove that ertain typesof singularities on the Pfa(cid:30)an always produ e genuine multiple poles, forinstan e the ordinary double point at the origin, as is the ase in the Pfa(cid:30)anhypersurfa e for the group U := h x , . . . , x , y , y , y : [ x , x ] = y , [ x , x ] = y , [ x , x ] = y i , produ es a double pole in the zeta fun tion.Our (cid:28)nal orollary bounds the number of poles for all these zeta fun tions.This follows immediately from Theorem 1.7.Corollary 1.14. Let G be a group as in Theorem 1.7, and assume that thePfa(cid:30)an is a generi hypersurfa e of degree d . Then the number of poles ofthe lo al zeta fun tion ζ ⊳G,p ( s ) is at most d + d ′ + m + r where r is the numberof irredu ible omponents of the Fano varieties on the Pfa(cid:30)an hypersurfa e.7inally, we observe the fun tional equations, proved by Voll [19℄.Corollary 1.15. Assume the Pfa(cid:30)an hypersurfa e of G is absolutely irre-du ible, and that the omponents of the Fano varieties appearing in the de- omposition of the zeta fun tion of G are smooth and absolutely irredu ible.Then the lo al normal zeta fun tion satis(cid:28)es a fun tional equation of theform ζ ⊳G,p ( s ) | p p − = ( − d + d ′ + m p ( d + d ′ + m ) − (2 d + d ′ + m ) s · ζ ⊳G,p ( s ) . It would be interesting to know if all the omponents of the Fano varieties orresponding to a smooth irredu ible Pfa(cid:30)an have the same dimension.For quadri s this is known, also for the Fano variety of lines on a ubi hypersurfa e (see [1℄). However, for singular Pfa(cid:30)ans this is not true. In [5℄Browning and Heath-Brown give an example of a ubi hypersurfa e de(cid:28)nedby the equation y y y + y y + y y = 0 , whose Fano variety of planes onsists of P and a point. This hypersurfa e an be en oded as the Pfa(cid:30)anof the group G = < x , . . . , x , y , . . . , y : [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y > . However, this Pfa(cid:30)an hypersurfa e is not smooth and hen e our theoremdoes not apply to this group. The splitting of the Fano variety of planes inthis ase is urious and deserves some further investigation.1.2 Layout of the paperOur results on subgroup growth and abs issae of onvergen e, namely Corol-laries 1.10-1.14, are proved in Se tion 2. In Se tion 3 we brie(cid:29)y ommenton how the fun tional equation des ribed in Corollary 1.15 an be dedu edfrom the main theorem.Next in Se tion 4 we explain how to enumerate latti es in Z mp , and al u-late the number of latti es of a (cid:28)xed elementary divisor type. In Se tion 5 wede(cid:28)ne Grassmannian and (cid:29)ag varieties, give oordinates for the Plü ker em-bedding of the Grassmannian and de(cid:28)ne (cid:29)ags in terms of these oordinates.We also determine when two di(cid:27)erent latti es lift the same (cid:29)ag. In Se tion 6we shall dis uss solution sets of systems of linear ongruen es, de(cid:28)ne Pfa(cid:30)anhypersurfa es and give a areful des ription of the geometry needed to (cid:28)nda solution set of general ongruen es. In Se tion 7 we ompute the p -adi valuations whi h arise in the solution sets.In the Se tions 8 to 10 we de ompose the zeta fun tion and apply ourearlier work to omplete the proof of Theorem 1.7. Se tion 11 is an expli itexample and illustration of the general theory.8 Abs issae of onvergen eIn this se tion we prove the Corollary 1.10 on the rate of subgroup growthand prove a result about the number of poles of the zeta fun tion in the asethat the Pfa(cid:30)an hypersurfa e is generi ; generi meaning that, there is anon-empty open subset U of the parameter spa e of hypersurfa es su h thatfor every point in U the answer to the property we are asking, is uniform.Our main theorem, Theorem 1.7, is stated in terms of lo al zeta fun tions.However, the degree of polynomial subgroup growth is equal to the abs issaof onvergen e of the global zeta fun tion. Thus we need to analyse the onvergen e of an in(cid:28)nite produ t. This an be done using the followingwell-known results.(A) An in(cid:28)nite produ t Q n ∈ J (1 + a n ) onverges absolutely if and only ifthe orresponding sum P n ∈ J | a n | onverges.(B) P p | p − s | onverges at s ∈ C if and only if ℜ ( s ) > .In view of (B), − p − ais + bi has abs issa of onverge b i a i , so (A) implies that Q p prime − p − ais + bi has abs issa of onvergen e b i +1 a i .If we an show that the abs issa of onvergen e is determined by thedenominator of the zeta fun tion, then we an use the above observation todedu e the desired result, namely Corollary 1.10.First, we note that the Igusa-type fa tors in the numerator do not a(cid:27)e tthe abs issa of onvergen e. Indeed, they are a ombination of the numeri aldata p − a i s + b i from the denominator, multiplied with sums of powers of p − whi h ome from the fa tors b I ( p − ) . In parti ular, all the Igusa type of fa -tors in the numerator will onverge no worse than the abs issa of onvergen e oming from the denominator.The se ond main ingredient in the expli it formula in the statement ofthe Theorem 1.7 are the ß thi ex eptional fa tors in variables X i and Y ß i , whi hare of the form E ß i ( X i , Y ß i ) = p − d ß i Y ß i − p − n i X i (1 − X i )(1 − Y ß i ) . To analyse this fa tor we need to take into a ount the numbers n ß i ( p ) . By theLang-Weil estimates [16℄, we have n ß i ( p ) = δp d ß i + O ( p d ß i − ) , where δ = ( d − d − . We an rewrite the numerator of the ex eptional fa tor as p − d ß i Y ß i − p − n i X i = p − d ß i ( Y ß i − p − c ß i X i ) , so the numerator of n ß i ( p ) E ß i ( X i , Y ß i ) is equalto δY ß i − δp − c ß i X i + O ( p − Y ß i ) + O ( p − c ß i − X i ) and this will onverge noworse than the denominator (1 − X i )(1 − Y ß i ) for large enough p .This gives us a (cid:28)rst estimate for the abs issa of onvergen e just byre ording the numeri al data from the poles, namely α ⊳G = max ≤ i ≤ d ′ − (cid:26) d + m, i ( d + d ′ + m −
