1. Introduction
The Burnside ring is an invariant of the group that detects solubility, as well as being a framework for the induction theorems and having different applications in topology, see [1, 2]. On the other hand, the zeta function is an invariant of the ring that detects the distribution of prime ideals, see [5].
According to the definition given by Solomon for the zeta function of an order, it is necessary to know all its ideals of finite index, which might be complicated. In this work, we use a method used by Bushnell C. J. and Reiner I. [4], which depends only on the finite set of the isomorphism classes of the ideals of finite index. From [11] we have that:
For Bp(Cp) there are 2 isomorphism classes of fractional ideals of finite index; For Bp(Cp2 ) there are 9 isomorphism classes of fractional ideals of finite index.
In both cases, we can see that this is a better alternative than the method used in [10], where the same results were obtained by computing all the ideals. However, as we will see in this paper, for Bp(Cp3 ) there are
isomorphism classes of fractional ideals of finite index. So for Bp(Cpn ), this method used by I. Reiner quickly becomes unmanageable. At the present, we are trying to discover a method that only depends on the conductors, and we conjecture that there are exactly n + 1 for the general case.
Throughout this paper, let G be a finite group. Its Burnside ring B(G) is the Grothendieck ring of the category of finite left G-sets. This is the free abelian group on the isomorphism classes of transitive left G-sets of the form G H for subgroups H of G, two of which are identified if their stabilizers H are conjugate in G; addition and multiplication are given by the disjoint union and Cartesian product, respectively.
In Section 2, we recall the Burnside ring B (G) of a finite group G, along with the Zeta function ζB(G)(s) of B(G) and the ideals of a fiber product of rings.
In Section 3, we determine ζB(Cp3 )(s). In [6] this zeta function was obtained via the calculation of all de ideals of finite index in B(Cp3 ). In this paper, we use a method employed by Bushnell C. J. and Reiner I. [4] which only requires the family of all isomorphism classes of the fractional ideals of the finite index of the Burnside ring for this group. First, we recall the ideals of the finite index in Bp Cp2 according to [11], to compute the family of all isomorphism classes of the fractional ideals of the finite index of Bp Cp3 via the fiber product of rings. Next, we determine the Zeta function ζ
Remark 1.1. In Section 3, we correct a mistake made in [6, pp 17] about the calculation of the index (B : M85(α))−s , according to which (B : M85(α))−s = ps. However, this index must be (B : M85(α))−s = p2s, and therefore, we correct ζB(Cp3 )(s). (In the present paper, M85(a) was reindexed by M81(α).)
2. Zeta Functions of Burnside Rings
Let X be a finite G-set and let [X] be its G isomorphism class. We define
which is a commutative semiring with the unit, with the binary operations of disjoint union and Cartesian product.
Definition 2.1. We define the Burnside ring B (G) of G as the Grothendieck ring of B+ (G) .
Now, for subgroup H of G, we write [H] for its conjugacy class. We observe that as an abelian group, B (G) is free, generated by elements of the form G H, where [H] belongs to the set of representatives of all conjugacy classes of subgroups of G, which we call C (G) . That is
For further information about the Burnside ring, see [3].
Let H ≤ G be a subgroup and X a G-set, we denote the set of fixed points of X under the action of H by
We define the mark of H on X as the number of elements of XH and we call it φH (X) . The reader can find some of the properties satisfied by φH in [12, pp 3].
We define e
which is a morphism of semirings that extends to a unique injective morphism of rings
Let R be a Dedekind domain with a quotient field K, and let β be a finite-dimensional K − algebra. For any finite-dimensional K − space V, a full R − lattice in V is a finitely generated R − submodule L in V such that KL = V, where
An R − order in ℬ is a subring Λ of ℬ, such that the center of Λ contains R and such that Λ is a full R − lattice in ℬ.
A fractional ideal of R is a full R − lattice I in K. We can see that there is a non-zero element r ∈ R, such that rI ⊆ R.
Let p ∈ ℤ be a rational prime and let ℤp be the ring of p − adic integers. We denote the following tensor products by
and
where we have that Bp(G) is a ℤp − order, being
Let A be a finite-dimensional semisimple algebra over the rational field ℚ or over a p − adic field ℚp, and let Λ be an order in A. When A is a ℚ −algebra, Λ is a ℤ−order in A; when A is a ℚp−algebra, Λ is a ℤp −order in A. Let I be a left ideal of Λ, such that the index (Λ : I) is finite. We use this index symbol in a general sense: if, for example, Y1 and Y2 are ℤp − lattices spanning the same ℚp − vector space, we put
The symbol (Y1: Y2) is therefore unambiguously defined as whether or not Y1 contains Y2.
Definition 2.2. We define the Solomon’s zeta function ζΛ(s) of an order Λ, as follows:
which is a generalization of the classical Dedekind Zeta function ζK (s) of an algebraic number field K. When Λ is a ℤ−order in A, ζΛ(s) is the global zeta function; when Λ is a ℤp−order in A, ζΛ(s) is the local zeta function.
For the commutative rings Bp(G) and
Let Λ and Λi be ℤ -orders, for i = 1, ..., n and let , which is an order over ℤp. We see that the function ζ satisfies the following properties:
For further information about Solomon’s Zeta function, see [9].
Theorem 2.3. Let G be a finite group and B(G) its Burnside ring, if q ∈ ℤ is a prime, we have
where fG (q−s) is a polynomial in ℤ [q−s] . See [9, Theorem 1].
Remark 2.4. If q does not divide |G| , then we have that Bq (G) =
Definition 2.5. Let M be a full Λ − lattice in A. We define the zeta function ZΛ (M; s) , as follows:
the sum extending over all full Λ − sublattices N in A, such that N, M are in the same isomorphism class.
So we can express
the finite sum extending over all the representatives of the isomorphism classes of the full Λ − lattices in A.
We define the conductor of M in Λ, as follows:
Let Φ{M:Λ} be the characteristic function in A of {M : Λ} . Now we choose a Haar measure d∗x on the unit group A∗. For measurable sets E ⊂ A, E′ ⊂ A∗, it will be convenient to write
We have that:
∥x∥A = (Lx : L) for x ∈ A∗, which is independent of the choice of full ℤp − lattice L in A, and we observe that it is multiplicative. Furthermore, we can see that ∥x∥A = 1 whenever x is a unit in some ℤp − order in A. For further details on this result, see [4, 2.1 The Local Case, pp 138-139].
We assume that
is a fiber product diagram of rings, where all the maps are ring surjections. By definition
Let I ≤ A and Ii ≤ Ai be left ideals, such that Ii = fi (I) for i = 1, 2. Let A2 be a PID. Then I2 = A2β for some β ∈ A2. We have α ∈ I1 such that (α, β) ∈ I. Let J= {c ∈ A1: (c, 0) ∈ I} , which is an ideal of A1. We have that
and then it is determined by the following data:
a generator β of a principal ideal A2β of A2,
an ideal J ≤ A1 such that g1 (J) = 0, and
an element α ∈ A1 such that g1 (α) = g2 (β) . Clearly, α is uniquely determined mod J.
Let D = {a ∈ A : f2 (a) β = 0} which is an ideal of A. We have that f1 (D) α ⊆ J. For further details on this result, see [8].
3. The Zeta function of B (Cp3 )
3.1.Isomorphism classes of the fractional ideals of Bp Cp3
Let p be a prime, and let Cpn = {α} be a ciclyc group of order pn for n ∈ N. We have that the conjugacy classes of Cpn are . Therefore, a basis for Bp (Cpn ) is
and so,
Furthermore is its maximal order.
On the other hand, we know that
and then, we have that φ induces the following inclusion:
Therefore, we can see Bp (Cpn ) in
and then we can give the following fiber product structure:
We observe that ℤp is a PID. Therefore, it has ideals of the form pr ℤp, for every integer r ≥ 0, and according to the structure of the fiber product, we have that the ideals of finite index in Bp Cp3 are ideals of the form:
where α is an element of Bp (Cp2) and J ≤ Bp (Cp2) is an ideal such that:
g1 (J) = 0,
g1 (α) = g2 (pr) , where α is uniquely determined mod J, and
if D = pℤp × p2ℤp × p3ℤp × {0} we have that f1 (D) α ⊆ J.
