ON THE DISTRIBUTION OF IWASAWA INVARIANTS ASSOCIATED TO MULTIGRAPHS

Abstract Let 
$\ell $
 be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian 
$\ell $
 -towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime 
$\ell $
 and the abelian 
$\ell $
 -tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa 
$\mu $
 and 
$\lambda $
 invariants.


Introduction
Classical Iwasawa theory.Let ℓ be a prime number, K be a number field and Z ℓ denote the ring of ℓ-adic integers.A Z ℓ -tower over K consists of a sequence of abelian extensions such that for all values of n, the Galois group Gal(K n /K) is isomorphic to Z/ℓ n Z.The union K ∞ := n≥1 K n is an infinite Galois extension of K with Galois group Gal(K ∞ /K) isomorphic to Z ℓ .We refer to K ∞ as a Z ℓ -extension of K.For n ≥ 0, we denote by Cl ℓ (K n ), the Sylow ℓ-subgroup of the ideal class group of K n , and by h ℓ (K n ) the cardinality of Cl ℓ (K n ).In his seminal work [Iwa73a], Iwasawa studied growth patterns for h ℓ (K n ), as n → ∞.More specifically, it is shown in loc.cit.that there are invariants µ, λ and ν, which only depend on the prime ℓ and the extension K ∞ (and independent of n), such that for n ≫ 0, (1.1) The quantities µ and λ are non-negative integers, while the quantity ν is an integer, possibly negative.Of special interest is the case when K ∞ /K is the cyclotomic Z ℓ -extension of K, in which case we simply denote the Iwasawa invariants by µ ℓ (K), λ ℓ (K) and ν ℓ (K).By the celebrated result of Ferrero and Washington [FW79], µ ℓ (K) is equal to 0 for all abelian extensions K/Q.Moreover, it is conjectured in [Iwa73b] that µ ℓ (K) is 0 for all number fields K .The λ-invariant is more subtle, and it is natural to study statistical questions about its average behavior.
Let n > 0 be an integer, let F be a family of number fields of fixed degree n.We do not provide an explicit definition of what constitutes a family, however it is natural to study number fields satisfying prescribed splitting conditions at a fixed set of primes.For instance, it is natural to consider the case when F is the family of imaginary quadratic fields in which p is inert (or split).
For a number field K, we let |∆ K | be the absolute discriminant.Given a real number x > 0, let F ≤x be the subset of F consisting of number fields K such that |∆ K | ≤ x.Given µ 0 ≥ 0 and λ 0 ≥ 0 and a real number x > 0, we let F ≤x (µ 0 , λ 0 , ℓ) consist of K ∈ F ≤x such that µ ℓ (K) = µ 0 and λ ℓ (K) = λ 0 .Note that both F ≤x and F ≤x (µ 0 , λ 0 , ℓ) are finite sets.The following question motivates the line of inquiry in this manuscript.
This can be regarded as an analogue of similar questions on classical Iwasawa invariants of number fields.The heuristics for the distribution of these arithmetic Iwasawa invariants have been studied by Ellenberg, Jain and Venkatesh [EJV11].One may attempt to compute the limit above for various choices of (µ 0 , λ 0 ).Of course, the µ = 0 conjecture would imply that d(µ 0 , λ 0 , ℓ) = 0 if µ 0 > 0. It turns out that the above question, though of considerable interest, leads to difficulties that are yet to be resolved systematically.
Iwasawa theory of multigraphs.In the present article, we consider graph-theoretic analogs of Iwasawa invariants, for which questions similar to Question 1.1 become tractable for certain families of multigraphs.What might be called the Iwasawa theory of multigraphs is a recent area of research.The reader may refer to [Val21,MV22a,MV22b] where the approach via Artin-Ihara L-functions is introduced and to [Gon22] where the module theoretical approach was initiated.Other papers on the subject where the reader can find more information are [LV23], [DV22], [KM22], [RV22], and [Kat23].However, at present, there is no standard text on the subject.