1) + 1 d + i , i ( d + m ) + d ß i + 1 d + i − , d + m + d + 1 d − , d + d ′ + md + 1 (cid:27) . (2)9n order to (cid:28)nish the proof of the Corollary 1.10 we need to estimate the d ß i appearing in the numeri
al data.We
an make some preliminary observations and redu
tions. In parti
u-lar, it su(cid:30)
es to
onsider only the
ases where d ≥ . Indeed, if d = 2 thenwe may assume G is the Heisenberg group and this is known to have a nor-mal zeta fun
tion with abs
issa of
onvergen
e 2, see [12℄. In the
ase d = 4 ,the Pfa(cid:30)an hypersurfa
e is a quadri
, and the smooth proje
tive quadri
sover F p have been
lassi(cid:28)ed, see e.g. [14℄. By inspe
ting an expli
it list we
an use Theorem 1.7 to dedu
e that the abs
issa of
onvergen
e is always 4.We also note that d ′ ≤ d ( d − , with equality for the free
lass two nilpotentgroups. Trivially we have ≤ i ≤ d ′ . If we assume that the Pfa(cid:30)an hypersurfa
e of degree d behaves like ageneri
hypersurfa
e in P d ′ − , then the dimension of the Fano variety of ( i − -planes is d ß i = i ( d ′ − i ) − (cid:18) d + ( i − i − (cid:19) , (3)for d ≥ (see e.g. [13, Theorem 12.8℄). Note that if d ß i < , then the Fanovariety of ( i − -planes on the Pfa(cid:30)an is empty.Lemma 2.1. Let P G be a smooth and generi
Pfa(cid:30)an hypersurfa
e asso
i-ated to a group G as in Theorem 1.7. Then F ( P G ) is the highest non-emptyFano variety.Proof. We need to determine for whi
h d, d ′ , i we have d ß i = i ( d ′ − i ) − (cid:18) d + ( i − i − (cid:19) ≥ , under the additional hypothesis d ≥ , d ′ ≤ d ( d − and ≤ i ≤ d ′ . Theseadditional
onditions are all trivial bounds
oming from the group stru
turethat governs the Pfa(cid:30)an hypersurfa
e. In general the binomial term growsin d i − , while d ′ grows at most in d , so asymptoti
ally the binomial termsurpasses the quadrati
term qui
kly. In order to make this pre
ise for small d and d ′ . We (cid:28)rst observe that i ( d ′ − i ) − (cid:18) d + ( i − i − (cid:19) ≥ if and only if d ′ ≥ i (cid:18) d + ( i − i − (cid:19) + i. Trivially, we have d ′ ≤ d ( d − so it is enough to show that d ( d − ≥ i (cid:18) d + ( i − i − (cid:19) + i d and i . It is easy to
al
ulate that if i = 6 then thisinequality holds only if d ≤ , whi
h means that d ≤ and we have alreadydealt with this
ase. Thus it is enough to
onsider i ≤ , as
laimedFor hypersurfa
es that are assumed to be smooth, but not ne
essar-ily generi
a proposition of Starr from the Appendix of [5℄ states that F i − ( P G ) = ∅ for i > d ′ . Beheshti [3℄ has given some bounds for thesedimensions in
hara
teristi
zero. However, we do not know enough aboutthese dimensions in
hara
teristi p and so we only
onsider generi
hyper-surfa
es.We have from (2) α ⊳G = max ≤ i ≤ d ′ − (cid:26) d + m, i ( d + d ′ + m − i ) + 1( d + i ) , i ( d + m ) + d ß i + 1 d + i − , d + m + d + 1 d − , d + d ′ + md + 1 (cid:27) . We shall prove the
orollary in two stages, (cid:28)rst when d ( d + m ) > d ′ and then d ( d + m ) ≤ d ′ . In the (cid:28)rst
ase d ( d + m ) > d ′ and generalisingthe results from [18℄ that max ≤ i ≤ d ′ − (cid:26) d + m, i ( d + d ′ + m − i ) + 1 d + i (cid:27) = d + m. It is an easy
ase analysis to show that α ⊳G = max ≤ i ≤ (cid:26) d + m, i ( d + m ) + d ß i + 1 d + i − , d + d ′ + md + 1 (cid:27) = d + m, using the estimate for the dimension and the inequality d ( d + m ) > d ′ , andwe leave this to the reader.In the se
ond
ase, d ( d + m ) ≤ d ′ . We again leave i = 1 to the reader,and we have from (2) α G = max ≤ i ≤ d ′ − (cid:26) i ( d + d ′ + m − i ) + 1( d + i ) , i ( d + m ) + d i + 1 d + i − (cid:27) . To prove the
orollary, by the Lemma 2.1, we need to show that max ≤ i ≤ (cid:26) i ( d + m ) + d i + 1 d + i − (cid:27) < max ≤ i ≤ d ′ − (cid:26) i ( d + d ′ + m − i ) + 1( d + i ) (cid:27) . Let us take i = d ′ on the right hand side of the above equation. Then max ≤ i ≤ d ′ − (cid:26) i ( d + d ′ + m − i ) + 1( d + i ) (cid:27) ≥ (cid:16) d ′ (cid:17) (cid:16) d + d ′ + m − d ′ (cid:17) + 1 d + d ′ = d ′ (cid:16) d + m + d ′ (cid:17) + 1 d + d ′ ≥ d ′ and we have a simple bound for the quantity in question from below. Usingthis bound, a
ase analysis for any i = 1 , . . . , , shows that i ( d + m ) + d i + 1 d + i − ≤ d ′ sin
e d ≥ . This (cid:28)nishes the
ase d ( d + m ) ≤ d ′ and hen
e the proof.11 The fun
tional equationIn this se
tion we
omment on the Corollary 1.15, namely that, the lo
alnormal zeta fun
tion of a group G as in Theorem 1.7, satis(cid:28)es a fun
tionalequation of the form ζ ⊳G,p ( s ) | p p − = ( − d + d ′ + m p ( d + d ′ + m ) − (2 d + d ′ + m ) s · ζ ⊳G,p ( s ) . By the Observation 2 on page 1013 [21℄, in our formulation, this is equivalentto A ( p, p − s ) | p p − = ( − d ′ − p ( d ′ ) · A ( p, p − s ) , where A ( p, p − s ) = W ( p, p − s ) + k X i =1 X ß i n ß i ( p ) W ß i ( p, p − s ) . Re
all from Theorem 1.7 that W ß i ( X , Y ) = I d ′ − i − ( X d ′ − , . . . , X i +1 ) E ß i ( X i , Y ß i )I i − ( Y ß i − , . . . , Y ß ) . To prove the Corollary 1.15 it is enough to show that ea
h of the sum-mands n ß i ( p ) W ß i ( X , Y ) , for i ≥ , satisfy the same fun
tional equation as A ( p, p − s ) . We shall (cid:28)rst establish inversion properties for the W ß i ( X , Y ) ,and se
ondly de(cid:28)ne what the formal inversion of p p − means for the
oe(cid:30)
ients n ß i ( p ) .Following Igusa [15℄, it is proved in [22, Theorem 4℄ that for U =( U , . . . , U n ) = ( p − a s + b , . . . , p − a n s + b n ) we have I n ( U ) | U i U − i = ( − n p ( n +12 ) I n ( U ) , where I n ( U ) as in De(cid:28)nition 1.5. Here the map U i U − i orresponds to p p − .So for ea
