ON LATTICES OF INTEGRAL GROUP ALGEBRAS AND SOLOMON ZETA FUNCTIONS

We investigate integral forms of certain simple modules over group algebras in characteristic 0 whose p-modular reductions have precisely three composition factors. As a consequence we, in particular, complete the description of the integral forms of the simple QSn-module labelled by the hook partition (n− 2, 12). Moreover, we investigate the integral forms of the Steinberg module of finite special linear groups PSL2(q) over suitable fields of characteristic 0. In the second part of the paper we explicitly determine the Solomon zeta functions of various families of modules and lattices over group algebra, including Specht modules of symmetric groups labelled by hook partitions and the Steinberg module of PSL2(q). Mathematics Subject Classification (2010): 20C10, 20C11, 20C30, 20C20, 11S45


Introduction
In this paper we continue our study of integral representations of symmetric groups, begun in [5].Let S n be the symmetric group of degree n 0.Moreover, for every partition λ of n, let S λ Q be the corresponding Specht QS n -module.As λ varies over the set of partitions of n, the Specht modules S λ Q yield representatives of the isomorphism classes of (absolutely) simple QS n -modules.Every Specht module Q is already equipped with a particular integral form, the Specht ZS n -lattice S λ Z .In light of the celebrated Jordan-Zassenhaus Theorem, it is, therefore, natural to ask for a description of all isomorphism classes of ZS n -lattices that are Z-forms of a given Specht module S λ Q , or at least for the number of these.To do so, a possible strategy is to consider, for each prime p, the p-adic completion S λ Qp := Q p ⊗ Q S λ Q and determine the Z p S n -lattices that are Z p -forms of S λ Qp .In general, this is way too difficult a task.
In [5] we investigated the case where λ is a hook partition of n 3, that is, a partition of the form (n − r, 1 r ), for some r ∈ {1, . . ., n − 2}.Specht modules (and Specht lattices) labelled by hook partitions have been studied a lot and much is known about their structure.Nevertheless, as far as the determination of the Z-forms of S (n−r,1 r ) Q as concerned, we so far only have complete information in the case where r = 1: By work of Plesken [13] and Craig [3], the number of isomorphism classes of Z-forms of S The case where p = 2 turned out to be considerably more difficult.In [5, Section 7], we were only able to give explicit representatives of the isomorphism , and only if n ≡ 0 (mod 4).One aim of the present paper is to settle the remaining case n ≡ 0 (mod 4).This will be achieved in Theorem 3.7, which then entails Corollary 3.9 on the number of isomorphism classes of Z-forms of S (n−2,1 2 ) Q .
In fact, Theorem 3.7 will turn out to be a special instance of the more general result in Theorem 3.2.The latter deals with the following situation: suppose that G is a finite group, R is a principal ideal domain with field of fractions K of characteristic 0 and residue field k := R/J(R) of characteristic p > 0. Suppose further that V is an absolutely simple KG-module with an R-form L whose modular reduction k ⊗ R L has, as kG-module, precisely three composition factors satisfying some additional properties.Then we shall determine all R-forms of the KG-module V up to isomorphism.
The hypotheses of Theorem 3.2 might at first seem rather special.In Proposition 3.18, we shall see a second application of this result to the case where G is a finite projective special linear group of degree 2 and V is the Steinberg module of KG, for suitable fields K of characteristic 0.
In light of this, we are tempted to ask whether Theorem 3.2 can be used to treat further finite groups and simple KG-modules arising as augmentation kernels of two-transitive permutation representations; see Question 3.19.At the moment we are, however, not able to answer this question.
In Section 4 we then investigate the (Solomon) zeta functions of various families of ZG-lattices, where G is a finite group.In [16] L. Solomon introduced a generalization of the Riemann zeta function with the aim to study enumerative problems in integral representation theory.Subsequently, Bushnell and Reiner intensively studied Solomon's zeta functions; see [2] for an overview of their theory.
In Section 4, we shall give a concise summary of Solomon's definitions and the properties of the Solomon zeta functions relevant to our applications.In the case where G is a finite group and L is a ZG-lattice, the zeta function of L is defined as where N varies over all ZG-sublattices of L of finite index.This will usually be viewed as a formal Dirichlet series, disregarding questions of convergence.
The concrete computation of zeta functions of ZG-lattices is in general a rather difficult problem, and not too much is known in this direction.The case where L is the regular ZG-lattice has been studied most intensively; for a list of known results see [6].In [6], the second author determined the zeta functions ζ ZSn (L, s), where L is a Z-form of the Specht QS n -module labelled by the hook partition (2, 1 n−2 ).In Section 4.5 of the present paper we shall generalize the results of [6], and determine global and local zeta functions of further Specht lattices labelled by hook partitions.As well, in Section 4.6 we again consider the projective special linear group PSL 2 (q), where q is a prime power, and the Steinberg module of Q[PSL 2 (q)].We shall determine the zeta function of a distinguished Z-form of this module.The key ingredient here will again be Theorem 3.2.
The present paper is organized as follows: In Section 2 we briefly summarize some properties of graduated orders that will be relevant in subsequent sections.
Section 3 is then devoted to establishing Theorem 3.2 and its applications to the study of integral forms of the Specht QS n -module S and the Steinberg module of Q[PSL 2 (q)], respectively.In Section 4 we recall Solomon's notion of global and local zeta functions of modules over group algebra.We then explicitly compute these zeta functions for various families of modules and lattices, including Specht modules of symmetric groups labelled by hook partitions, and the Steinberg module of PSL 2 (q).