A finite multigraph X consists of a finite set of vertices joined by directed edges and an inversion map satisfying prescribed properties.For the precise definition of a multigraph, the reader is referred to Definition 2.1.In this article, we allow there to be more than one edge joining any two vertices.We also allow for loops based at a single vertex.It is assumed that there are only finitely many edges, and the multigraph is connected.Given a prime ℓ, a connected Z ℓ -tower over X consists of a sequence of covers of connected multigraphs such that for every integer n ≥ 0, the Galois group Gal(X n /X) is isomorphic to Z/ℓ n Z.
Note that such a tower is not necessarily unique.The multigraphs X n can be constructed from a Z ℓ -valued voltage assignment on the edges of X.For further details, please refer to §2.2 where this construction is summarized.We establish a sufficient condition for X n to be connected for all n ≥ 0 (see Theorem 2.15).This particular criterion is potentially of independent interest.The inverse limit lim ← −n Gal(X n /X) is isomorphic to Z ℓ , and thus a Z ℓ -tower of a multigraph is analogous to a Z ℓ -extension of a number field.The Picard group Pic 0 (X n ) has cardinality equal to the number of spanning trees in X n , for more details, cf.[Sun13,Theorem 10.8].The cardinality of the Sylow ℓ-subgroup of Pic 0 (X n ) is denoted by κ ℓ (X n ).It is shown by McGown and the fourth named author (cf.[Val21,MV22a,MV22b]) that there are invariants µ, λ ∈ Z ≥0 and ν ∈ Z such that (1.2) κ ℓ (X n ) = ℓ (µℓ n +λℓ+ν) for n ≫ 0. This establishes an analog of Iwasawa's formula (1.1) in the setting of multigraphs.The formula (1.2) above was also independently obtained by Gonet, see [Gon21b,Gon22].We prove that when ℓ is odd, the invariant λ is in fact always an odd integer (see Theorem 2.18), which again may be of independent interest.
Main results.The main goal of the present paper is to study the distribution of Iwasawa µ and λ-invariants for multigraphs that feature in (1.2).We study two related questions in this context.First, we fix a multigraph X and allow the Z ℓ -tower over X to vary in a natural way and study the variation of Iwasawa invariants.These investigations are carried out for multigraphs with one or two vertices as well as certain families of Z ℓtowers of complete graphs.We expect that the methods developed in the present paper can be generalized to many more families of multigraphs.However, the calculations may become cumbersome for general multigraphs.The second theme is more closely related to arithmetic statistics.Here, we consider all multigraphs with a given number of vertices, and their connected Z ℓ -towers and study the variation of µ and λ-invariants for this entire family.This question can be formulated for multigraphs with a fixed number of vertices N for any number N .However, we find that the computations are significantly involved even in the case when N = 1.We refer to question (2) for the precise formulation of the problem in this setting.Below is a more detailed description of the results we prove.
(A) In §4, we give a detailed analysis of the setting where X is a bouquet (a multigraph with a single vertex).In particular: (A.1) We first establish several sufficient and/or necessary conditions for µ = 0.This allows us to prove that µ is always zero and give an upper bound on λ under the hypothesis that ℓ is larger than the number of loops in a bouquet (see Corollary 4.3 and Theorem 4.13 for the precise statements).Without this hypothesis, we show that both µ and λ can be arbitrarily large (see Lemma 2.17), which mirrors similar results in the setting of Iwasawa theory of number fields and elliptic curves [Iwa73b, Mat07, Kim09, KR22].(A.2) Let t be the number of loops in a bouquet.We study the distribution of µ and λ as the Z ℓ -tower varies.As we shall see in §2.2, where we review the Iwasawa theory of multigraphs, the connected Z ℓ -towers of a bouquet with t loops are in one-one correspondence with Z t ℓ \ (ℓZ ℓ ) t .In particular, the Haar measure on Z ℓ gives a measure on the set of all connected Z ℓ -towers of a given bouquet.Under this measure, we are able to obtain an upper bound on the probability for µ > 0, and for µ = 0 with a fixed choice of λ which is sufficiently small (see Theorems 4.6 and 4.16).For µ = 0 and λ = 1, we are able to prove a precise formula (see Theorem 4.18).We also investigate the cases where t = 2, 3 separately in §4.3.(A.3) Specializing to the case where the Z ℓ -towers correspond to the vectors α = (α 1 , . . ., α t ) ∈ Z t \ (ℓZ) t , we show in Theorem 4.21 that as both t and α vary, the probability of µ = 0 and λ = 1 is equal to 1 − ℓ −1 , mirroring a similar result obtained by the first and third named authors in the context of 2-bridge links (see [DR22, Theorem 5.4]).(B) In §5, we study multigraphs with two vertices.In particular, we determine a precise formula for the probability of a Z ℓ -tower of a given multigraph with two vertices to have µ = 0 and λ = 1 via the theory of quadratic forms over finite fields.See in particular Theorem 5.3.(C) Finally, we study Z ℓ -towers of complete graphs.In particular, Theorem 6.4 gives a precise description of the distribution of µ and λ for certain families including the so-called cyclic voltage covers of the complete graph K u with u vertices (which were initially studied by Gonet [Gon21a]) as u varies.