h W ß i ( X , Y ) = I d ′ − i − ( X d ′ − , . . . , X i +1 ) E ß i ( X i , Y ß i ) I i − ( Y ß i − , . . . , Y ß ) we get fun
tional equation
oe(cid:30)
ients ( − d ′ − i − p ( d ′− i ) and ( − i − p ( i ) from the Igusa fa
tors. It remains to determine the fun
tional equationfor E ß i . This is now easy, sin
e we have the expli
it formula E ß i ( X i , Y ß i ) = p − d ß i Y ß i − p − n i X i (1 − X i )(1 − Y ß i ) , whi
h yields E ß i ( X i , Y ß i ) | p p − = p n i + d ß i E ß i ( X i , Y ß i ) . n i = i ( d ′ − i ) is the dimension of the Grassmannian G ( n − , d ′ − ,and d ß i is the dimension of the
omponent F ß i on F i − ( P G ) .Re
all, that n ß i ( p ) denoted the number of F p -points on
ertain varieties.We shall formally de(cid:28)ne n ß i ( p ) p p − = p − d ß i n ß i ( p ) . This
an be thoughtof
oming from the fa
t that the Hasse-Weil zeta fun
tion has a fun
tionalequation if the variety is smooth and absolutely irredu
ible. In that setting,together the rationality and the Riemann hypothesis for the Hasse-Weil zetafun
tion imply that if the
omponents of the Fano varieties are smooth andabsolutely irredu
ible then n ß i ( p ) = p d ß i + · · · + 1 + ( − m X j π j , where π j ∈ C and | π j | = √ p . Further, the fun
tional equation of the zetafun
tion implies that π j p d ß i − π j indu
es a permutation of the set { π j } . Itfollows that the inversion n ß i ( p ) | p p − = p − d ß i n ß i ( p ) is well-de(cid:28)ned. Thisyields W ß i ( X , Y ) | p p − = ( − d ′ − i − i − p ( d ′− i ) + n i + ( i ) W ß i ( X , Y ) . Sin
e, (cid:0) d ′ − i (cid:1) + n i + (cid:0) i (cid:1) = (cid:0) d ′ (cid:1) , we get W ß i ( X , Y ) | p p − = ( − d ′ p ( d ′ ) W ß i ( X , Y ) , as required.A paper by Debarre and Manivel [6℄ shows that if P G is a generi
hyper-surfa
e, and moreover a
omplete interse
tion over an algebrai
ally
losed(cid:28)eld, then the Fano variety is smooth,
onne
ted and has the expe
ted di-mension, i.e., d ß i = i ( d ′ − i ) − (cid:18) d + ( i − i − (cid:19) . Λ ′ is an additive subgroup of Z d ′ p . We say that Λ ′ is maximal inits homothety
lass if Λ ′ ≤ Z d ′ p but p − Λ ′ Z d ′ p . The latti
es whi
h aremaximal in their
lass are enumerated by elementary divisor types.The type of Λ ′ ∼ = diag ( p r i + ··· + r il , . . . , p r i + ··· + r il | {z } i , p r i + ··· + r il , . . . , p r i + ··· + r il | {z } i − i , . . . , , . . . , | {z } d ′ − i l ) is denoted by ν = ( I, r I ) , where I = { i , . . . , i l } < ⊆ { , . . . , d ′ − } , with i < i < · · · < i l , and the ve
tor r I = ( r i , . . . , r i l ) re
ords the values of the13 i j . If we are only interested in the indi
es appearing in the type, and notthe exa
t values of the r i j , we say that Λ ′ has (cid:29)ag type I .Let Λ ′ be a maximal latti
e of type ν = ( I, r I ) , as above. The group GL d ′ ( Z p ) a
ts transitively on the set of maximal latti
es, and the Orbit-Stabiliser Theorem gives a 1-1
orresponden
e between { maximal latti
es of type ν } − ←→ GL d ′ ( Z p ) /G ν , (4)where G ν is the stabiliser of the diagonal matrix diag ( p r i + ··· + r il , . . . , p r i + ··· + r il | {z } i , p r i + ··· + r il , . . . , p r i + ··· + r il | {z } i − i , . . . , , . . . , | {z } d ′ − i l ) in GL d ′ ( Z p ) .The matri
es in the stabiliser G ν are of the form GL i ( Z p ) ∗ ∗ . . . ∗ p r i Z p GL i − i ( Z p ) ∗ . . . ∗ p r i + r i Z p p r i Z p GL i − i ( Z p ) . . . ∗ ... ... ... . . . ... p r i + ··· + r il − Z p p r i + ··· + r il − Z p p r i + ··· + r il − Z p . . . GL d ′ − i l ( Z p ) , where ∗ indi
ates an arbitrary matrix.By identifying Λ ′ with a
oset representative βG ν , we
an think of the
olumns of β as points in P d ′ − ( Z p ) and then list them as β i j ,k where i j ∈ I indi
ates the blo
k that β i j ,k belongs to, and k ∈ { i j − + 1 , . . . , i j } is therunning index a
ross the
olumns of the matrix. Moreover, by the a
tionof the stabiliser, we
an multiply any
olumn β i j ,k by a unit, add multiplesof β i j ,k to β i j ,k , whenever k > k (and ne
essarily i j ≥ i j ), and alsoadd p r ij + ··· + r ij − β i j ,k to β i j ,k when j > j . If b i j ,mi j ,n denotes the ( n, m ) -entry of β , the above operations imply that b i j ,mi j ,n ∈ Z p / ( p r ij + ··· + r ij − ) . This observation enables us to
ompute the number of latti
es of a (cid:28)xedelementary divisor type.De(cid:28)nition 4.1. For ea
h latti
e Λ ′ of type ( I, r I ) we de(cid:28)ne the multipli
ityof Λ ′ , whi
h we denote by µ (Λ ′ ) , to be the number of latti
es of (cid:28)xed type ( I, r I ) , divided by the fa
tor b I ( p ) whi
h
ounts the number of F p -points onthe
orresponding (cid:29)ag variety.We now de(cid:28)ne an expression whi
h en
odes this multipli
ity as a fun
tionof ( I, r I ) . One
an
he
k that in order to
ompute µ (Λ ′ ) we may assume that b i j ,mi j ,n ∈ p Z p / ( p r ij + ··· + r ij − ) . The fun
tion µ : N × N −→ N will measurethe size of the set of x ∈ p Z p / ( p a ) of a (cid:28)xed p -adi
valuation as follows:14e(cid:28)nition 4.2. Let a, b be (cid:28)xed positive integers. We de(cid:28)ne a binaryfun
tion µ : N × N −→ N as follows µ ( a, b ) := | { x ∈ p Z p / ( p a ) : v p ( x ) = b } | = if a = bp a − b (1 − p − ) if a > b otherwise.This de(cid:28)nition extends naturally to a ( n + 1) -ary fun
tion µ : N × N n −→ N ; if b = ( b , . . . , b n ) is a ve
tor then µ ( a ; b ) := | { x ∈ ( p Z p / ( p a )) n : v p ( x i ) = b i } | . We note that µ ( a ; b , b , b , . . . , b n ) = µ ( a, b ) µ ( a, b ) . . . µ ( a, b n ) and a X b =1 µ ( a, b ) = p a − . (5)The next result re
ords the multipli
ity of a latti