Notation and prerequisites
In this section we fix some notation, briefly recall the notion of a graduated order, and summarize the known results that will be relevant in Section 3 later.We follow the work of Plesken on the subject, and refer the reader to [14] and [15] for further background.
Notation 2.1.(a) Let F be any field, and let A be a finite-dimensional F -algebra.
An A-module is always supposed to be a finitely generated left module.For an Amodule V , we denote by Rad(V ) the Jacobson radical of V , and by Hd(V ) := V / Rad(V ) the head of V .The socle of V will be denoted by Soc(V ).
(b) Let R be a principal ideal domain with field of fractions K, let m = (π) be a maximal ideal in R, and let k := R/m be the corresponding residue field.By an R-order we understand a finitely generated R-algebra Λ that is free over R of finite R-rank.One has a k-algebra isomorphism k ⊗ R Λ ∼ = Λ/mΛ; for convenience, we shall often identify these algebras and denote them simply by kΛ.A Λ-lattice is then a finitely generated left Λ-module L that is R-free of finite R-rank, which we denote by rk R (L).The factor module L := L/mL naturally carries the structure of a kΛ-module, and L/mL If A is a finite-dimensional K-algebra and if an R-order Λ is a subring of A with KΛ = A, then one calls Λ an R-order in A. In this case, we also identify the As usual, we shall often work with an R-form L of V such that L ⊆ V .Moreover, recall that every R-form of V is isomorphic to a Λ-sublattice of any given R-form L.
(c) With the notation as in (b), suppose that L is a Λ-lattice, and let L ⊆ L be a Λ-sublattice of L with rk R (L) = rk R (L ).If L ⊆ π i L, for some i ∈ N, then we denote by L /π i the Λ-sublattice {π −i x : x ∈ L } of L, which satisfies For simplicity, for the remainder of this section, R will be a local principal ideal domain with maximal ideal m = (π), field of fractions K and residue field k Remark 2.3.Suppose that A is any semisimple K-algebra, and let ε be a block idempotent of A, that is, a projection onto one of the Wedderburn components of A.
If K is a splitting field of the simple K-algebra εAε, then εAε ∼ = K n×n , where n is the dimension of the (up to isomorphism uniquely determined) simple εAε-module.
This is the situation we shall investigate in the following.
is a (graduated) order in K n×n if and only if, for all i, j, k ∈ {1, . . ., r}, one has and If, moreover, one has whenever i = j, one says that Λ is in standard form, and calls M the exponent matrix of Λ.By [15, Remark (II.3)], every graduated order in K n×n is isomorphic to a graduated order in standard form.
Moreover, let V be an absolutely simple A-module with εV = V , and let L ⊆ V be a Γ-lattice that is an R-form of V .Then εΓε is a graduated order in εAε if and only if the following conditions are satisfied: (i) every composition factor of the kΓ-module L/mL occurs with multiplicity 1; (ii) every composition factor of the kΓ-module L/mL is absolutely simple.
In the course of this paper we shall apply Theorem 2.5 in the case where A = KG is the group algebra of a finite group G over K, and Γ is the R-order RG in A.
Therefore, we recall how to obtain a graduated order in standard form in εKGε that is isomorphic to εRGε.
2.6.Sublattices and exponent matrices.We keep the notation of Theorem 2.5, and suppose that conditions (i) and (ii) are satisfied.Denote the R-order εΓε of εAε by Λ.Let D 1 , . . ., D r be the pairwise non-isomorphic composition factors of the kΓ-module L/mL, with k-dimensions d 1 , . . ., d r .
Since L ⊆ V and ε acts as the identity on V , it also acts as the identity on L and all its sublattices.In particular, L = εL is also a Λ-lattice, and the Γ-sublattices of L are just the inflations of the Λ-sublattices of L, along the surjective R-algebra homomorphism Γ → Λ = εΓε , a → εa.
We may also view L/mL both as kΓ and kΛ-module.Since V is, up to isomorphism, the only simple εAε-module and since Λ is an R-order in εAε, the simple kΛ-modules arise precisely as the composition factors of the kΛ-module L/mL.
On the other hand, if D is a simple Λ-module, then D also becomes a simple Γ-module via inflation along the surjective R-algebra homomorphism Γ → Λ = εΓε , a → εa.
Thus, altogether, the simple kΓ-modules D 1 , . . ., D r may also be viewed as simple kΛ-modules.As such they are the composition factors of the kΛ-module L/mL.
(d) Now suppose that A = KG and Γ = RG, for a finite group G. Then kΓ ∼ = kG.Consider the simple A-module V * , that is, the K-linear dual of V , let ε * be the block idempotent of A with ε * V * = V * , and let T ⊆ V * be an R-form of V * .
Then V * and T also satisfy the hypotheses of Theorem 2.5.The composition factors of the kG-module T /mT are isomorphic to the simple kG-modules D * 1 , . . ., D * r .If V is a self-dual KG-module, and if D 1 , . . ., D r are self-dual kG-modules, then we may take T ∼ = L and, by [14,Satz (III.1)] and [15, Proposition (IV.1)], we obtain for i, j ∈ {1, . . ., r}.It should be mentioned that, at the beginning of [14, Chapter III], L is assumed to be projective, when viewed as εRGε-module.In our applications, we shall usually work with lattices that do not have this property.The assertion of [14, Satz(III.1)(i)]is, however, valid without any restrictions on L.