Let Fp = {0, 1, ..., p − 1} and Fp ∗ = {1, ..., p − 1} , from [11] we have that the following is a complete list of representatives of isomorphism classes of fractional ideals of Bp Cp2 :
J1 = Z3 p
J2 = (x, y, z) ∈ ℤ3 p: (y − x) ∈ pℤp
J3 = (x, y, z) ∈ ℤ3 p: (z − x) ∈ pℤp
J4 = (x, y, z) ∈ ℤ3 p: (z − y) ∈ pℤp
J5 = (x, y, z) ∈ ℤ3 p: (z − y) ∈ p2ℤp
J6= (x, y, z) ∈ ℤ3: (y − x) ∈ pZ , (z − y) ∈ pℤp}
J7 = Bp (Cp2)
J8 = (x, y, z) ∈ ℤ3 p: x − y + z ∈ pℤp
J9 = (x, y, z) ∈ ℤ3 p: px − y + z ∈ p2ℤp}
Based on the previous paragraph, we will study Eq. (1), for the nine cases above. We will denote Bp Cp3 by B.
1). From Eq. (1) for J1, we obtain the following list of representatives of isomorphism classes of fractional ideals of B :
M1=B = ((1,1,1,1), (0,p,p,p), (0, 0,p2,p2), (0, 0, 0,p3)) , for which: {Mi : B} = B, AutBMi = B*, (B : M1)-s =1, ((ℤ * p)4: B*) = p3(p - 1)3, (ℤ 4p : B) = p6.
M2 = {(u,v,w,t) ∈ℤ P : (w - v) ∈ p2 ℤp, (t - w) ∈ p3 ℤp} = ((1, 0, 0, 0), (0,p2, 0, 0), (0,1,1,1), (0, 0, 0,p3)) , for which: {M2: B} = (p,p,p,p) Ms, AutBM2 = M2 *, (B : M2)-s = ps, ((ℤ*p)4: M2*) = p3(p - 1)2, (ℤ4p : M2) = p5.
M3 = {(u, v, w, t) ∈ ℤ4 p: (w - u) , (w - v) ∈ pℤp, (t - w) ∈ p3ℤp} = ((p, 0, 0, 0), (0,p, 0, 0), (1,1,1,1), (0, 0, 0,p3)) , for which: {M3: B} ( p, p, p, p) Ms, AutBM3 = M3*, (B : M3)-s = ps, ((ℤ*p)4: M3*) = p2(p - 1)3, }ℤ4 p: M3) = p5.
M4 = (u, v,w, t) ∈ ℤ4 P: (w - v) ∈ pℤp, (t - w) ∈ p3ℤp} = (1 , 0, 0, 0) , (0, p, 0, 0) , (0, 1 , 1 , 1)1, (0, 0, 0, p3) ), for which: { M4: B} = ( p, p, p, p) Ms, AutBM4 = M4*, (B : M4)-s = p2s, ((ℤ*p)4: M4*) = p2(p - 1)2, (ℤ4p : M4) = p4.
M5 = {(u, v, w, t) ∈ ℤ4 p: (w - u) ∈ pℤp, (w - v) , (t - w) ∈ p2Zp} = ((p, 0, 0, 0), (0,p2, 0, 0), (1,1,1,1), (0, 0, 0,p2)) , for which: {M5: B} = (p,p,p,p) Ms, AutBM5 = M5 *, (B : M5)-s = ps, ((ℤ*p)4: M5 *) = p2(p - 1)3, (ℤ4 p: M5) = p5.
M6 = {(u, v, w, t) ∈ ℤ4 p: (w - v) , (t - w) ∈ p2ℤp} = 0, 0, 0), (0,p2, 0, 0), (0,1,1,1), (0, 0, 0,p2)}, for which: {M6: B} = (p,p,p,p) Ms, ( p, p, p, p) Ms, ( p, p, p, p) Ms, AutBM6 = M6* (B : M6)-s = p2s, ((ℤ*p)4: M6*) = p2(p - 1)2, (ℤ4 p: M6) = p4.
M7 = {(u, v, w, t) ∈ ℤ4 p: (w - u) , (w - v) ∈ pℤp, (t - w) ∈ p2ℤp} = ((p, 0, 0, 0), (0,p, 0, 0), (1,1,1,1), (0, 0, 0,p2)), for which: {M7: B} = AutBM7 = M7*, (B : M7)-s = p2s, ((ℤ *p)4: M7*) = p(p - 1)3, (ℤ4 p: M7) = p4.
M8 = {(u, v, w, t) ∈ ℤ4 p: (w - v) ∈ pℤp, (t - w) ∈ p2ℤp} = ( (1 , 0, 0, 0) , (0, p, 0, 0) , (0, 1 , 1 , 1)1, (0, 0, 0, p2 ) )), for which: { M8: B} = AutBM8 = M8*, (B : M8)-s = p3s, ((ℤ*p)4: M8*) = p(p - 1)2, (ℤ4 p: M8) = p3.
M9 = {(u, v, w, t) ∈ ℤ4 p: (w - u) ∈ pℤp, (t - w) ∈ p3ℤp} ((p, 0, 0, 0), (0,1, 0, 0), (1, 0,1,1),(0, 0, 0,p3)), for which: {M9: B} (p,p2,p2,p2) M16, AutBM9= M9*, (B : M*9)-s = p2s, ((ℤ *p: M9*) = p2(p - 1)2, (ℤ4 p: M9) = p4.
M10 = {(u,v,w,i) ∈ ℤP : (t - w) ∈ p3 ℤp} = ((1, 0, 0, 0), (0,1, 0, 0), (0, 0,1,1), (0,0, 0,p3)) , for which: {M10: B} = (p,p2,p2,p2) M16 AutBM10 = M10, (B : M*10)-s = p3s, ((ℤ* p)4: M10) = p2(p - 1), (ℤ4 p: M10 )= p3.
M11 = {(u, v, w, t) ∈ ℤ4 p: (w - u) ∈ pℤp, (t - w) ∈ p2 ℤp} = ((p, 0, 0, 0), (0,1, 0, 0), (1, 0,1,1), (0,0, 0,p2)) , for which: {M11: B} = (p,p2,p2,p2) M16, AutBM11 = M11, (B : M*11)-s = p3s, ((ℤ*p)4: M11 *) = p(p - 1)2, (ℤ4 p: M11) = p3.
M12 = {(u,v,w,t) ∈ ℤ4 p: (t - w) ∈ p2ℤp} = 0, 0, 0), (0,1, 0, 0), (0, 0,1,1), (0,0, 0,p2)) , for which: {M12: B} = (p,p2,p2,p2) M16, AutBM12 = M12, (B : M12)-s = p4s, ((ℤ* p)4: M* 12) = p(p - 1), (ℤ4 p: M12) = p2.
M13 = {(u, v, w, t) ∈ ℤ4 p: (w - u) , (w - v) , (t - w) G p ℤ p} = ((p, 0, 0, 0), (0,p, 0, 0), (1,1,1,1), (0,0, 0,p)) , for which: {M13: B} = (p,p2,p2,p2) M16 AutBM13 = M13, (B : M*13)-s = p3s, ((ℤ* p)4: M13 *) = (p - 1)3, (ℤ4 p: = p3.
M14 = {(u, v, w, t) ∈ ℤ4 p: (w - v) , (t - w) ∈ pℤp} = ((1, 0, 0, 0), (0,p, 0, 0), (0,1,1,1), (0,0, 0,p)) , for which: {M14: B} = (p,p2,p2,p2) M16, AutBM14 = M14, (B : M*14)-s = p4s, ((ℤ* p)4: M14 *) = (p - 1)2, (ℤ4 p: M14) = p2.
M15 = {(u, v, w, t) ∈ ℤ4 p: (w - u) , (t - w) ∈ pℤp} = ((p, 0, 0, 0), (0,1, 0, 0), (1, 0,1,1), (0,0, 0,p)) , for which: {M15: B} = (p,p2,p2,p2) M16, AutBM15 = M15, (B : M*15)-s = p4s, ((ℤ* p)4: M15 *) = (p - 1)2, (ℤ4 p: M15) = p2.
M16 = {(u,v,w,t) ∈ ℤ4 p: (t - w) ∈ p ℤ p} = ((1, 0, 0, 0), (0,1, 0, 0), (0, 0,1,1), (0,0, 0,p)) , for which: {M16: B} = (p,p2,p2,p2) M16, AutBM16 = M16, (B : M16)-s = p5s, ((ℤ* p)4: M16 *) = (p - 1), (ℤ4 p: M16) = p.