e of given type ν =( I, r I ) .Proposition 4.3. Let Λ ′ be a maximal latti
e of type ν = ( I, r I ) with a
oset representative βG ν under the 1-1
orresponden
e in (4), where I = { i , . . . , i l , i l +1 } and β ∈ GL d ′ ( Z p ) . Write b i j ,mi j ,k for the ( k, m ) entry of β ,where k ∈ { i j − + 1 , . . . , i j } and m ∈ { i j − + 1 , . . . , i j } , so that the pair ( i j , i j ) indi
ates the blo
k of this entry. Then µ (Λ ′ ) = Y j ,j ∈{ ,...,l,l +1 } j
22n order to determine the index of g we need to know the rank, thedeterminant and the elementary divisor type of the matri es M ( α i j ,k ) , andthese do not depend on the parti ular basis hosen. So it su(cid:30) es to solve g M ( α i , ) − det M ( α i , ) 0 , J, . . . , J ! ≡ p r i g M ( α i , ) − det M ( α i , ) 0 , J, . . . , J ! ≡ p r i ... g M ( α i ,i ) − det M ( α i ,i ) 0 , J, . . . , J ! ≡ p r i simultaneously, and we read o(cid:27) w ′ (Λ ′ , i , r i ) = dr i − min { r i , v p (det ( M ( α i , ))) , v p (det ( M ( α i , ))) , . . . , v p (det ( M ( α i ,i ))) } . (11)Sin e we are assuming the orank is 1, we do not have the oe(cid:30) ient two inthe minimum-fun tion that we had in Lemma 6.5.The di(cid:27)erent omponents of the Fano variety may give di(cid:27)erent weightfun tions. This is either due to the rank ondition, or the possibility thatthe omponents have di(cid:27)erent dimensions. We shall denote the F p -points onthe omponent F ß i by n ß i . If a omponent has orank 0, we get the sameweight fun tion as the empty interse tion ase, and then we set δ ß i = 0 .Case III: The general interse tionHere we assume that the interse tion of the (cid:29)ag of type I = { i , . . . , i l } < withthe Pfa(cid:30)an is not just a hyperplane but a (cid:29)ag with subspa es ontained onthe Pfa(cid:30)an. For simpli ity, let us ex lude the orank 0 ases, sin e theseagain will give the same weight fun tion as the ase of an empty interse tion.As before, we shall onsider the ongruen e onditions separately for di(cid:27)erentlatti es lifting (cid:28)xed (cid:29)ags on the Pfa(cid:30)an. Moreover, as in the ase of ahyperplane interse tion we onsider the points α i j ,k de(cid:28)ned in Se tion 5equation (8), instead of the β i j ,k . In order to solve the equations against thesame modulus, we onsider the set of equations p r i + ··· + r ij − g M ( α i j ,k ) ≡ p r i + ··· + r ij − + r ij + ··· + r il , where i j ∈ J and k ∈ { i j − + 1 , . . . , i j } .For ea h (cid:28)xed i j , we get an augmented matrix as in the hyperplane ase: p r i + ··· + r ij − g ( M ( α i j ,i j − +1 ) | . . . | M ( α i j ,i j )) ≡ p r i + ··· + r ij − + r ij + ··· + r il . p r i + ··· + r ij − g M ( α i j ,k ) − det M ( α i j ,k ) 0 ! , J, . . . , J ! ≡ p r i + ··· + r ij + ··· + r il . (12)Moreover, as we vary i j ∈ J , we get the whole system of ongruen es on-sisting of augmented matri es of the same form as those whi h arose in the ase J = { i } . As before, the elementary row and olumn operations donot hange the elementary divisor type of the matri es M ( α i j ,k ) , so withoutloss of generality we may assume all of the matri es are of the form above(12). We an then read o(cid:27) the weight fun tion: w ′ (Λ ′ , J , r J ) = d ( r i + · · · + r i l ) − min { r i + · · · + r i l , v p (det M ( α i , )) , . . . , v p (det M ( α i ,i )) ,r i + v p (det M ( α i ,i +1 )) , . . . , r i + v p (det M ( α i ,i )) , . . . ,r i + · · · + r i l − + v p (det M ( α i l ,i l − +1 )) , . . . , r i + · · · + r i l − + v p (det M ( α i l ,i l )) } . However, if the (cid:29)ag ontains the subspa es of points, i.e. if J = { , i , . . . , i l } , then w ′ (Λ ′ , J , r J ) = d ( r + · · · + r i l ) − min { r , v p (det M ( α , )) }− min { r + · · · + r i l , v p (det M ( α , )) , r + v p (det M ( α i , )) , . . . , r + v p (det M ( α i ,i )) ,r + r i + v p (det M ( α i ,i +1 )) , . . . , r + r i + v p (det M ( α i ,i )) , . . . ,r + r i + · · · + r i l − + v p (det M ( α i l ,i l − +1 )) , . . . , r + r i + · · · + r i l − + v p (det M ( α i l ,i l )) } . (13)7 p -adi valuations of determinantsIn order to get an expli it expression for the weight fun tion w ′ (Λ ′ ) we are leftwith the determination of v p (det M ( α i j ,k )) . This is easy from the followinglemma:Lemma 7.1. Let α i j ,k be the olumns of α that we onstru ted in Se tion 5,equation (8), with the appropriate redu tions. Then v p (det M ( α i j ,k )) = min { v p ( a ) , . . . , v p ( a c ß k ) } , where c ß k is the odimension of the omponent F ß k of F k − ( P G ) in G ( k − , d ′ − .Proof. First we note that we an expand det M ( α i j ,k ) as a bihomogeneouspolynomial of bidegree ( d , d ) in the entries of α i j ,k , and in the λ n , where24 n are the p -adi unit entries of the matrix B (see equation (7)) whi h weused in order to move between latti es lifting the same (cid:29)ag. Writing this asa polynomial in the λ n , the oe(cid:30) ients whi h appear are the de(cid:28)ning idealsof the Fano variety on the Grassmannian. Furthermore, the p -adi valuationof any monomial v p ( λ ε · · · λ ε n n ) = 1 , where P ni =1 ε i = d . Hen e v p (det M ( α i j ,k )) ≥ min { v p ( a ) , . . . , v p ( a c ß k ) } , where a i are polynomials in the entries of α . To see that equality holds,suppose for a ontradi tion that the left hand side is stri tly bigger than theright hand side, so we have the ongruen e X ε + ··· + ε n = d λ ε . . . λ ε n n a ε ,...,ε n ≡ p κ +1 , where κ = min { v p ( a ) , . . . , v p ( a c ß k ) } . Then p κ ( X ε + ··· + ε n = d λ ε . . . λ ε n n a ′ ε ,...,ε n ) ≡ p κ +1 , where p κ a ′ ε ,...,ε n = a ε ,...,ε n and thus X ε + ··· + ε n = d λ ε . . . λ ε n n a ′ ε ,...,ε n ≡ p for all possible p -adi units λ n . It follows that a ′ ε ,...,ε n ≡ p for all possible hoi es of { ε i } with P ni =1 ε i = d . But this is a ontradi tion,sin e at least one of the a ′ ε ,...,ε n is a p -adi unit.We have set up the new oordinates su h that the set of valuations v p (det M ( α i j ,k )) is ontained in the set of valuations v p (det M ( α i j ,k )) whenever k < k . By applying Lemma 7.1, we an expand the determi-nants in the weight fun tion.We start with the ase I = { i } so the interse tion of the (cid:29)ag varietywith the Pfa(cid:30)an hypersurfa e is a hyperplane. In view of, (11) we need to ompute min { r i , v p (det ( M ( α i , ))) , v p (det ( M ( α i , ))) , . . . , v p (det ( M ( α i ,i ))) } . (14)From the de(cid:28)nition of the α i j ,k , we get v p (det ( M ( α i ,k ))) = min { v p ( a ) , v p ( a ) , . . . , v p ( a c ß k ) } k ∈ { , . . . , i } , where the a n are polynomials in the oordinates ofthe points α i ,k , and hen e in the oordinates of the β i ,k . Substituting in(14) and an elling the extra minima and any terms that appear more thantwi e, we obtain w ′ (Λ ′ , i , r i ) = dr i − min { r i , v p ( a ) , . . . , v p ( a c ß i ) } . Finally, we onsider the ase of a general latti e with weight fun tion asin (13). Here we need to expand min { r i + · · · + r i l , v p (det M ( α i , )) , . . . , v p (det M ( α i ,i )) ,r i + v p (det M ( α i ,i +1 )) , . . . , r i + v p (det M ( α i ,i )) , (15) . . . , r i + · · · + r i l − + v p (det M ( α i l ,i l − +1 )) , . . . , r i + · · · + r i l − + v p (det M ( α i l ,i l )) } . Again, by expanding out the valuations of the determinants, we set v p (det ( M ( α i j ,k ))) = min { v p (ˆ a ) , v p (ˆ a ) , . . . , v p (ˆ a c ß ij − ) , v p ( a c ß ij − ) , . . . , v p ( a c ß k ) } for ea h k ∈ { i j − + 1 , . . . , i j } , where ˆ a n denotes the redu tion mod p r i + ··· + r ij − as explained in Se tion 5, in parti ular in onne tion with equa-tion (8). Now we an substitute the above expansions ba k inside (15), and an el the redundant minima inside the expression. We an also an el allthe sums with redu tions in them, as the minimum annot be attained atthese points be ause r i + · · · + r i j − + v p (ˆ a n ) ≥ v p ( a n ) (16)for any n ∈ { , . . . , c ß ij − } . Let r i + r i + · · · + r i j − + a c ß ij denote the ( c ß ij − c ß ij − ) -tuple ( r i + r i + · · · + r i j − + v p ( a c ß ij − ) , . . . , r i + r i + · · · + r i j − + v p ( a c ß ij )) , where c ß ij is the odimension of F ß ij in G ( i j − , d ′ − . Then our weightfun tion be omes w ′ (Λ ′ , J , r J )= d ( r i + · · · + r i l ) − min { r i + · · · + r i l , v p ( a c ß i ) , r i + a c ß i2 , r i + r i + a c ß i3 , . . . , r i + · · · + r i l − + a c ß il } where the highest dimensional linear subspa e ontained in the Pfa(cid:30)an, orthe dimension of the subspa e in a (cid:29)ag where the orank of the augmentedmatrix of relations hanges to , is i l − . Re all Remark 6.7.26y applying Lemma 6.4, we dedu e that the general weight fun tion fora latti e Λ ′ of type I = { i , . . . , i m } < is given by w ′ (Λ ′ , I, r I )= d ( r i + · · · + r i m ) − min { r i + · · · + r i l , v p ( a c ß i ) , r i + a c ß i2 , r i + r i + a c ß i3 , . . . , r i + · · · + r i l − + a c ß il } where the highest dimensional linear subspa e ontained in the Pfa(cid:30)an, orthe dimension of the subspa e in a (cid:29)ag where the orank of the augmentedmatrix of relations hanges to , is i l − . Re all Remark 6.7.If ∈ I then we have w ′ (Λ ′ , I, r I )= d ( r i + · · · + r i m ) − min { r , v p ( a ) }− min { r i + · · · + r i l , v p ( a c ß i ) , r i + a c ß i2 , r i + r i + a c ß i3 , . . . , r i + · · · + r i l − + a c ß il } . ζ ⊳G ( s ) = X H⊳ f G | G : H | − s , of a (cid:28)nitely generated, torsion-free, lass-two-nilpotent group G . We shallnow put the previous se tions ba k into this ontext.The zeta fun tion admits the following Euler produ t de omposition ζ ⊳G ( s ) = Y p prime ζ ⊳G,p ( s ) , where ζ ⊳G,p ( s ) ounts subgroups of (cid:28)nite p -power index only.Let us again write G = Γ × Z m . Let be the orresponding Lie ring onstru ted as an image of G under the log -map using the Mal' ev orre-sponden e. Set p := ⊗ Z p . Then for almost all primes p we have from[12℄ ζ ⊳G,p ( s ) = ζ ⊳,p ( s ) = ζ ⊳ p ( s ) . Therefore, it su(cid:30) es to ount ideals in the asso iated Lie ring. Let ′ p denotethe derived Lie ring.Lemma 8.1. [12, Lemma 6.1℄ Suppose p / ′ p ∼ = Z d + mp and ′ p ∼ = Z d ′ p as rings.For ea h latti e Λ ′ ≤ ′ p put X (Λ ′ ) / Λ ′ = Z ( p / Λ ′ ) . Then ζ ⊳,p ( s ) = ζ ⊳ p ( s ) = ζ Z d + mp ( s ) X Λ ′ ≤ ′ p | ′ p : Λ ′ | d + m − s | p : X (Λ ′ ) | − s = ζ Z d + mp ( s ) ζ p (( d + d ′ ) s − d ′ ( d + m )) A ( p, p − s ) , A ( p, p − s ) = X Λ ′ ≤ ′ p Λ ′ maximal | ′ p : Λ ′ | d + m − s | p : X (Λ ′ ) | − s . This lemma allows us to restri t to latti es in the entre only. We enu-merate su h latti es a ording to the elementary divisor type of the latti e,as explained in Se tion 4. It is enough to onsider only maximal latti esof p -power index, sin e Λ ′ = p r d ′ Λ ′ max if Λ ′ is not maximal, where Λ ′ max ismaximal in its lass. Now | ′ p : Λ ′ | = p d ′ r d ′ | ′ p : Λ ′ max | and | p : X (Λ ′ ) | = p dr d ′ | p : X (Λ ′ max ) | . We de(cid:28)ne the weight fun tions w (Λ ′ ) := log p ( | ′ p : Λ ′ | ) w ′ (Λ ′ ) := log p ( | p : X (Λ ′ ) | ) , where Λ ′ is a maximal latti e. Then A ( p, p − s ) = X Λ ′ ≤ ′ p Λ ′ maximal p ( d + m ) w (Λ ′ ) − s ( w (Λ ′ )+ w ′ (Λ ′ )) . Note that the w ′ (Λ ′ ) is pre isely the fun tion we onsidered in Se tion 6,while w (Λ ′ ) is easily read o(cid:27) from the type of the latti e. Indeed, if the typeof Λ ′ is ν = ( I, r I ) , then w (Λ ′ ) = X i ∈ I ir i . A ( p, p − s ) further to run overlatti es of (cid:28)xed (cid:29)ag type, and write it as A ( p, p − s ) = X I ⊆{ ,...,d ′ − } A I ( p, p − s ) , (17)where A I ( p, p − s ) = X ν (Λ ′ )= I p ( d + m ) w (Λ ′ ) − s ( w (Λ ′ )+ w ′ (Λ ′ )) µ (Λ ′ ) . (18)Here Λ ′ is a representative latti e of (cid:29)ag type I and multipli ity µ (Λ ′ ) , as inProposition 4.3. 