On simple KG-modules with three modular composition factors
Throughout this section, let R be a principal ideal domain with maximal ideal m = (π), residue field k of characteristic p > 0, and field of fractions K of characteristic 0.Moreover, let G be a finite group.and head isomorphic to (e) there is some t ∈ N and, for each i ∈ {1, . . ., t}, there is some sublattice S 3i+1 of S 1 such that (i) for each i ∈ {0, . . ., t − 1}, πS 3i+1 ⊆ S 3(i+1)+1 ⊆ S 3i+1 and the dimen- Theorem 3.2.Suppose that Hypotheses 3.1 hold.Then one has the following: (e) πS 3(t−1)+1 is the unique maximal sublattice of S 3t+1 , and is the unique maximal sublattice of S 2 as well as the unique maximal sublattice of S 3 , then S 1 , . . ., S 3t+1 are representatives of the isomorphism classes of R-forms of V , and S 1 has the following full-rank sublattices: (g) if D 1 , D 2 and D 3 are absolutely simple, let ε be the block idempotent of KG where in particular, πS 2 is maximal in S 5 and πS 2 = S 7 .
To complete the proof of the theorem, it remains to settle (f) and (g).To do so we shall apply [13,Proposition 2.3].We first note that neither of the lattices S 1 , . . ., S 3t+1 is contained in πS 1 .Namely, by Hypotheses 3.1(d) and assertions (a)-(d) above, for j ∈ {1, . . ., 3t + 1}, every composition factor of S 1 /S j is isomorphic to D 2 or D 3 , while S 1 /πS 1 has a composition factor isomorphic to D 1 .Hence, by [13, Proposition 2.3], the R-forms S 1 , . . ., S 3t+1 are pairwise non-isomorphic RG-lattices.By construction, (7) is part of the submodule lattice of S 1 .Now suppose that both S 2 and S 3 have a unique maximal sublattice, which then has to be equal to S 4 .We again argue by induction on i to show that each of S 3i+2 and S 3i+3 has precisely two maximal sublattices, for all i ∈ {1, . . ., t − 1}.
So let i = 1, and assume that S 5 has a maximal sublattice T with S 7 = T = πS 2 .Then we must have But S 2 /πS 2 has a simple head isomorphic to D 3 , hence does not have a factor This proves the assertions concerning S 5 , and the lattice S 6 is treated analogously.Now let i > 1. Assume that we have a maximal sublattice T of S 3i+2 with πS 3(i−1)+2 = T = S 3(i+1)+1 .Then, as in the case where i = 1, we get S 3i+2 /πS 3i+2 ∼ = and Note that since we are assuming S 3i+2 /πS 3i+2 to be semisimple, this implies that This proves the assertions concerning S 3i+2 , and the lattice S 3i+3 is treated analogously.Consequently, we have now verified that (7) is the lattice of fullrank sublattices of S 1 .Moreover, S 1 , . . ., S 3t+1 are precisely those sublattices of S 1 of full rank that are not contained in πS 1 , which are then representatives of the isomorphism classes of R-forms of V , by [13,Proposition 2.3].This settles (f).
Hence, by 2.6(b), with respect to the chosen ordering on D 1 , D 2 , D 3 , we must have Consider the uniquely determined sublattices P 2 and P 3 of S 1 not contained in πS 1 such that P 2 / Rad(P 2 ) ∼ = D 2 and P 3 / Rad(P 3 ) ∼ = D 3 .That is, P 2 is a projective cover of D 2 and P 3 is a projective cover of D 3 , when viewed as Λ-modules.Then, by 2.6(b), we deduce that S 1 /P 2 has only composition factors isomorphic to D 3 , and the number of these is b.Similarly, S 1 /P 3 has only composition factors isomorphic to D 2 , and the number of these is a.Since S 2 and S 3 are the only maximal sublattices of S 1 , this forces P 2 ⊆ S 3 and P 3 ⊆ S 2 .Therefore, we have a = b = 1 if and only if P 2 = S 3 and P 3 = S 2 .This in turn is equivalent to S 2 and S 3 having a unique maximal sublattice, which then has to be the common sublattice S 4 .
3.2.Application I: symmetric groups.Our first application of Theorem 3.2 will be concerned with the symmetric group S n of degree n 0. We begin by setting up some notation that will be chosen in accordance with [5].For details on the representations of symmetric groups and the well-known properties of these used below, we refer the reader to [9].k .Recall also that D µ k is isomorphic to the head of S µ k .Every simple KS n -module as well as every simple kS n -module is self-dual.
In [5], we considered the case where for some prime number p.We studied the R-forms of the Specht KS n -modules labelled by hook partitions (n − r, 1 r ), for r ∈ {0, . . ., n − 1}.In [5, Theorem 6.1] we determined a set of representatives of the isomorphism classes of Z p -forms of the , for p > 2 and r ∈ {0, . . ., n − 1}; see also [14,Satz (III.8)].The main ingredient in the proof of our result were the results of Plesken [13] and Craig [3] on the case r = 1.The case where p = 2 turned out to be much more difficult.In In what follows, for every prime number p and every n ∈ N, we denote by ν p (n) the p-adic valuation of n, that is, ν p (n) = max{l ∈ N 0 : p l | n}.
By Theorem 3.2, we also know that the exponent matrix of Λ with respect to S 1 equals for some a, b ∈ N. In order to complete the proof of the theorem, it suffices to show that a = b = 1 or, equivalently, that each of the lattices S 2 and S 3 has a unique maximal sublattice, namely the common maximal sublattice S 4 .We examine S 2 .By construction, S 2 is the unique maximal sublattice of S 1 such that if n ≡ 0 (mod 4), n > 4 .