M17 = {(u, v, w, t) ∈ ℤ4 p: (v - u) , (w - v) ∈ pℤp, (t - v) ∈ p2 ℤp} = ((p, 0, 0, 0), (1,1,1,1), (0, 0,p, 0), (0,0, 0,p2)) , for which: {M17: B} = (p,p2,p2,p2) M16, AutBM17 = M*17, (B : M17)-s = p2s, ((ℤ* p)4: M17) = p(p - 1)3, (ℤ4 p: M17) = p4.
M18 = {(u, v, w, t) ∈ ℤ4 p: (w - v) ∈ pℤ p, (t - v) ∈ p2 ℤp} = 0, 0, 0), (0,1,1,1), (0, 0,p, 0), (0,0, 0,p2)) , for which: {M18: B} = (p,p2,p2,p2) M16, AutBM18 = M18, (B : M*18)-s = p3s, ((ℤ* p)4: M18 *) = p(p - 1)2, (ℤ4 p: M18) = p3.
M19 = {(u,v,w,t) ∈ ℤ4 p: (v - u) ∈ pℤp, (t - v) ∈ p2 ℤ p} = ((p, 0, 0, 0), (1,1, 0,1), (0, 0,1, 0), (0,0, 0,p2)) , for which: {M19: B} = (p,p2,p3,p3) M24, AutBM19 = M19, (B : M19 *)-s = p3s, ((ℤ* p)4: M19 *) = p(p - 1)2, (ℤ* p: M*19) = p3.
M20 = {(u,v,w,t) ∈ ℤ4 p: (t - v) ∈ p2 ℤ p) = ((p 0, 0, 0), (0,1, 0,1), (0, 0,1, 0), (0,0, 0,p2)) , for which: {M20: B} = (p,p2,p3,p3) M24, AutBM20 = M* 20, (B : M20)-s = p4s, ((ℤ* p)4: M*20) =p(p - 1), (ℤ4 p: M* 20) = p2.
M21 = {(u, v, w, t) ∈ ℤ4 p: (v - u) , (t - v) ∈ pZp} , ((p 0, 0, 0), (0,1, 0,1), (0, 0,1, 0), (0,0, 0,p)) , for which: {M21: B} = (p,p2,p3,p3) M24, AutBM21 = B∗ 21, (B : M21)−s = p4s, ((ℤ* p)4: M*21) =p(p - 1), (ℤ4 p: M* 21) = p2..
M22 = (u, v, w, t) ∈ ℤ4 p: (w − v) ∈ p2 ℤ p, (t − w) ∈ p3 ℤp} = ((p 0, 0, 0), (0,1, 0,1), (0, 0,1, 0), (0,0, 0,p)) for which: {M22: B} = (p,p2,p3,p3) M24, AutBM22 = M22 ∗, (B : M22)−s = p4s, ((ℤ* p)4: M*22) =p(p - 1), (ℤ4 p: M* 22) = p.
M23 = (u, v, w, t) ∈ ℤ4 p: (w − v) ∈ p2 ℤ p, (t − w) ∈ p3 ℤp} = = ⟨(1, 0, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, p)⟩ for which: {M23: B} = (p,p2,p3,p3) M23, AutBM23 = M23 ∗, (B : M23)−s = p4s, ((ℤ* p)4: M*23) =p(p - 1), (ℤ4 p: M* 23) = p.
M24 = ℤ4 p = = ⟨(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)⟩ for which: {M24: B} = (p,p2,p3,p3) M24, AutBM24 = M24 ∗, (B : M24)−s = p4s, ((ℤ* p)4: M*24) =p(p - 1), (ℤ4 p: M* 24) = 1.
2). From Eq. (1) for J2, we obtain the following list of representatives of isomorphism classes
M25 = (u, v, w, t) ∈ ℤ4 p: (w − v) ∈ p2 ℤ p, (t − w) ∈ p3 ℤp} = (1, p, 0, 0), (0, p2, 0, 0), (0, 0, p2, 0), (0, 1, 1, 1) for which: {M25: B} = (p,p,p,p) M8, AutBM25 = M25 ∗, (B : M25)−s = p4s, ((ℤ* p)4: M*25) =p(p - 1)3, (ℤ4 p: M25) = p5.
M26 = (u, v, w, t) ∈ ℤ4 p: (w − v) ∈ p2 ℤ p, (t − w) ∈ p3 ℤp} = (1, p, 0, 0), (0, p2, 0, 0), (0, 0, p2, 0), (0, 1, 1, 1) for which: {M26: B} = (p,p,p,p) M26, AutBM26 = M26 ∗, (B : M26)−s = p4s, ((ℤ* p)4: M*5) =p2(p - 1)3, (ℤ4 p: M26) = p4.
M27 = {(u,v,w,t) ∈ ℤ4 p: (u - v + t) ∈ p2 ℤ p, (t - w) ∈ p3 ℤ p} = ( (1 , 1 , 0, 0), (0, p, 0, 0), (0, 0, p3 , 0) , (10, 1 , 1 , 1) ) , f)or which: { M27: B} = (p,p2,p2,p2) M16 AutBM27 = M3 *, (B : M27)-s = p2s, ((ℤ* p)4: M* 3) = p2(p - 1)3, (ℤ4 p: M27) = p4.
M28 = {(u,v,w,t) ∈ ℤ4 p: (v - u) ∈ p2 ℤ p, (t - w) ∈ p3 ℤ p} = (1 , 1 , 0, 0), (0, p, 0, 0), (0, 0, 1 , 1) , (0,10, 0, p3 ) , for which: { M28: B} =(p,p2,p2,p2) M16 AutBM28 = M28*, (B : M28)-s = p2s, ((ℤ*)4: M28*) = p2(p - 1)2, (ℤ4 p: M28)
M29 = {(u,v,w,t) ∈ ℤ4 p: (u - v + t) ∈ p2 ℤ p, (t - w) ∈ p3 ℤ p } =(1, 1, 0, 0), (0, p, 0, 0), (0, 0, p2, 0), (0, 1, 1, 1) , , for which: {M29: B} = (p,p2,p2,p2) M16 AutBM29 = M7*, (B : M29)-s = p3s, ((ℤ* p)4: M7*) = p(p - 1)3, (ℤ4 p: M29) = p3.
M30 = {(u, v, w, t) ∈ ℤ4 p: (v - u) ∈ p2 ℤ p, (t - w) ∈ p3 ℤ p } = ((1, 0, 0), (0,p, 0, 0), (0, 0,p2, 0), (0, 0,1,1)) , for which: {M30: B} (p,p2,p2,p2) M16 AutBM30 = M* 30, (B : M30)-s = p3s, ((ℤ* p)4: M30*) = p(p - 1)2, (ℤ4 p: M30) = p3.
M31 = {(u, v, w, t) ∈ ℤ4 p: (pu - v + t) ∈ p2 ℤ p (t - w) ∈ p3 ℤ p } = ((1,p, 0, 0), (0,p2, 0, 0), (0, 0,p, 0), (0,1,1,1)) , for which: {M31: B} (p,p2,p2,p2) M16 AutBM31 = M17 *, (B : M31)-s = p3s, ((ℤ* p)4: M17 *) = p(p - 1)3, (ℤ4 p: M31) = p3.
M32 = {(u,v,w,t) ∈ ℤ4 p: (u - v + t) , (t - w) ∈ p2 ℤ p} = ((1,1, 0, 0), (0,p, 0, 0), (0, 0,p, 0), (0,1,1,1)) , for which: {M32: B} (p,p2,p2,p2) M16 AutBM32 = M* 13, (B : M32)-s = p4s, ((ℤ* p)4: M13*) = (p - 1)3, (ℤ4 p: M32) = p2.
M33 = {(u, v, w, t) ∈ ℤ4 p: (u - v) , (t - w) ∈ p2 ℤ p } = ((1,1, 0, 0), (0,p, 0, 0), (0, 0,p, 0), (0,0,1,1)) , for which: {M33: B} (p,p2,p2,p2) M16, AutBM33 = M* 33, (B : M33)-s = p4s, ((ℤ* p)4: M33 *) = (p - 1)2, (ℤ4 p: M33) = p2.
M34 = {(u,v,w,t) ∈ ℤ4 p: (pu - v + t) ∈ p2 ℤ p } = ((1,p, 0, 0), (0,p2, 0, 0), (0, 0,1, 0), (0,1, 0,1)) , for which: {M34: B} = (p,p2,p3,p3) M24, AutBM34 = M19 *, (B : M34)-s = p4s, ((ℤ* p)4: M19 *) = p(p - 1)2, (ℤ4 p: M34) = p2.