28 more subtle de omposition is needed in order to reveal the dependen eon the underlying geometry.If the ( i j − -dimensional subspa e of the (cid:29)ag of type I = { i , . . . , i k } < is ontained on the Pfa(cid:30)an, we shall write i ∗ j in the indexing set I to indi atethis fa t. For example A ∗ ( p, p − s ) = X ν (Λ ′ )=2 ∗ p ( d + m ) w (Λ ′ ) − s ( w (Λ ′ )+ w ′ (Λ ′ )) µ (Λ ′ ) where the sum is taken over latti es Λ ′ of (cid:29)ag type { } su h that theirasso iated (cid:29)ag them onsists only of a line whi h lies ompletely on thePfa(cid:30)an hypersurfa e.Thus with this notation we have I = { i , . . . , i k } ∈{ , . . . , d ′ − , ∗ , ∗ , . . . , l ∗ } , where l denotes the highest dimensional linearsubspa e on the Pfa(cid:30)an.The indexing and further de omposition of A ( p, p − s ) are done via ad-missible subsets I ⊆ { , . . . , d ′ − , ∗ , ∗ , . . . , l ∗ } . We all I admissible if thefollowing onditions hold:(i) Only i j or i ∗ j an belong to I , but not both;(ii) If i ∗ j ∈ I then i k I for all k ≤ j .Then we have A ( p, p − s ) = X I ⊆{ ,...,d ′ − } c I,p A I ( p, p − s ) + X I =1 ∗ ∪ J J ⊆{ ,...,d ′− } c I,p A I ( p, p − s )++ X I = J ∪ ∗ ∪ J J ⊆{ ∗} J ⊆{ ,...,d ′− } c I,p A I ( p, p − s ) + · · · + X I = J ∪ l ∗ ∪ J J ⊆{ ∗ ,...,l − ∗} J ⊆{ l +1 ,...,d ′− } c I,p A I ( p, p − s ) . It is possible to give an expli it des ription of the oe(cid:30) ients c I,p . First let n ß ij ( p ) denote the number of F p -points on the omponent F ß i of F i − ( P G ) ,and let δ ß i be zero or one depending whether the orank of this omponentis zero or one. Also let b I ( p ) be the number of F p -points on the (cid:29)ag varietyde(cid:28)ned by latti es of (cid:29)ag type I . This is equal to b I ( p ) = (cid:18) d ′ i l (cid:19)(cid:18) i l i l − (cid:19) . . . (cid:18) i i (cid:19)(cid:18) i i (cid:19) , for I = { i , i , . . . , i l } ⊆ { , . . . , d ′ } . We de(cid:28)ne the (cid:29)ag type I − k := { i j − k : i j ∈ I } ⊆ { , . . . , d ′ − k } . We laim that if I = { i , . . . , i n } then c I,p = b I ( p ) − b I − i ( p )( X ß i δ ß i n ß i ( p )) , (19)29hile for I = { i ∗ , . . . , i ∗ n , k ∗ , j , . . . , j r } . c I,p = b J − k ( p )( X ß k δ ß i n ß k ( p )) b J ( p ) − b J − ( k − j ) ( p )( X ß j δ ß i n ß j ( p )) b J ∪ k ( p ) . (20)This essentially omes from the formulae for the number of points on (cid:29)agvarieties. In (19) we have a (cid:29)ag variety where no part of the (cid:29)ag interse tsthe Pfa(cid:30)an hypersurfa e, so at the level of ( i − -spa es we need to subtra tthe number P ß i δ ß i n ß i ( p ) of ( i − -spa es on the Pfa(cid:30)an, for whi h theweight fun tion is di(cid:27)erent. The oe(cid:30) ient in (20) is derived in a similarfashion. As ( k − -dimensional spa es lie on the Pfa(cid:30)an we need to ompute, P ß k δ ß i n ß k ( p ) . Restri ting to any ( k − -dimensional spa e of these it is lear that the (cid:29)ag variety with the highest subspa e being this (cid:28)xed ( k − -dimensional spa es, lies ompletely on the Pfa(cid:30)an. In higher dimensions we ount only those spa es that are o(cid:27) the Pfa(cid:30)an at the level j , and thus wesubtra t P ß j δ ß i n ß j ( p ) .Rearranging su h that the we an pull out the oe(cid:30) ients n ß ij ( p ) , weobtain the following de omposition A ( p, p − s ) = W ( p, p − s ) + l X i =1 X ß i δ ß i n ß i ( p ) W ß i ( p, p − s ) where W ( p, p − s ) = X I ⊆{ ,...,d ′ − } b I ( p ) A I ( p, p − s ) and W ß i ( p, p − s ) = X I = J ∪ i ∗ ∪ J J ⊆{ ∗ , ∗ ,...,i − ∗} J ⊆{ i +1 ,...,d ′− } b J − i ( p ) b J ( p ) A I ( p, p − s ) − X I = J ∪ i ∪ J J ⊆{ ∗ , ∗ ,...,i − ∗} J ⊆{ i +1 ,...,d ′− } b J − i ( p ) b J ( p ) A I ( p, p − s )= X J ⊆{ ∗ , ∗ ,...,i − ∗ } J ⊆{ i +1 ,...,d ′− } b J − i ( p ) b J ( p )( A J ∪ i ∗ ∪ J ( p, p − s ) − A J ∪ i ∪ J ( p, p − s )) . Here i ∗ is a latti e that gives a point on F ß i , and J runs over all latti esthat are ontained in this parti ular omponent F ß i .10 Igusa fa torsNow that we have an expression for the generating fun tion, and all theterms appearing in it, the rest of the proof of Theorem 1.7 is basi ally ageneralisation of the summation formulae that appear in [17℄, see Lemmas4.2 to 4.16, in parti ular. 30emma 10.1. W ( p, p − s ) = X I ⊆{ ,...,d ′ − } b I ( p ) A I ( p, p − s )= X I ⊆{ ,...,d ′ − } b I ( p − ) Y i ∈ I X i − X i = I d ′ − ( X , . . . , X d ′ − ) , where X i = p ( d + d ′ + m − i ) i − ( d + i ) s . Proof. Using the Lemma 4.3 we an write X I ⊆{ ,...,d ′ − } b I ( p ) A I ( p, p − s )= X I ⊆{ ,...,d ′ − } b I ( p ) Y i ∈ I ∞ X r i =1 p ( d + m ) ir i − ( d + i ) r i s p − dim F I p ( d ′ − i ) ir i = X I ⊆{ ,...,d ′ − } b I ( p − ) Y i ∈ I ∞ X r i =1 p i ( d + d ′ + m − i ) r i − ( d + i ) r i s = X I ⊆{ ,...,d ′ − } b I ( p − ) Y i ∈ I p i ( d + d ′ + m − i ) − ( d + i ) s − p i ( d + d ′ + m − i ) − ( d + i ) s . Lemma 10.2. W ß i ( p, p − s )= X J ⊆{ ∗ , ∗ ,...,i − ∗ } J ⊆{ i +1 ,...,d ′− } b J − i ( p ) b J ( p )( A J ∪ i ∗ ∪ J ( p, p − s ) − A J ∪ i ∪ J ( p, p − s ))= X J ⊆{ i +1 ,...,d ′ − } b J − i ( p − ) Y j ∈ J X j − X j X J ⊆{ ∗ , ∗ ,...,i − ∗ } b J ( p )( A J ∪ i ∗ ( p, p − s ) − A J ∪ i ( p, p − s ))= I d ′ − i − ( X i +1 , . . . , X d ′ − ) X J ⊆{ ∗ , ∗ ,...,i − ∗ } b J ( p )( A J ∪ i ∗ ( p, p − s ) − A J ∪ i ( p, p − s )) , where X i = p ( d + d ′ + m − i ) i − ( d + i ) s . Again i ∗ is a latti e that gives a point on F ß i ,and