3.3.
Application II: projective special linear groups.Our second application of Theorem 3.2 will involve the Steinberg module of the projective special linear group PSL 2 (q) over suitable local fields of characteristic 0. We begin by setting up the necessary notation.Let further M R be the corresponding permutation RG-lattice with R-basis Ω, and let which is an RG-sublattice of M R of rank r − 1.Then the KG-module V K := KL R is absolutely simple, and KM R ∼ = V K ⊕ K; see [7,Satz V.20.2].Since KM R and K are a self-dual KG-modules, so is V K .Note that we also have (b) Now suppose that K is a finite extension of the field Q p of p-adic numbers.Let R be the valuation ring of K with respect to the extension of the p-adic valuation, and let m = (π) be the maximal ideal in R.
Note that, in this case, the permutation kG-module kΩ ∼ = M R /πM R has composition factors D (with multiplicity 1) and k (with multiplicity 2).Since kΩ and k are self-dual, so is D.

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The following properties of the RG-lattices introduced above are certainly well known.They will be important in the proofs of Lemma 3.17 and Proposition 3.18 below, so we include their proofs here.Lemma 3.11.In the notation of 3.10(a), let R be a principal ideal domain with field of fractions K, let m := (π) be a maximal ideal in R, let k := R/(π) be the corresponding residue field.Then the kG-module L R /πL R does not have a trivial factor module.
trivial factor module, and let ϕ : L R /πL R → k be a non-trivial kG-homomorphism.
Via restriction, β induces a symmetric G-invariant bilinear form on V K , and one easily checks that this is still non-degenerate.Let N ⊆ V K be any R-form of V K , let (b 1 , . . ., b r−1 ) be an R-basis of N , and consider Then N # is also an R-form of V K .Moreover, N ⊆ N # and, since R is local with finite residue field k, one has defines an RG-isomorphism.
Next observe that we have , by the choice of t.So, in order to complete the proof of the lemma, it suffices to show that [M 1 : On the other hand, using [5, Proposition 2.5], we also see that This completes the proof of the lemma.

(Projective) general linear and (projective) special linear groups.
Keep the notation as in 3.10.
(a) Let n ∈ N with n > 1, let p be a prime, and let q be a power of a prime.
The general linear group GL n (q) acts two-transitively on the set of one-dimensional subspaces of F n q ; as above, we simply denote this set by Ω = {ω 1 , . . ., ω r }, where r = 1 + q + • • • + q n−1 .Suppose that p q.In the notation of 3.10 above, the k[GL n (q)]module L R /πL R is absolutely simple if p r; see [10,p.16,p.47,Theorem 20.3] So, in this case L R is up to isomorphism the unique R-form of V K ; this follows from [4,Proposition (16.16)], see also [ The same is true when replacing GL n (q) by the projective general linear group PGL n (q).So, in these cases, Plesken's result [15, Theorem (VI.1)] determines the R-forms of V K listed in (8).
(b) One has SL n (q) GL n (q), and one can also regard PSL n (q) as a normal subgroup of PGL n (q) in the obvious way.Via restriction, the K[GL n (q)]-module V K in (a) becomes a K[SL n (q)]-module (and also a K[PSL n (q)]-module), which is still absolutely simple, since also SL n (q) acts two-transitively on Ω.
(c) In the following, we shall focus on the case n = 2. Then V K is the Steinberg module of the (projective) general and (projective) special linear groups under consideration; see [1,Chapter 9].Moreover, we shall from now on suppose that K is a finite unramified extension of Q p and R is the valuation ring in K with respect to the extension of the p-adic valuation.In this case we may take π := p.The residue field k is isomorphic to F p f , where f is the degree of K over Q p .We shall use the results of Plesken in [15, Theorem (VI.1);Chapter VII] and Theorem 3.2 to obtain representatives of the isomorphism classes of R-forms of Res PGL2(q) PSL2(q) (V K ).