M35 = {(u,v,w,t) ∈ ℤ4 p: (u - v + t) ∈ p2 ℤ p } = ( (1 , 1 , 0, 0) , (0, p, 0, 0) , (0, 0, 1 , 0) , (0,11 , 0, 1)) , for which: { M35: B} (p,p2,p3,p3) M24 AutBM35 = M21 *, (B : M35)-s = p5s, ((ℤ* p)4: M21 *) = (p - 1)2, (ℤ4 p: M35) = p.
M36 = {(u, v, w,t) ∈ ℤ4 p: (u - v) ∈ p2 ℤ p } = ( (1 , 1 , 0, 0) , (0, p, 0, 0) , (0, 0, 1 , 0) , (0,10, 0, 1)) , for which: {M36: B} (p,p2,p3,p3) M24 AutBM36 = M* 36, (B : M36)-s = p5s, ((ℤ* p)4: M* 36) = (p - 1), (ℤ4 p: M36) = p.
3). From Eq. (1) for J3, we obtain the following list of representatives of isomorphism classes
M37 = {(u,v,w,t) ∈ ℤ4 p: (p2u - w + t) ∈ p3 ℤ p, (t - v) ∈ p2 ℤ p } = (( 1,0,p2, 0), (0,p2, 0, 0), (0, 0,p3,0), (0,1,1,1)) , for which: {M37: B} (p, p, p, p)M8, AutBM37 = B*, (B : M37)-s = ps, ((ℤ* p)4: B *) = p3(p - 1)3, (ℤ4 p: M37) = p5.
M38 = {(u,v,w,t) ∈ ℤ4 p: (p2u - w + t) ∈ p3 ℤ p, (t - v) ∈ p ℤ p} ={(1,0,p2, 0), (0,p, 0, 0), (0, 0,p3, 0), (0,1,1,1)) , for which: {M38: B} = (p,p,p,p) M8, AutBM38 = M3*, (B : M38)-s = p2s, ((ℤ* p)4: M3*) = p2(p - 1)3, (ℤ4 p: M38) = p4.
M39 = {(u, v, w,t) ∈ℤ4 p: (p2u - w + t) ∈ p3 ℤ p = {(1, 0,p2, 0), (0,1, 0, 0), (0, 0,p3, 0),(0, 0,1,1)) , for which: {M39: B} = (p,p2,p2,p2) M16, AutBM39 = M9*, (B : M39)-s = p3s, ((ℤ* p)4: M9*) = p2(p - 1)2, (ℤ4 p: M39) = p3.
M40 = {(u, v, w, t) ∈ℤ4 p: (pu - w + t) , (t - v) ∈ p2 ℤ p} = {(1,0,p, 0), (0,p2, 0, 0), (0, 0,p2, 0),(0,1,1,1)) , for which: {M40: B} = (p,p2,p2,p2) M16, AutBM40 = M5*, (B : M40)-s = p2s, ((ℤ* p)4: M5 *) = p2(p - 1)3, (ℤ4 p: M40 ) = p4.
M41 = {(u, v, w, t) ∈ℤ4 p: (pu - w + t) ∈p2 ℤ p, (t - v) ∈ p ℤ p} = {(1, 0,p, 0), (0,p, 0, 0), (0, 0,p2, 0), (0,1,1,1)) , for which: {M41: B} = (p,p2,p2,p2) M16, AutBM41 = M7*, (B : M41)-s = p3s, ((ℤ* p)4: M7*) =p(p - 1)3, (ℤ4 p: M41) = p3.
M42 = {(u, v, w, t) ∈ℤ4 p: (pu - w + t) ∈ p2 ℤ p} ={(1,0,p, 0), (0,1, 0, 0), (0, 0,p2, 0), (0, 0,1,1)) , for which: {M42: B} = (p,p2,p2,p2) M16, AutBM42 = M* 11, (B : M42)-s = p4s, ((ℤ* p)4: M* 11) = p(p - 1)2, (ℤ4 p: M42) = p2.
M43 = {(u,v,w,t) ∈ℤ4 p: (u - w + t) ∈ p ℤ p, (t - v) ∈ p2 ℤ p} = ( (1, 0, 1 , 0), (0, p2 , 0, 0) , (0, 0, p, 0) , (0, , 1 , 1 , 1) ) , for which: { M43: B} = ( p, p2 , p3 , p3) M24, AutBM43 = M*17, (B : M43)-s = p3s, ((ℤ* p)4: M17 *) = p(p - 1)3, (ℤ4 p: M43) = p3.
M44 = {(u, v, w, t) ∈ℤ4 p: (u - w) ∈ p ℤ p, (t - v) ∈ p2 ℤ p} = ( (1 , 0, 1 , 0), (0, p2 , 0, 0) , (0, 0, p, 0) , (0, , 1 , 0, 1) ) , for which: { M44: B} = ( p, p2 , p3 , p3) M24, AutBM44 = M* 44, (B : M44)-s = p3s, ((ℤ* p)4: M14) = p(p - 1)2, (ℤ4 p: M44) = p3.
M45 = {(u, v, w, t) ∈ℤ4 p: (u - w + t) , (t - v) ∈ p ℤ p} = ((1, 0,1, 0), (0,p, 0, 0), (0, 0,p, 0), (0,1,1,1)) , for which: {M45: B} = (p,p2,p3,p3) M24, AutBM45 = M* 13, (B : M45)-s = p4s, ((ℤ* p)4: M13 *) = (p - 1)3, (ℤ4 p: M45) = p2.
M46 = {(u, v, w, t) ∈ℤ4 p: (u - w) , (t - v) ∈ p ℤ p } = ((1, 0,1, 0), (0,p, 0, 0), (0, 0,p, 0), (0,1, 0,1)) , for which: {M46: B} = (p,p2,p3,p3) M24, AutBM46 = M*46, (B : M46)-s = p4s, ((ℤ* p)4: M*46) = (p - 1)2, (ℤ4 p: M46 ) = p2.
M47 = { (u, v, w, t) ∈ℤ4 p: (u - w + t) ∈ p ℤ p} = ( (1 , 0, 1 , 0) , (0, 1 , 0, 0) , (0, 0, p, 0) , (0,, 0, 1 , 1)) , for which: {M47: B} = ( p, p2 , p3 , p3) M24, AutBM47 = M*15, (B : M47)-s = p5s, ((ℤ* p)4: M15*) = (p - 1)2, (ℤ4 p: M47 ) = p.
M48 = {(u, v, w,t) ∈ℤ4 p: (u - w) ∈ p ℤ p } = ( (1 , 0, 1 , 0) , (0, 1 , 0, 0) , (0, 0, p, 0) , (0,, 0, 0, 1)) , for which: {M48: B} = (p, p2 , p3 , p3) M24, AutBM48 = M* 48, (B : M48)-s = p5s, ((ℤ* p)4: M*48) = (p - 1), (ℤ4 p: M48 ) = p.
4). From Eq. (1) for J4, we obtain the following list of representatives of isomorphism classes of fractional ideals of B :
M49 (α) = {(u, v, w, t) ∈ℤ4 p: (pav - (1 + pa)w + t) ∈ p3 ℤ p, (u - t) , (v - t) ∈ p ℤ p} = ((p, 0, 0, 0), (0,p,p2(1+ pa)-1a, 0), (0, 0, (1 + pa)-1p3, 0), (1,1,1,1)) , for a e Fp* and for which: {M49 (a): B} = (p,p,p,p) M8, AutB M49 (a) = M*49 (a) , (B : M49 (a))-s = ps, = ((ℤ* p)4 4: M49 ∗(a) = p2(p − 1)3, ℤ 4 p: M49(a) = p5.
M50 (α) = {(u, v, w, t) ∈ℤ4 p: (pav - (1 + pa)ω + t) ∈ p3 ℤ p, (v - t) ∈ p ℤ p} = ((1, 0, 0, 0), (0,p,p2(1+ pa)-1a, 0), (0, 0, (1 + pa)-1p3, 0), (0,1,1,1)) , for a e Fp* and for whicli: {M50 (a): B} = (p,p,p,p) M8, AutB M50 (a) = M_b (a) , (B : M50 (a))-s = p2s, ((ℤ* p)4: M* 50,(a)) = p2(p - 1)2, (ℤ 4 p: M50(a)) = p4.