J runs over all latti es that are ontained on this parti ular omponent.Lemma 10.3. X J ⊆{ ∗ , ∗ ,...,i − ∗ } b J ( p )( A J ∪ i ∗ ( p, p − s ) − A J ∪ i ( p, p − s ))= X J ⊆{ ∗ , ∗ ,...,i − ∗ } b J ( p − ) Y j ∈ J Y ß j − Y ß j ( A i ∗ ( p, p − s ) − A i ( p, p − s ))= I i − ( Y , . . . , Y ß i − )( A i ∗ ( p, p − s ) − A i ( p, p − s )) , Y ß i = p i ( d + m )+ c ß i − ( d + i − s for ß > , while Y = p d + m + c − ( d − s . It remains to al ulate A i ∗ ( p, p − s ) − A i ( p, p − s ) and justify the terms X i and Y ß i appearing in the formulae above.Proposition 10.4. Let G = Γ × Z m as before. Let d = h ( G/Z ( G )) , and d ′ = h ([ G, G ]) . Assume d is even. Let n i be the dimension of G ( i − , d ′ − , d ß i the dimension of F ß i and c ß i its odimension, so n i = c ß i + d ß i . Then A i ∗ − A i = ∞ X r i =1 r i X a =1 r i X a =1 · · · r i X a ni =1 µ ( r i , a , a , . . . , a n i ) p i ( d + m ) r i − ( d + i ) r i s ( p st i min { r i ,a ,a ,...,a c ß i } − p i ( d + m ) − ( d + i − t i ) s (1 − p − st i )(1 − p i ( d + m )+ d ß i − ( d + i − t i ) s )(1 − p i ( d + m )+ n i − ( d + i ) s ) = p − d ß i Y ß i − p − n i X i (1 − Y ß i )(1 − X i ) , where X i = p i ( d + d ′ + m − i ) − ( d + i ) s and Y ß i = p ( i ( d + m )+ d ß i ) − ( d + i − t i ) s . Further-more, t = 2 and t i = 1 for i ≥ .Proof. The proof is given in Proposition 4.12 in [17℄ and is essentially amanipulation of in(cid:28)nite series.We on lude that the W ß i ( p, p − s ) are of the form given in Theorem 1.7.This on ludes the proof of Theorem 1.7.11 ExampleIn this se tion show an expli it appli ation of the main theorem, by de(cid:28)ninga lass two Lie ring (re all, that there is also a lass two nilpotent group withthe same presentation) whose Pfa(cid:30)an is the quadri Segre surfa e in P .Let G S have the presentation G S = h x , x , x , x , y , y , y , y : [ x , x ] = y , [ x , x ] = y , [ x , x ] = y , [ x , x ] = y i . The matrix of relations M ( y ) ij = [ x i , x j ] , is M ( y ) = y y y y − y − y − y − y , and the Pfa(cid:30)an is thus de(cid:28)ned by S : y y − y y = 0 . Sin e
S ∼ = P × P , the number of F p -rational points on S is |S ( F p ) | = ( p +1) .Furthermore, over F p there are p + 1) lines on this surfa e and no higherdimensional linear subspa es, see e.g. [14℄.Note that in this group Z ( G S ) = [ G S , G S ] , so in the appli ation of theTheorem 1.7, we have m = 0 . 32heorem 11.1. For almost all primes p , the lo al normal zeta fun tion of G S is given by ζ ⊳G S ,p ( s ) = ζ Z ,p ( s ) · ζ p (8 s − · ( W ( p, p − s )+( p +1) W ( p, p − s )+2( p +1) W ( p, p − s )) where W ( p, T ) = I ( X , X , X ) W ( p, T ) = I ( X , X ) E ( X , Y ) W ( p, T ) = I ( X ) E ( X , Y ) I ( Y ) with X i = p i (8 − i ) − (4+ i ) s , Y = p − s and Y = p − s . A I ( p, T ) = X ν (Λ ′ )= I p dw (Λ ′ ) − s ( w (Λ ′ )+ w ′ (Λ ′ )) µ (Λ ′ ) . The elementary divisor types of the latti e Λ ′ in this are are ( p r + r + r , p r + r , p r , , where r i ≥ , the (cid:29)ag type is I := { i : r i > } ⊆ { , , } . Immediately itfollows that w (Λ ′ ) = P i ∈ I ir i , and the multipli ity µ (Λ ′ ) an be al ulatedusing the 1-1 orresponden e between latti es and osets.Example 11.2. Let Λ ′ have elementary divisors ( p r + r , p r , , so thattype onsists of I = { , } and r I = ( r , r , , . The stabiliser of Λ ′ takesthe form GL ( Z p ) ∗ ∗ p r Z p GL ( Z p ) ∗ p r + r Z p p r Z p GL ( Z p ) . If we then ex lude the terms oming from the (cid:29)ag varieties, a oset of thestabiliser takes form b b a b a , where a , a ∈ p Z p / ( p r ) , b , b ∈ p Z p / ( p r + r ) , b ∈ p Z p / ( p r ) . We need to al ulate how many hoi es we an make for ea h a i , b i . Thus the multipli ityof Λ ′ is µ (1 ,
2) = X µ ( r ; b ) µ ( r , b ) µ ( r + r ; a ) µ ( r + r ; a ) µ ( r ; a )= p − p r +4 r a i , b i are variables, and the ranges of summations are the obviousones from equation (5).For w ′ (Λ ′ ) we need to onsider the (cid:29)ags asso iated with Λ ′ , de(cid:28)ned inSe tion 5. Some of the (cid:29)ags in F p interse t with the Segre surfa e and in thelight of the ongruen e onditions, the weight fun tion w ′ (Λ ′ ) hanges.As in (9) Se tion 6, in order to determine w ′ (Λ ′ ) we want to understandthe di(cid:27)erent solutions to the set of ongruen es g M ( β ) ≡ p r + r + r g M ( β ) ≡ p r + r g M ( β ) ≡ p r . Note that we only insist that r i ≥ .Sin e the Segre surfa e does not ontain any planes, the ve tor β analways be hosen su h that det ( M ( β )) is a p -adi unit and the third on-dition redu es to g ≡ p r , and we get an immediate redu tion, as in Lemma 6.4, w ′ (1 ,
3) = w ′ (1) + w ′ (3) w ′ (2 ,
3) = w ′ (2) + w ′ (3) w ′ (1 , ,
3) = w ′ (1 ,
2) + w ′ (3) . (21)This indu es a similar redu tion in generating fun tions. A ∗ , = p A ∗ · A A ∗ , , = p A ∗ , · A A ∗ , = p A ∗ · A A ∗ , ∗ , = p A ∗ , ∗ · A , (22)be ause w ( J ∪