Remark 3.14.(a) Keep the notation from 3.13(c).Suppose first that p is odd and that q is power of a prime different from p. Let G := PGL 2 (q) and H := PSL 2 (q).If p does not divide the order of H, then the kH-module Res G H (L R /pL R ) is absolutely simple.Thus, in this case, Res G H (L R ) is up to isomorphism the unique R-form of Res G H (V K ); see [4,Proposition (16.16)], [5, Proposition 2.12].Hence, we may suppose that p | |H|.Recall that |H| is a divisor of (q +1)q(q −1).
has a trivial submodule with (absolutely) simple quotient.So, we are in the situation of 3.10(b), and representatives of the isomorphism classes of R-forms of V K are given by the RH-lattices in (8), for t = ν p (q + 1).
(b) Now let p = 2, and let q be odd.If q ≡ ±3 (mod 8) and if the degree of K over Q 2 is odd, then the kH-module Res G H (L R /2L R ) has two composition factors; see [15, p. 110].In consequence of 3.13(a) and Lemma 3.11, we are then again in the situation of 3.10(b), and representatives of the isomorphism classes of R-forms of Res G H (V K ) are given by the lattices in ( 8), for t = ν 2 (q + 1).If q ≡ ±1 (mod 8), or if q ≡ ±3 (mod 8) and the degree of K over Q 2 is even, then Res G H (L R /2L R ) has three (absolutely) simple composition factors; see [15, p. 110], [1,Section 9.4.4].These cases will be dealt with in the following.Remark 3.15.As for the case of equal characteristic, that is, in the case where p | q, note that the kH-module Res G H (L R /pL R ) is projective and absolutely simple; see [1,Lemma 10.2.4].So, also in this case, Hypotheses 3.16.For the remainder of this subsection, we suppose that q is an odd prime power and p = 2.We set G := PGL 2 (q), and let H := PSL 2 (q).Let (K, R, k) be as in 3.13(c).If q ≡ ±1 (mod 8), then we may take K to be any finite unramified extension of Q 2 .If q ≡ ±3 (mod 8), then let K be a finite unramified extension of Q 2 of even degree; in particular, k then contains the field with four elements.Lastly, let V K be the absolutely simple KG-module with R-form L R as defined in 3.13(a).
Lemma 3.17.Let q be an odd prime power, and let t := ν 2 (q + 1).With Hypotheses 3.16 one has the following: Proposition 3.18.Let q be an odd prime power, and let t := ν 2 (q + 1).In the notation of Lemma 3.17, the RH-lattice L R has submodule lattice (7).The lattices S 1 , S 2 , . . ., S 3t+1 are representatives of the isomorphism classes of R-forms of the absolutely simple KH-module Res G H (V K ).
Proof.Let ε be the block idempotent of KH corresponding to the absolutely simple module Res G H (V K ).Consider the graduated R-order Λ := εRHε in εKHε.By Lemma 3.17 and Theorem 3.2, we know that ( 7) is part of the lattice of full-rank sublattices of L R = S 1 .Moreover, S 1 , S 2 , . . ., S 3t+1 are pairwise non-isomorphic R-forms of the KH-module V K .On the other hand, by [15, Chapter VII], there is a KH-lattice L ⊆ V K that is an R-form of V K and the exponent matrix of Λ with respect to L is From this and 2.6(c) one easily deduces that the isomorphism classes of R-forms of the KH-module V K are in bijection with the set {(0, 0, 0), (j, 0, 0), (j, 0, 1), (j, 1, 0) : j ∈ {1, . . ., t}} , which has cardinality 3t + 1.Hence, (7) has to be the complete lattice of full-rank sublattices of L R = S 1 , and S 1 , S 2 , . . ., S 3t+1 are representatives of the isomorphism classes of R-forms of the KH-module V K .Question 3.19.To conclude this section, suppose again that G is any finite group acting two-transitively on a finite set Ω = {ω 1 , . . ., ω r } with r > 2. In the notation of 3.10, we know that the absolutely simple KG-module V K admits at least t + 1 pairwise non-isomorphic R-forms, representatives of which are given by the RG- (8).We also know, by Lemma 3.12, that M π t is isomorphic to the dual lattice M * 1 = L * R .Consider the kG-module L k := L R /πL R and its trivial submodule D 1 as in 3.10.Suppose that L k /D 1 is the direct sum of two non-trivial pairwise non-isomorphic simple kG-modules D 2 and D 3 .For i ∈ {0, . . ., t}, set S 3i+1 := M π i .Do these RG-lattices then satisfy Hypotheses 3.1?If so, is S 4 the only maximal RG-sublattice of each of the maximal sublattices S 2 and S 3 of S 1 ?If this was the case, then Theorem 3.2 would be applicable to determine representatives of the isomorphism classes of R-forms of V K , generalizing Proposition 3.18.