M51 = {(u,v,w,t) ∈ℤ4 p: (p2v - w + t) ∈ p3 ℤ p, (u - t) ∈ p ℤ p} = ((p, 0, 0, 0), (0,1,p2, 0), (0, 0,p3, 0),(1, 0,1,1)) , for which: {M51: B} = (p,p2,p2,p2) M16, AutBM51 = M3*, (B : M51)-s= p2s, ((ℤ* p)4: M3*) =p2(p - 1)3, (ℤ 4 p: M51) = p4.
M52 = {(u,v,w,t) ∈ℤ4 p: (p2v - w + t) ∈ p3 ℤ p} =(1, 0, 0, 0),, (0,1,p2, 0), (0, 0,p3, 0),(0, 0,1,1)) , for which: {M52: B} = (p,p2,p2,p2) M16, AutBM52 = M4*, (B : M52)-s = p3s, ((ℤ* p)4: M4*) = p2(p - 1)2, (ℤ 4 p: M52 ) = p3.
M53 = {(u, v, w, t) ∈ℤ4 p: (v - w) ∈ p2 ℤ p, (v - u) , (t - v) ∈ p ℤ p}= ((p, 0, 0, 0), (1,1,1,1), (0, 0,p2, 0), (0, 0, 0,p)) , for which: {M53: B} = (p,p2,p2,p2) M16, AutBM53 = M* 53, (B : M53)-s = p2s, ((ℤ* p)4: M53*) = p(p - 1)3, (ℤ 4 p: M53) = p4.
M54 (a) = {(u, v, w, t) ∈ℤ4 p: (av - (1 + a)w + t) ∈ p2 ℤ p, (t - u) , (v - t) ∈ p ℤ p} = ((p, 0, 0, 0), (0,p,p(1+ a)-1a, 0), (0, 0, (1 + a)-1p2, 0), (1,1,1,1)) , for a e {1,...,p - 2} and for which: {M54 (a) : B} = (p,p2,p2,p2) M16, AutBM54 (a) = M* 54 (a) , (B : M54 (a))-s = p2s, ((ℤ* p)4: M54*)) = p(p - 1)3, (ℤ 4 p: M54(a)) = p4.
M55 = {(u,v,w,t) ∈ℤ4 p: (v - w) ∈ p2 ℤ p, (t - v) ∈p ℤ p} = ( (1 , 0, 0, 0), (0, 1 , 1 , 1) , (0, 0, p2 , 0) , (01, 0, 0, p) ) , for which: { M55: B} = ( p, p2 , p2 , p2 ) M16 , AutBM55 = M*55, (B : M55)-s = p3s, ((ℤ* p)4: M*55) = p(p - 1)2, (ℤ 4 p: M55) = p3.
M56 (a) = {(u, v, w, t) ∈ℤ4 p: (av - (1 + a)w + t) ∈ p2 ℤ p, (v - t) ∈ p ℤ p} = 0, 0, 0), (0,p,p(1 + a)-1a, 0), (0, 0,(1 + a)-1p2, 0), (0,1,1,1)) , for a ∈ {1,...,p - 2} , and for which: {M56 (a) : B} = (p,p2,p2,p2) M16, AutBM56 (a) = M*56 (a) , (B : M*56 (a))-s = p3s, ((ℤ* p)4: M*56(a) = p(p - 1)2, (ℤ 4 p: M56(a)) = p3.
M57 = {(u, v, w, t) ∈ ℤ4 p: (pv - w + t) ∈ p2 ℤ p, (t - u) ∈ p ℤ p} = ((p, 0, 0, 0), (0,1,p, 0), (0, 0,p2, 0), (1, 0,1,1)) , for which: {M57: B} = (p,p2,p2,p2) M16, AutBM57 = M7*, (B : M57)-s = p3s, ((ℤ* p)4: M7*) = p(p - 1)3, (ℤ 4 p: M57) = p3.
M58 = {(u,v,w,t) ∈ ℤ4 p: (pv - w + t) ∈ p2 ℤ p} = ((1, 0, 0, 0), (0,1,p, 0), (0, 0,p2, 0), (0, 0,1,1)) , for which: {M58: B} = (p,p2,p2,p2) M16, AutBM58 = M8*, (B : M58)-s = p4s, ((ℤ* p)4: M8*) = p(p - 1)2, (ℤ 4 p: M58) = p2.
M59 = {(u, v, w, t) ∈ ℤ4 p: (v - pw + t) ∈ p2 ℤ p, (t - u) ∈ p ℤ p} = ((p, 0, 0, 0), (0,p2, 0, 0), (0,p, 1, 0), (1, -1, 0,1)) , for which: {M59: B} = (p,p2,p3,p3) M24, AutBM59 = M*17, (B : M59)-s = p3s, ((ℤ* p)4: M*17) = p(p - 1)3, (ℤ 4 p: M59) = p3.
M60 = { (u, v, w, t) ∈ ℤ4 p: (v - pw + t) ∈ p2 ℤ p, } = ( (1 , 0, 0, 0) , (0, p2 , 0, 0) , (0, p, 1 , 0) , (0, - 1 , 0, 1) ) , for which: { M60: B} = ( p, p2 , p3 , p3) M24, AutBM60 = M*18, (B : M60)-s = p4s, ((ℤ* p)4: M18*) = p(p - 1)2, (ℤ 4 p: M60) = p2.
M61= {(u, v, w, t) ∈ ℤ4 p: (v - w + t) , (t - u) ∈ p ℤ p}= ((p, 0, 0, 0), (0,p, 0, 0), (0,1,1, 0), (1,-1, 0,1)) , for which: {M61: B} = (p,p2,p3,p3) M24, AutBM61 = M*13, (B : M61)-s = p4s, ((ℤ* p)4: M*13 )= (p - 1)3, (ℤ 4 p: M61) = p2.
M62 = {(u, v, w, t) ∈ ℤ4 p: (v - w + t) ∈ p ℤ p} = ((1, 0, 0, 0), (0,p, 0, 0), (0,1,1, 0), (0,-1, 0,1)) , for which: {M62: B} = (p,p2,p3,p3) M24, AutBM62 = M*14, (B : M62)-s = p5s, ((ℤ* p)4: M14*) = (p - 1)2, (ℤ 4 p: M62 ) = p.
M63 = {(u, v, w, t) ∈ ℤ4 p: (v - w) , (t - u) ∈ p ℤ p }= ((p, 0, 0, 0), (0,p, 0, 0), (0,1,1, 0), (1,0, 0,1)) , for which: {M63: B} = (p,p2,p3,p3) M24, AutBM63 = M63, (B : M63)-s = p4s, ((ℤ* p)4: M63) = (p - 1)2, (ℤ 4 p: M63 ) = p2.
M64 = {(u, v, w,t) ∈ ℤ4 p: (v - w) ∈ p ℤ p } = ((1, 0, 0, 0), (0,p, 0, 0), (0,1,1, 0), (0,0, 0,1)) , for which: {M64: B} = (p,p2,p3,p3) M24, AutBM64 = M*64, (B : M64)-s = p5s, ((ℤ* p)4: M64) = (p - 1), (ℤ 4 p: M64 ) = p.
5). From Eq. (1) for J5, we obtain the following list of representatives of isomorphism classes of fractional ideals of B :
M65 = {(u,v,w,t) ∈ ℤ4 p: (pv - w + t) ∈ p3 ℤ p, (t - u) ∈ p ℤ p} = ((p, 0, 0, 0), (0,1,p, 0), (0, 0,p3, 0),(1, 0,1,1)) , for which: {M65* : B} = (p,p2,p2,p2) M16, AutBM65 = B *, (B : M65*)-s = p2s, ((ℤ* p)4: B *) =p3(p - 1)3, (ℤ 4 p: M65*) = p4.
M66 = {(u, v, w,t) ∈ ℤ4 p: (pv - w + t) ∈ p3 ℤ p} = ( (1 , 0, 0, 0), (0, 1 , p, 0), (0, 0, p3 , 0) , (,0, 0, 1 , 1) ) , for which: { M66: B} = ( p, p2 , p2 , p2 ) M16 , AutBM66 = M2*, (B : M66)-s = p3s, ((ℤ* p)4: M2*) = p3(p - 1)2, (ℤ 4 p: M66) = p3.