3) = w (3) + w ( J ) , for J ⊆ { ∗ , ∗ , , } and the µ -fun tionhas enough multipli ative properties, e.g., µ ( r i + r ; a ) = p r µ ( r i ; a ) .There are four additional ases we need to onsider re(cid:29)e ting the fourdi(cid:27)erent geometri on(cid:28)gurations how a (cid:29)ag an interse t the Segre surfa e.1. M ( β ) and M ( β ) are both non-singular mod p .2. M ( β ) is singular mod p , but M ( β ) is not singular mod p , and theline h β , β i does not lie on the Segre surfa e.3. M ( β ) and M ( β ) are both singular mod p , and the line h β , β i lieson the Segre surfa e.4. M ( β ) and M ( β ) are both singular mod p , and the (cid:29)ag h β i < h β , β i onsisting of a point and a line lies on the Segre surfa e.34e are now justi(cid:28)ed to use the same indexing as in Se tion 9 and if wedenote by n ( p ) the number of points on the Pfa(cid:30)an, and similarly by n ( p ) the number of lines, the generating fun tion takes the expli it form as a sumover all admissible subsets of { ∗ , ∗ , , , } : A ( p, p − s ) = A ∅ + (cid:18) (cid:19) p − n ( p ) ! A + (cid:18) (cid:19) p − n ( p ) ! A + (cid:18) (cid:19) p A + (cid:18) (cid:19) p (cid:18) (cid:19) p − n ( p ) ! A , + (cid:18) (cid:19) p (cid:18) (cid:19) p − n ( p ) ! A , + (cid:18) (cid:19) p (cid:18) (cid:19) p − n ( p ) ! A , + (cid:18) (cid:19) p (cid:18) (cid:19) p (cid:18) (cid:19) p − n ( p ) ! A , , + n ( p ) A ∗ + (cid:18) (cid:19) p n ( p ) − n ( p ) (cid:18) (cid:19) p ! A ∗ , + (cid:18) (cid:19) p n ( p ) A ∗ , + (cid:18) (cid:19) p (cid:18) (cid:19) p n ( p ) − n ( p ) (cid:18) (cid:19) p ! A ∗ , , + n ( p ) A ∗ + n ( p ) (cid:18) (cid:19) p A ∗ , ∗ + (cid:18) (cid:19) p n ( p ) A ∗ , + (cid:18) (cid:19) p n ( p ) (cid:18) (cid:19) p A ∗ , ∗ , . We rearrange this sum in order to see exa tly whi h parts depend on n ( p ) and n ( p ) , and obtain A ( p, p − s ) = W ( p, p − s ) + n ( p ) W ( p, p − s ) + n ( p ) W ( p, p − s ) , (23)where W ( p, p − s ) = X I ⊆{ , , } b I ( p ) A I ( p, p − s ) ,W ( p, p − s ) = X I ⊆{ , } b I ( p )( A ∗ ∪ I ( p, p − s ) − A ∪ I ( p, p − s )) W ( p, p − s ) = (cid:18) (cid:19) p p A ! ( A ∗ − A ) + (cid:18) (cid:19) p ( A ∗ , ∗ − A ∗ , ) ! . The third formula follows from the redu tions in generating fun tions (22).By Lemma 10.1 W ( p, p − s ) = I ( X , X , X ) , as wanted.Now we need to determine the generating fun tions A ∗ − A , A ∗ − A and A ∗ , ∗ − A ∗ , , whi h will (cid:28)nish this example.The easiest ase is A ∗ − A . By Lemma 6.5 the latti es ( p r , , , lifting a point on Pfa(cid:30)an are in 1-1 orresponden e with the ve tor β = , a , a , a ) t , a i ∈ p Z p /p r , and the weight fun tion is w ′ (Λ ′ ) = 4 r − { r , v p (det M ( β )) } . By Lemma 7.1 on p-adi valuations, this is equalto w ′ (Λ ′ ) = 4 r − { r , v p ( a ) } and hen e we may apply Proposition 10.4and get A ∗ − A = p − s − p − s (1 − p − s )(1 − p − s ) = E ( X , Y ) , where X = p − s and Y = p − s .Next we al ulate the weight fun tion over latti es that lift lines on theSegre surfa e. There are two rulings of lines on the Segre surfa e, but in fa tboth of them behave similarly. We onsider the line de(cid:28)ned by y = y = 0 . Latti es Λ ′ of type { ∗ } lifting this line are in one-to-one orresponden e withpairs of ve tors β = ( a , a , , t , β = ( b , b , , t where a i , b i ∈ p Z / ( p r ) . We will de(cid:28)ne an equivalent latti e as in De(cid:28)nition 5.2 α = λ a λ a λ , α = λ a + λ b λ a + λ b λ λ . Now the ongruen es to be satis(cid:28)ed by g are g λ a λ a λ − λ a − λ a − λ ≡ p r g λ a + λ b λ a + λ b λ λ − λ a − λ b − λ − λ a − λ b − λ ≡ p r . The orank of this system of linear equations is 1, as suitable row and olumnoperations show. Thus we get the result as in the equation (11) the weightfun tion is w ′ (Λ ′ ) = 4 r − min { r , v p (det ( M ( α ))) , v p (det ( M ( α ))) } . Nowapplying the redu tion Lemma 7 in the p -adi valuations, we have to evaluate min { r , v p ( λ a ) , v p ( λ a + λ λ ( b − a ) − λ λ b ) } . Sin e λ , λ are p -adi units, this redu es to min { r , v p ( a ) , v p ( b ) , v p ( b ) } . By Proposition 10.4 A ∗ − A = ∞ X r =1 r X a i ,b i =1 p r T r − min { r ,b ,b ,a } µ ( r , a , a , b , b )= p − s (1 − p − s )(1 − p − s )(1 − p − s ) = E ( X , Y ) , for X = p − s and Y = p − s . 36inally, we need to al ulate the generating fun tion A ∗ , ∗ over a latti eof type ( p r + r , p r , , . A latti e Λ ′ lifting a (cid:29)ag of type { ∗ , ∗ } , is inone-to-one orresponden e with pairs of ve tors β = ( a , a , a , t β = ( b , b , , t where a , a ∈ p Z / ( p r + r ) , a ∈ p Z / ( p r ) and b , b ∈ p Z / ( p r ) . Any other hoi e of a (cid:29)ag and its a(cid:30)ne neighbourhood behaves in the same way. Again, hange variables α = λ a λ a λ a λ , α = λ ¯ a + λ b λ ¯ a + λ b λ ¯ a + λ λ , where ¯ a i denotes the redu tion mod p r . As in the previous ase, the orankof the system of equations is 1, and thus using the equation (13) the weightfun tion is w ′ (Λ ′ ) =4 r + 4 r − min { r , v p (det ( M ( α ))) }− min { r + r , v p (det ( M ( α ))) , r + v p (det ( M ( α ))) } , and an appli ation of Lemma 7 yields min { r , v p ( λ ( a − a a )) } = min { r , v p ( a ) } and min { r + r , v p ( λ ( a − a a )) , r + v p ( λ (¯ a − ¯ a ¯ a ) + λ λ ( b − ¯ a − b ¯ a ) + λ ( b )) } = min { r + r , v p ( a − a a ) , r + min { v p (¯ a − ¯ a ¯ a ) , v p ( b − ¯ a − b ¯ a ) , v p ( b ) }} = min { r + r , v p ( a − a a ) , min { r + v p (¯ a − ¯ a ¯ a ) , r + v p ( b − ¯ a − b ¯ a ) , r + v p ( b ) }} = min { r + r , v p ( a − a a ) , r + v p (¯ a − ¯ a ¯ a ) , r + v p ( b − ¯ a − b ¯ a ) , r + v p ( b ) } = min { r + r , v p ( a ) , r + v p ( b ) , r + v p ( b ) } . Now we an write A ∗ . ∗ − A ∗ , in a form where we an (cid:28)rst apply Lemma10.3 to get A ∗ . ∗ − A ∗ , = I ( Y )( A ∗ − A ) and after that Proposition 10.4, to obtain the (cid:28)nal result as above.11.2 A losed expression ζ ⊳G S ,p ( s ) = ζ Z ,p ( s ) ζ p (8 s − ζ p (5 s − ζ p (6 s − ζ p (7 s − ζ p (3 s − ζ p (9 s − · W ( p, p − s ) , W ( p, T ) = 1 + p T + 2 p T − p T + 2 p T + p T − p T − p T + 2 p T + p T + p T + p T + p T − p T − p T − p T + p T − p T − p T + 2 p T + p T + p T − p T − p T + p T + 2 p T + 2 p T − p T − p T − p T − p T − p T − p T − p T + p T + p T + p T − p T − p T − p T − p T + p T + p T + p T − p T − p T − p T − p T − p T − p T − p T + 2 p T + 2 p T + p T − p T − p T + p T + p T + 2 p T − p T − p T + p T − p T − p T − p T + p T + p T + p T + p T + 2 p T − p T − p T + p T + 2 p T − p T + 2 p T + p T + p T . 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