Zeta functions
In this section we briefly review the notion of zeta functions of modules.We follow Solomon [16], who introduced these objects to study enumerative problems in integral representation theory.After introducing the general zeta function, we focus on the case over local principal ideal domains and determine zeta functions of various types of lattices, including the ones from Section 3.  Let P ⊂ N be the set of all prime numbers.By [16], one has In [16] it is also shown that there exists a complex function 4.2.Uniserial reductions.Throughout this subsection, let R be a local principal ideal domain with maximal ideal m = (π), field of fractions K and finite residue field k = R/m of cardinality q.Assume further that G is a finite group and M is an R-form of an absolutely simple KG-module V such that the lattice of RGsublattices of M of full R-rank is totally ordered.This happens, for instance, if the reduction modulo m of every R-form of V is a uniserial kG-module; see [5,Proposition 3.7].

Denote by πM
By [13, Proposition 2.3], we know that M 1 , . . ., M r form a set of representatives of the R-forms of V .
Proof.Since the lattice of full-rank RG-sublattices of M is totally ordered, we have For i, j ∈ {1, . . ., r} and L ∈ Φ(M i , M j ), we have µ(M i , L) = 1 if L = M i , and µ(M i , L) = −1 otherwise.Thus the claim follows.
Proposition 4.5.The matrix B = (B ij ) 1 i,j r defined by Proof.It is sufficient to show that BA M is the identity matrix.To this end, let A j be the jth column of A M and B i the ith row of B. Then and if j > i, then By summing up the entries of B M row-wise, we obtain: Corollary 4.6.For i ∈ {1, . . ., r}, one has

Modular reductions with two non-isomorphic composition factors.
In this section, let R be a local principal ideal domain with maximal ideal m = (π), field of fractions K, and finite residue field k = R/m of cardinality q.Let V be an absolutely simple KG-module of dimension d such that the reduction modulo m of any R-form of V has two non-isomorphic composition factors D 1 and D 2 .Assume further that the Jordan-Zassenhaus theorem holds for R-forms of V , that is, up to isomorphism there are only finitely many R-forms of V .By [12, Satz (I.6)], there exists an R-form M of V such that M/πM is indecomposable.We fix such an R-form M of V , for the remainder of this subsection.We shall suppose that the head of M/πM is isomorphic to D 1 .In fact, [12,Satz (I.6)] is stated in the case where R = Z and m is any maximal ideal in Z.The proof, however, generalizes literally to our situation.Alternatively, see also [13,Theorem 3.22].Lemma 4.8.The matrix A M = (A ij ) ∈ Z[X] (t+1)×(t+1) satisfies and Φ(M i ) = {M i , M i+1 , πM i−1 , πM i }, for i ∈ {1, . . ., t − 1}.Since M 0 , . . ., M t are representatives of the isomorphism classes of R-forms of V , we conclude that Φ(M 0 , M j ) = {M j }, for j ∈ {0, 1}, and Φ(M 0 , M j ) = ∅ otherwise.Moreover, for i ∈ {1, . . ., t}, we have From this the assertion of the lemma follows.