M67 = {(u,v,w,t) ∈ ℤ4 p: (v - w + t) ∈ p2 ℤ p, (t - u) ∈ p ℤ p} = ((p, 0, 0, 0), (0,1,1, 0), (0, 0,p2, 0), (1, 0,1,1)) , for which: {M67: B} = (p,p2,p3,p3) M24, AutBM67 = M5*, (B : M67)-s = p3s, ((ℤ* p)4: M5*) = p2(p - 1)3, (ℤ 4 p: M67) = p3.
M68 = {(u, v, w, t) ∈ ℤ4 p: (v - w + pt) ∈ p2 ℤ p, (t - u) ∈ p ℤ p} = ((p, 0, 0, 0), (0,1,1, 0), (0, 0,p2, 0), (1, 0,p, 1)) , for which: {M68: B} = (p,p2,p3,p3) M24, AutBM68 = M53, (B : M68)-s = p3s, ((ℤ* p)4: M53) = p(p - 1)3, (ℤ 4 p: M68 ) = p3.
M69 = {(u, v, w, t) ∈ ℤ4 p: (v - w) ∈ p2 ℤ p, (t - u) ∈ p ℤ p} = ((p, 0, 0, 0), (0,1,1, 0), (0, 0,p2, 0), (1, 0, 0,1)) , for which: {M69: B} = (p,p2,p3,p3) M24, AutBM69 = M69, (B : M69)-s = p3s, ((ℤ* p)4: M69*} = p(p - 1)2, (ℤ 4 p: M69) = p3.
M70 = {(u,v,w,t) ∈ ℤ4 p: (v - w + t) ∈ p2 ℤ p} = ((1,0 0, 0, 0), (0,1,1, 0), (0, 0,p2, 0), (0, 0,1,1)) , for which: {M70: B} = (p,p2,p3,p3) M24, AutBM70 = M6*, (B : M70)-s = p4s, ((ℤ* p)4: M6*) = p2(p - 1)2, (ℤ 4 p: M70) = p2.
M71 = {(u,v,w,t) ∈ ℤ4 p: (v - w + pt) ∈ p2 ℤ p} = ( (1 , 0, 0, 0), (0, 1 , 1 , 0) , (0, 0, p2 , 0) , (0, , 0, p, 1) ) , for which: { M71: B} = ( p, p2 , p3 , p3) M24, AutBM71 = M55, (B : M71)-s = p4s, ((ℤ* p)4: M*72) = p(p - 1)2, (ℤ 4 p: M71) = p2.
M72 = {(u,v,w,t) ∈ ℤ4 p: (v - w) ∈ p2 ℤ p} = 0, 0, 0), (0,1,1, 0), (0, 0,p2, 0), (0, 0, 0,1)) , for which: {M72: B} = (p,p2,p3,p3) M24, AutBM72 = M*72, (B : M72)-s = p4s, ((ℤ* p)4: M72*) = p(p - 1), (ℤ 4 p: M72) = p2
6). From Eq. (1) for J6, we obtain the following list of representatives of isomorphism classes of fractional ideals of B:
M73 (a) ={(u, v, w,t) ∈ ℤ4 p: (pau - v + t) ∈ p2 ℤ p, (p2u - w + t) ∈ p3 ℤ p} = ((1,pa,p2, 0), (0,p2, 0, 0), (0, 0,p3, 0), (0, 1, 1,1)) , for a ∈ Fp*. {M73 (a) : B} = (p,p,p,p) M8, AutBM73 (a) = B *, (B : M73 (a))-s = ps, ((ℤ* p)4: B *) = p3(p - 1)3, (ℤ 4 p: M73(a)) = p5.
M74 (a) = {(u, v, w, t) ∈ ℤ4 p: (p2u - w + t) ∈ p3 ℤ p, (u - v + at) ∈ p ℤ p} = ((1, 1,p2, 0), (0,p, 0, 0), (0, 0,p3, 0), (0,a, , for a ∈ Fp. {M74 (a) : B} = (p,p2,p2,p2) M16, AutBM74 (a) = M3*, (B : M74 (a))-s = p2s, ((ℤ* p)4: M3*) = p2(p - 1)3, (ℤ 4 p: M74(a)) = p4.
M75 (a) ={(u, v, w, t) ∈ ℤ4 p: (pau - v + t) ∈ p2 ℤ p, (pu - w + t) ∈ p2 ℤ p} = ((1,pa,p, 0), (0,p2, 0, 0), (0, 0,p2, 0), (0,1,1,1)) for a ∈ Fp*. {M75 (a) : B} = (p,p2,p2,p2) M16, AutBM75 (a) = M5*, (B : M75 (a))-s = p2s, ((ℤ* p)4: M5*) = p2(p - 1)3, (ℤ 4 p: M7*(a)) = p4.
M76 (a) = {(u, v, w, t) ∈ ℤ4 p: (pu - w + t) ∈ p2 ℤ p, (u - v + at) ∈ p ℤ p} ={(1, 1,p, 0), (0,p, 0, 0), (0, 0,p2, 0), (0,a, 1,1)) , for a e Fp. {M76 (a) : B} = (p,p2,p2,p2) M16, AutBM76 (a) = M7*, (B : M76 (a))-s = p3s, ((ℤ* p)4: M7*) = p(p - 1)3, (ℤ 4 p: M76(a)) = p3.
M77 (a) = {(u, v, w, t) ∈ ℤ4 p: (pu - v + t) ∈ p2 ℤ p, (u - w + at) ∈ p ℤ p} = ((1,p, 1, 0), (0,p2, 0, 0), (0, 0,p, 0), (0,1,a, 1)) , for a ∈ Fp. {M77 (a) : B} = (p,p2,p3,p3) M24, AutBM77 (a) = M*17, (B : M77 (a))-s = p3s, ((ℤ* p)4: M17*) = p}p - 1)3, (ℤ 4 p: M77(a)) = p3.
M78 = {(u, v, w, t) ∈ ℤ4 p: (u - w) ∈ p ℤ p, (u - v) ∈ p ℤ p} = ((1,1,1, 0), (0,p, 0, 0), (0, 0,p, 0), (0,0, 0,1)) , for which: {M78: B} = (p,p2,p3,p3) M24, AutBM78 = M*78, (B : M78)-s = p4s, ((ℤ* p)4: M78*) = (p - 1)2, (ℤ 4 p: M78) = p2.
M79 (a) = {(u, v, w, t) ∈ ℤ4 p: (u - w + t) ∈ p ℤ p, (u - v + at) ∈ p ℤ p} = ((1,1,1, 0), (0,p, 0, 0), (0, 0,p, 0), (0,a, 1,1), , for a ∈ Fp. {M79 (a) : B} = (p,p2,p3,p3) M24, AutBM79 (a) = M*13, (B : M79 (a))-s = p4s, ((ℤ* p)4: M*13) = (p - 1)3, (ℤ 4 p: M79(a)) = p2.
M80 = {(u, v, w, t) ∈ ℤ4 p: (u - w) ∈ p ℤ p, (u - v + t) ∈ p ℤ p} = ((1,1,1, 0), (0,p, 0, 0), (0, 0,p, 0), (0, 0, 0,1)) , for which: {M80: B} = (p,p2,p3,p3) M24, AutBM80 = M*13, (B : M80)-s = p4s, ((ℤ* p)4: M*13) = (p - 1)3, (ℤ 4 p: M80) = p2.
7). From Eq. (1) for J7, we obtain the following list of representatives of isomorphism classes of fractional ideals of B :
M81 (a) ={ (u, v, w, t) ∈ ℤ4 p: (pv - w + t) ∈ p3 ℤ p, (u - v + at) ∈ p ℤ p} = ((p, 0, 0, 0), (1,1,p, 0), (0, 0,p3, 0), (-a, 0,1,1)) for a ∈ Fp. {M81 (a) : B} = (p,p2,p2,p2) M16, AutBM81 (a) = B *, (B : M81 (a))-s = p2s, ((ℤ* p)4: B *) = p3(p - 1)3, (ℤ 4 p: M81(a)) = p4.
M82 = {(u, v, w, t) ∈ ℤ4 p: (v - w) ∈ p2 ℤ p, (u - w) ∈ p ℤ p} = ((p, 0, 0, 0), (0,p2, 0, 0), (1,1,1, 0), (0, 0, 0,1)) , for which: {M82: B} = (p,p2,p3,p3) M24, AutBM82 = M*82, (B : M82)-s = p3s, ((ℤ* p)4: M82*) = p(p - 1)2, (ℤ 4 p: M82) = p3.