Outlook.The present paper initiates the study of distributions of Iwasawa invariants of multigraphs and answers several statistical questions in some special cases.Our main results rely crucially on studying the power series in Z ℓ [[T ]] that governs the growth formula (1.2).For example, in order to study when µ = 0 and λ = 1 occurs, we are faced with the task of analyzing whether the leading term of the aforementioned power series is divisible by ℓ.In the cases being treated, we are able to reformulate this question in terms of explicit quadratic forms over F ℓ and employ results on finite fields (e.g. the Chevalley-Warning theorem).On developing further results on systems of equations over finite fields in several variables, one may study more than one coefficient of a power series in Z ℓ [[T ]] simultaneously.This should then allow us to extend our results on µ = 0 and λ = 1 to more general combinations of Iwasawa invariants.As mentioned above, carrying out similar calculations beyond the cases we have studied so far can potentially be cumbersome.It would be desirable to develop a systematic approach to study the questions inaugurated in the present paper for more general families of graphs and multigraphs.For example, it would be very interesting to extend our methods to study distributions of Iwasawa invariants that arise from Erdös-Rényi random graphs.
We plan to study some of these questions in the future.
Acknowledgement.CD's research is supported by the NSERC Canada Graduate Scholarships -Doctoral program.AL's research is supported by the NSERC Discovery Grants Program RGPIN-2020-04259 and RGPAS-2020-00096.DV would like to thank Kevin Mc-Gown for several stimulating discussions.We thank the anonymous referee for valuable comments and suggestions.

Preliminaries
This section is dedicated to the discussion of preliminary notions in this paper.
2.1.Basic properties of multigraphs.Fix a prime number ℓ.We introduce basic notions in the theory of multigraphs and Iwasawa theory of Z ℓ -towers.For further details, the reader may refer to [MV22b, §4] and [LV23,§2].A finite undirected graph X consists of a finite set V X of vertices {v 1 , . . ., v n }, and a finite number of edges E X joining the vertices.For such a graph, there is at most one edge which joins v i and v j .Thus, in this context, we may view E X as a subset of V X × V X .More specifically, if there is an edge joining v i to v j , then (v i , v j ) belongs to E X .Since the graph X in question is undirected, we have that The graph X is fully described by the pair (V X , E X ).On the other hand, a directed graph is a graph for which there is at most one directed edge from v i to v j .In this case, the set of edges is denoted E + X , and is a subset of V X × V X .More specifically, there is a directed edge joining v i to v j precisely when (v i , v j ) ∈ E + X .A multigraph is a graph with possibly multiple directed edges joining any two vertices, subject to some further conditions.In this case, the set E + X is not a subset of V X × V X , but simply admits a map to V X × V X .The notion of a multigraph is more general than that of a graph.We give the precise definition below.