Proposition 4.9.The matrix B = (B ij ) 1 i,j t+1 defined by if j i, X (i−j)d2 if i j.
We aim to show that B is the inverse of A. To do this, we partition (1 − X d ) • B into blocks: First we define B 1 = (b ij ) 1 i,j 7 ∈ Z[X] 7×7 , that is, .

(n− 1
,1) Q equals the number of positive divisors of n, and one can give explicit representatives.So, one may focus on the case where r > 1.If p is an odd prime, then theQ p S n -module S (n−r,1 r ) Qp admits precisely ν p (n)+1 isomorphism classes of Z p -forms,where ν p (n) denotes the p-adic valuation of n.Explicit representatives of these isomorphism classes have been determined in[5, Theorem 6.1]; see also the work of Plesken in[14, Satz (III.8)] and [15, Theorem (VI.2)], who studied these modules using different methods.

3. 1 .
Submodule lattices.In this subsection, we shall investigate R-forms of particular absolutely simple KG-modules with three modular composition factors.Theorem 3.2 below will subsequently be applied to two examples on finite symmetric and projective special linear groups, respectively.Throughout this subsection suppose that R is local.Hypotheses 3.1.Let V be an absolutely simple KG-module, and let S 1 := L ⊆ V be an R-form of V satisfying the following properties: (a) there are pairwise non-isomorphic simple kG-modules D 1 , D 2 and D 3 with k-dimensions d 1 ,d 2 and d 3 , respectively, such that d 2 = d 1 = d 3 and d 1 = d 2 + d 3 , and such that the kG-module S 1 /πS 1 has radical isomorphic to D 1 2 and S 3 are the maximal sublattices of S 1 with S 1 /S 2 ∼ = D 2 and S 1 /S 3 ∼ = D 3 as kG-modules;

Notation 3 . 4 .
Suppose that R is a principal ideal domain with field of fractions K of characteristic 0.Moreover, let (π) be a maximal ideal in R such that the residue field k := R/(π) has characteristic p > 0. The isomorphism classes of (absolutely) simple KS n -modules are labelled by the partitions of n: for every partition λ of n, there is a simple KS n -module S λ K , called the Specht KS n -module labelled by λ, which carries a distinguished R-form S λ R , called the Specht RS n -lattice labelled by λ.The kS n -module S λ R /πS λ R shall be denoted by S λ k .It is well known and easily deduced from the explicit construction of Specht modules in[8, Sections 4 and 8]    thatS λ R ∼ = R ⊗ Z S λ Z ,for every partition λ of n.The isomorphism classes of (absolutely) simple kS n -modules are labelled by the p-regular partitions of n, that is, partitions λ of n each of whose parts occurs with multiplicity at most p−1.The simple kS n -module labelled by a p-regular partition µ is usually denoted by D µ

3. 5 .
Specht modules labelled by hook partitions.(a) Suppose now that p = 2 and n 4. Let t