M83 = {(u, v, w, t) ∈ ℤ4 p: (v - w) ∈ p2 ℤ p, (u - w + t) ∈ p ℤ p} = ((p, 0, 0, 0), (0,p2, 0, 0), (1,1,1, 0), (-1, 0, 0,1)) , for which: {M83: B} = (p,p2,p3,p3) M24, AutBM83 = M*53, (B : M83)-s = p3s, ((ℤ* p)4: M*53) = p(p - 1)3, (ℤ 4 p: M83) = p3.
M84 (a) = {(u, v, w, t) ∈ ℤ4 p: (v - w + pt) ∈ p2 ℤ p, (u - w + at) ∈ p ℤ p} = ((p, 0, 0, 0), (0,p2, 0, 0), (1,1,1, 0), (-a, -p, 0,1)) for a ∈ Fp. {M84 (a): B} = (p,p2,p3,p3) M24, AutBM84 (a) = M*53, (B : M84 (a))-s = p3s, ((ℤ* p)4: M53) = p(p - 1)3, (ℤ 4 p: M84( a)= p3.
M85 (a) = {(u, v, w, t) ∈ ℤ4 p: (v - w + t) ∈ p2 ℤ p, (u - w + at) ∈ p ℤ p} = ( ( p, 0, 0, 0) , (0, p2 , 0, 0) , (1 , 1 , 1 , 0) , ( - a, - 1 , 0, 1) ) for a ∈ Fp. { M85 ( a) : B} = (p,p2,p3,p3) M24, AutBM85 (a) = M*, (B : M85 (a))-s = p3s, ((ℤ* p)4: M5*) = p2(p - 1)3, (ℤ 4 p: M85 ( a) ) = p3.
8). From Eq. (1) for J8, we obtain the following list of representatives of isomorphism classes of fractional ideals of B :
M86 (a) = {(u, v, w, t) ∈ ℤ4 p: (pav - (1 + pa)w - p2u + t) ∈ p3 ℤ p, (v - t) ∈ p ℤ p} = 0, -p2(1+ pa)-1, 0), (0,p,p2a(1+ pa)-1, 0), (0, 0, 0,p3), (0,1,1,1)) , for a ∈Fp*. ,M86(a): B} = (p,p,p,p) M8, AutBM86 (a) = M49* (a) , (B : M86 (a))-s = p2s, ((ℤ* p)4: M49*(a) = p2(p - 1)3, (ℤ 4 p: M86(a)) = p4.
M87 (a) = {(u, v, w, t) ∈ ℤ4 p: (p2u - w- p2u+t) ∈ p3 ℤ p, = (1, 0,−p2, 0), (0, 1, p2, 0), (0, 0, p3, 0), (0, 0, 1, 1), for which: {M87: B} = (p, p2, p2, p2) M16, AutBM87 = M3*, (B : M87)-s = p3s, ((ℤ* p)4: M3*) = p2(p - 1)3, (ℤ 4 p: M87) = p3.
M88 (a) = { (u, v, w, t) ∈ ℤ4 p: (av - (1 + a)w - pu + t) ∈ p2 ℤ p, (v - t) ∈ p ℤ p} = {(1, 0, -p(1+ a)-1, 0), (0,p,pa(1+ a)-1, 0), (0, 0,p2, 0), (0,1,1,1)) , for a e {1,...,p - 2} {M88(a): B} = (p,p2,p2,p2) M16, AutBM88 (a) = M*54 (a) , (B : M88 (a))-s = p ((ℤ* p)4: M*54(a) = p(p - 1)3, (ℤ 4 p: M88(a)) = p3.
M89= {(u, v, w, t) ∈ ℤ4 p: (v - w - pu) ∈ p2 ℤ p, (v - t) ∈ p ℤ p} = {(1, 0, -p, 0), (0,1,1,1), (0, 0,p2, 0),(0, 0, 0,p)) , for which: {M89: B} = (p,p2,p2,p2) M16, AutBM89 = M*53, (B : M89)-s = p3s, ((ℤ* p)4: M53) = p(p- 1)3, (ℤ 4 p: M89) = p3.
M90 = {(u, v, w, t) ∈ ℤ4 p: (pv - pu - w + t) ∈ p2 ℤ p} = {(1, 0, -p, 0), (0,1,p, 0), (0, 0,p2, 0),(0, 0,1,1)) , for which: {M90: B} = (p,p2,p2,p2) M16, AutBM90 = M7*, (B : M90)-s = p4s, ((ℤ* p)4: M7*) = p(p - 1)3, (ℤ 4 p: M90) = p2.
M91 = {(u, v, w, t) ∈ ℤ4 p: (pu - v + pw + t) ∈ p2 ℤ p} = ((1,p, 0, 0), (0,p2, 0, 0), (0,p, 1, 0), (0,1, 0,1)) , for which: {M91: B} = (p,p2,p3,p3) M24, AutBM91 = M*17, (B : M91)-s = p4s, ((ℤ* p)4: M*17) =p(p - 1)3, (ℤ 4 p: M91 ) = p2.
M92 = {(u, v, w, t) ∈ ℤ4 p: (v - u - w + t) ∈ p ℤ p} = ((1,1, 0, 0), (0,p, 0, 0), (0,1,1, 0), (0,-1, 0,1)) , for which: {M92: B} = (p,p2,p3,p3) M24, AutBM92 = M*13, (B : M92)-s = p5s, ((ℤ* p)4: M*13} = (p - 1)3, (ℤ 4 p: M92) = p.
M93 = {(u, v, w,t) ∈ ℤ4 p: (v - w - u) ∈ p ℤ p} = ( (1 , 1 , 0, 0) , (0, p, 0, 0) , (0, 1 , 1 , 0), (0,, 0, 0, 1) ) , for which: { M93: B} = ( p, p2 , p3 , p3) M24, AutBM93 = M78*, (B : M93)-s = p5s, ((ℤ* p)4: M78*) = (p - 1)2, (ℤ 4 p: M93 ) = p.
9). From Eq. (1) for J9, we obtain the following list of representatives of isomorphism classes of fractional ideals of B :
M94 = {(u, v, w, t) ∈ ℤ4 p: (pv - w - p2u + t) ∈ p3 ℤ p} = {(1, 0, -p2, 0), (0,1,p, 0), (0, 0,p3, 0), (0, 0,1,1)}, for which: {M94: B} = (p,p2,p2,p2) M16, AutBM94 = B *, (B : M94)-s = p3s, ((ℤ* p)4: B *) = p3(p- 1)3, (ℤ 4 p: M94) = p3.
M95 = {(u, v, w, t) ∈ ℤ4 p: (pu + w - v + t) ∈ p2 ℤ p} = ( (1 , 0, - p, 0) , (0, 1 , 1 , 0) , (0, 0, p2 , 0) ,, (0, 0, - 1 , 1)}, for which: { M95: B} = ( p, p2 , p3 , p3) M24, AutBM95 = M5*, (B : M95*)-s = p4s, ((ℤ* p)4: M*5) = p2(p- 1)3, (ℤ 4 p: M95) = p2.
M96 = {(u, v, w, t) ∈ ℤ4 p: (pu + w - v + pt) ∈ p2 ℤ p} = ( (1 , 0, - p, 0) , (0, 1 , 1 , 0) , (0, 0, p2 , 0) , ,(0, 0, - p, 1) ) for which: { M96: B} = ( p, p2 , p3 , p3) M24, AutBM96 = M53*, (B : M96)-s = p4s, ((ℤ* p)4: M53) = p(p - 1)3, (ℤ 4 p: M96) = p2.
M97 = {(u, v, w, t) ∈ ℤ4 p: (pu + w - v) ∈ p2 ℤ p} = ( (1 , 0, - p, 0) , (0, 1 , 1 , 0) , (0, 0, p2 , 0) , ,(0, 0, 0, 1) ) , for which: { M97: B} = ( p, p2 , p3 , p3) M24, AutBM97 = M82*, (B : M97)-s = p4s, ((ℤ* p)4: M*82) = p(p - 1)2, (ℤ 4 p: M97) = p2.
Remark 3.1. We have that M1,...,M97 form a set of representatives of all the isomorphism classes of fractional ideals of finite index in Bp (Cp3( . Inn the previous list we observe that (lie only conductors are Bp (Cp3) , (p,p,p,p) (ℤ p x Bp (Cp2)) , (p,p2,p2,p2) (ℤ p x Bp (Cp)) and ((p,p2,p3,p3)) ℤ p. Therefore, in the following subsection (The Local Zeta Function for Bp (Cp 3)) it will be sufficient to compute the integrals corresponding to these four conductors.