Remark 3 . 8 .
Therefore, S 2 has to be the Z 2 S n -sublattice of S 1 constructed in[5, Lemma 7.5]; in particular, S 2 does not have a maximal sublattice T such that S 2 /T ∼ = D 2 , by [5, Lemma 7.7(a)].Let L ⊆ S 1 be the unique sublattice of S 1 such that L ⊆ 2S 1 and L/ Rad(L) ∼ = D 3 .Then, due to the structure of M , we must have S 1 = L ⊆ S 3 .Hence L ⊆ S 2 , since S 2 and S 3 are the only maximal sublattices of S 1 .If L = S 2 , then a > 1 and so there would be a maximal sublattice T of S 2 such that S 2 /T ∼ = D 2 , a contradiction.Therefore, L = S 2 and a = 1.Since S (n−2,1 2 ) Q2 is a self-dual Q 2 S n -module and D 1 , D 2 , D 3 are self-dual F 2 S nmodules, (6) gives 0 + a + t = m 12 + m 23 + m 31 = m 21 + m 32 + m 13 = t + b + 0 , that is, b = a = 1 as well.So, by Theorem 3.2(f), S 4 is also the unique maximal sublattice of S 3 , and the assertion of the theorem follows.For completeness, we also comment on the simple Q 2 S 4 -module S (2,1 2 ) Q2 .It is well known that, for every r ∈ {0, . . ., n − 1} and every prime p, one has S (r+1,1 n−r−1 ) Qp ∼ = S (n−r,1 r ) Zp ⊗ sgn Qp , where sgn Qp denotes the one-dimensional sign module of Q p S n ; see [9, Theorem 6.7].Thus, in particular, if M 1 , M 2 , M 3 denote the Z 2 S 4 -sublattices of S (3,1) Z2 mentioned in 3.5, then M 1 ⊗ sgn Z2 , M 2 ⊗ sgn Z2 and M 3 ⊗ sgn Z2 are representatives of the isomorphism classes of Z 2 -forms of S Let h(V ) be the number of isomorphism classes of Z-forms of V , and let d(n) be the number of divisors of n in N. Then one has

3. 10 .
Two-fold transitive permutation lattices.(a) Let R be a principal ideal domain with field of fractions K of characteristic 0.Moreover, let r ∈ N with r > 2 and let G be a finite group acting two-transitively on a set Ω := {ω 1 , . . ., ω r }.

4. 1 .
Local and global zeta functions.Notation 4.1.Let R be a unitary ring, and let M be a left R-module such that, for all n ∈ N, the number a n of R-sublattices of M with index n is finite.One is uniquely determined by M , and we denote it by A M .By [16, Lemma 3], the matrix A M is the inverse of B M .4.3.Global zeta functions.Assume that M is a Z-form of a QG-module V , for which we want to determine the zeta function ζ ZG (M, s).Then, for every prime p, the p-adic completion M giving rise to a local zeta function ζ ZpG (M p , s).By 4.2, we know that ζ ZpG (M p , s) = Z(M p )(p −s ) .
Our next aim is to determine, for each i ∈ {1, . . ., r}, the zeta function ζ RG (M i , s), by determining B M and Z(M i ) ∈ Z[[X]].For i ∈ {1, . . ., r}, we denote by d i the k-dimension of M i /M i+1 , and we set d := d 1 + • • • + d r .Lemma 4.4.With the above notation, the matrix A
2(D 1 ) must have composition factors D 2 and D 3 .On the other hand, recall that D 1 is isomorphic to the head of S s), for all primes p not dividing the group order |G|.In particular, if P is a finite set of prime numbers containing all prime divisors of |G|, thenζ ZG (M, s) = ζ V (s) Lastly, note that if V is simple and if p ∈ P is such that the F p G-module M/pM ∼ = M p /pM p is also simple, then ζ ZpG (M p , s) = ζ V,p(s).Namely, in this case, pM p is the unique maximal sublattice of M p and {p i M p Thus, when determining the global zeta function ζ ZG (M, s), it is sufficient to determine ζ V (s) as well as the local zeta functions ζ ZpG (M p , s), for all prime divisors of |G|.The task of determining ζ V (s) is straightforward, once the structure of the blocks of QG containing the indecomposable direct summands of V are known; see [16, (1.2)].If V is absolutely simple of dimension d, then ζ V (s) = ζ Q (ds), where ζ Q is the Riemann zeta function.In particular, in this case, we have ζ V,p (s) = (1 − p −ds ) −1 , for all p ∈ P.