Remark 3.2. The way to get {Mi: B} , AutBMi, (B : Mi) -s , ((ℤ *p)4: (ℤ p 4: Mi ), for i = 1,... , 97, is very similar. As an example, we present a sketch of proof for M81:
We choose a Haar measure d*x on (ℚp *)4 , such as
and then, we have that μ∗(AutBM81)−1 =(( ℤ ∗p)4: AutBM81).
Now, if M81 (a) =(u, v,w, t) ∈ ℤ4 p: (pv − w + t) ∈ p3 ℤp, (u − v + at) ∈ p ℤp for a ∈ Fp, and (u, v,w, t) ∈ M81(a), then w = t+pv +p3w′ and u = v −at+pu′ for w′, u′ ∈ ℤp. It follows that.
and it is easy to see that as ℤp−module
First, let’s see how to calculate {M81(a) : B} . We know that
If (x, y,w, z) ∈ {M81(a) : B} , then from Eq. (2) we have that
From Eq. (6) it follows that w ∈ p2ℤp and z −w ∈ p3ℤp, then z ∈ p2ℤp. From Eq. (4) it follows that pw − y ∈ p2ℤp and y − x ∈ pℤp, then y ∈ p2ℤp and x ∈ pℤp. From Eq. (3) and Eq. (5) it follows that x ∈ ℤp and w ∈ ℤp respectively, then y, z ∈ ℤp. So, it is easy to see that
Next, let’s see how to calculate AutBM81(a). We know that
If (x, y, w, z) ∈ EndBM81(a), then from Eq. (2) we have that
From Eq. (8) and Eq. (10), it follows that y − x ∈ pℤp, w − y ∈ p2ℤp and z − w ∈ p3ℤp. From Eq. (7) and Eq. (9), it follows that x, w ∈ ℤp respectively, and then y, z ∈ ℤp. So, it is easy to see that EndBM81(a) = M1 = B, and then
To calculate µ∗(AutBM81(a))−1 = ((ℤ∗p)4: B∗). Let
where x = x0 + px1 + p2x2 + · · · ; y = y0 z = z0 + pz1 + p2z2 + · · · , for x0, y0, w0, z0 + py1 + p2y2 + · · · ; w = w0 + pw1 + p2w2 + · · · ; ∈ Fp ∗ and xi, yi, wi, zi ∈ Fp for i = 1, 2, ...; It is easy to see that φ is a surjective homomorphism of multiplicative groups, such that ker(φ) = B∗. The first isomorphism theorem gives
and then
To calculate ℤ 4 p: M81(a) . Let
where x = x0 + px1 + p2x2 + · · · ; y = y0 + py1 + p2y2 + · · · ; w = w0 + pw1 + p2w2 + · · · ; z= z0 + pz1 + p2z2 + · · · , and xi, yi, wi, zi ∈ Fp for i = 0, 1, 2, ...; It is easy to see that φ is a surjective homomorphism of additive groups, such that ker(φ) = M81(a). The first isomorphism theorem gives
and then
Finally, we know that (M81(a) : B) = (M81(a) : ℤp 4) (ℤ4 p: B = p6 (ℤ4: M81(a) −1 then
3.2. The Zeta function of B Cp3
Proposition 3.3. Let G be a finite group and let
It follows that
Now, by the Euler product, we have that
Finally, from the Theorem 2.3 we obtain that
Corollary 3.4. Let n ∈ ℕ and B (Cp
n) be the Burnside ring for a cyclic group of order pn, and let e
Proof. We have that there are n + 1 conjugacy classes of Cpn , therefore , then from the above proposition, it follows that
Now, by the Euler product, it follows that
Now, by Remark 2.4, since fCpn (q−s )=1, when q # p, according to Theorem 2.3 we obtain:
The Local Zeta Function for Bp Cp3
Remember that:
Hence, to compute the zeta function of Bp Cp3 , it is necessary to compute ZBp Cp3 (Mi; s) for i = 1, . . . , 97. According to the previous subsection, we only need to compute the integrals that we will study in the following four Remarks.
Remark 3.5. We choose a Haar measure d∗x on (ℚ∗p)4 , such that d∗x = (d∗α)4 , where d∗α is a Haar measure of ℚ∗p such that . Thus
Besides, we have
Thus, from Eq. (11), we obtain:
Remark 3.6. We choose a Haar measure d* z on (ℚ∗p)2, such that d* z = (d* α)2. We know that, Bp (Cp) is local, where rad (Bp (Cp)) = (p, p) ℤp 2. Thus and then
Besides. We have:
Thus, from Eq. (11) and Eq. (12), we obtain: .
Remark 3.7. We choose a Haar measure d* y on (ℚ∗p)3 such that d* y = (d*α)3. We know that, Bp (Cp2) is local, where rad (Bp (Cp2)) = (p,p,p)[ ℤp x Bp (Cp)]. Thus and then
Besides, we have:
Thus, from Eq. (11) and Eq. (13), we obtain:
Remark 3.8. We know that, Bp (Cp3) is local, where rad (Bp (Cp3) ) (p,p,p,p) { ℤp x Bp Cp2)]. Thus and
Proposition 3.9. Let p be a rational prime and let B = Bp (Cp3) be the Burnside ring for a cyclic group Cp3 of order p3. Therefore, the zeta function will be:
Proof. Remember that:
Hence, from case 1) in subsection 3.1, along with Remark 3.8, we obtain:
From case 1) in subsection 3.1, along with Remark 3.7, we obtain:
From case 1) in subsection 3.1, along with Remark 3.6, we obtain:
From case 1) in subsection 3.1, along with Remark 3.5, we obtain:
From case 2) in subsection 3.1, along with Remark 3.7, we obtain:
From case 2) in subsection 3.1, along with Remark 3.6, we obtain:
From case 2) in subsection 3.1, along with Remark 3.5, we obtain:
From case 3) in subsection 3.1, along with Remark 3.7, we obtain:
From case 3) in subsection 3.1, along with Remark 3.6, we obtain:
From case 3) in subsection 3.1, along with Remark 3.5, we obtain:
From case 4) in subsection 3.1, along with Remark 3.7, we obtain:
From case 4) in subsection 3.1, along with Remark 3.6, we obtain:
From case 4) in subsection 3.1, along with Remark 3.5, we obtain:
From case 5) in subsection 3.1, along with Remark 3.6, we obtain:
From case 5) in subsection 3.1, along with Remark 3.5, we obtain:
From case 6) in subsection 3.1, along with Remark 3.7, we obtain:
From case 6) in subsection 3.1, along with Remark 3.6, we obtain:
From case 6) in subsection 3.1, along with Remark 3.5, we obtain:
From case 7) in subsection 3.1, along with Remark 3.6, we obtain:
From case 7) in subsection 3.1, along with Remark 3.5, we obtain:
From case 8) in subsection 3.1, along with Remark 3.7, we obtain:
From case 8) in subsection 3.1, along with Remark 3.6, we obtain:
From case 8) in subsection 3.1, along with Remark 3.5, we obtain:
From case 9) in subsection 3.1, along with Remark 3.6, we obtain:
From case 9) in subsection 3.1, along with Remark 3.5, we obtain:
From the 97 partial zeta functions above, we obtain that:
and finally, from Remarks 3.5 to 3.8, we obtain that
3.3. Some relations for ZB (Mi; s)
Lastly, we will study a couple of relations that satisfy the zeta functions ZB (Mi; s) .
Let τ be the mapping such that
We will denote by
a). For each
we have that {Mi : B} = p, p2, p3, p3 ℤ4p which satisfy . Thus
Hence, according to the functional equation given in [12, Theorem 2.3] the following relations are fulfilled:
Where besides, therefore, from Eq. (14) we obtain
b). For each i ∈ {9, ..., 18, 27, ..., 33, 39, ..., 42, 51, ..., 58, 65, 66, 74, ..., 76, 81, 87, ..., 90, 94} we have that from which we obtain thus
Hence, for we have that , besides, , therefore, from Eq. (14) we obtain
c). For each i ∈ {2, ..., 8, 25, 26, 37, 38, 49, 50, 73, 86} we have that
from which we obtain therefore, the condition required in functional equation given in [12, Theorem 2.3], is not fulfilled.
d). Finally, for i = 1 we have that {M1: B} = B which satisfies therefore, the condition required in the functional equation given in [12, Theorem 2.3], is not fulfilled.