Definition 2.1.A finite multigraph X is the data consisting of a finite set of vertices V X and a finite set of directed edges E + X along with functions Here, i and ι are the incidence and inversion functions, respectively, and are subject to the following conditions (1) (2) ι(e) ̸ = e for all e ∈ E + X , (3) i(ι(e)) = τ (i(e)) for all e ∈ E + X , where τ (x, y) := (y, x).
Given a directed edge e ∈ E + X , the incidence function maps e to i(e) = (v, v ′ ).The edge starts at v and ends at the vertex v ′ .We define functions o, t : E + X → V X , such that i(e) = (o(e), t(e)).In other words, o(e) (resp.t(e)) is the projection of i(e) to the first (resp.second) coordinate.We refer to the function o (resp.t) as the source (resp.target) function.The inversion map ι satisfies the property that i(ι(e)) = (t(e), o(e)).
We shall occasionally denote the inverse ι(e) by ē.These two conventions will be used interchangeably throughout.
The set of directed edges e starting at a vertex v and ending at a vertex v ′ are the edges which satisfy the relation i(e) = (v, v ′ ).We note that there may be any number of directed edges joining two vertices v and v ′ , thus the graph X is referred to as a multigraph.We allow for there to be edges to start and end at the same vertex v, and such edges are referred to as loops based at v. We note that for distinct vertices v and v ′ , there are as many directed edges starting at v and ending at v ′ as there are from v ′ to v.Moreover, given a vertex v, there are an even number of directed loops starting and ending at v.
Example: Let us illustrate the above definitions through an example.The set of vertices is V X = {v 1 , v 2 }.There are a total of 6 directed edges E + X = {a, b, c, ā, b, c}, as depicted in the figure below.
2.2.Iwasawa theory of multigraphs.In this section, we discuss the Iwasawa theory of multigraphs.The basic object of study is a Z ℓ -tower of multigraphs.Let X be a multigraph.Recall from the previous section that the data associated with X consists of a finite set of vertices V X = {v 1 , . . ., v n } and finitely many directed edges E + X , along with an incidence map i : E + X → V X × V X , and an inversion map ι : E + X → E + X .These maps are subject to further conditions.Before introducing Z ℓ -towers of multigraphs, we first discuss the notion of a Galois cover of multigraphs.For v ∈ V X , let for all e ∈ E + X .We will often write f for both f V and f E .Definition 2.2.Let X and Y be two graphs and let f : Y → X be a morphism of graphs.If f satisfies the following conditions then f is said to be a cover.
If Y /X is a cover and Y is connected, then there exists a positive integer d such that |f −1 (v)| = d for all v ∈ V X .The integer d is called the degree of Y /X and will be denoted by Definition 2.3.The cover f : Y → X is called Galois if the following two conditions are satisfied.
(1) The graphs X and Y are connected.
(2) The group Aut f (Y /X) acts transitively on the fiber If Y /X is a Galois cover, then [Y : X] = |Aut f (Y /X)|, and we write Gal f (Y /X), or Gal(Y /X) if f is understood, instead of Aut f (Y /X).One has the usual Galois correspondence between subgroups of Gal(Y /X) and equivalence classes of intermediate covers of Y /X.
Galois covers of a multigraph can be constructed from certain combinatorial data associated to a multigraph, as we now explain.Define an equivalence relation on E + X , by setting e ∼ e ′ if e = e ′ or e = ē′ .The set of equivalence classes E X := E + X /ι is the set of undirected edges, and let π : E + X → E X be the 2 : 1 natural quotient map, mapping a directed edge to the associated undirected edge.Choose a section γ : E X → E + X to π, and set S := γ(E X ).Our main reference for voltage assignments and derived multigraphs is [GT01], but we rephrase everything in the formalism used in [Sun13].
Definition 2.4.Let G be a finite abelian group.With respect to notation above, a voltage assignment is a function α : S → G.
We extend α to all of E + X by setting α(s) = α(s) −1 .To this data, one can associate a multigraph X(G, S, α) as follows.The set of vertices is V = V X ×G and the set of directed edges is E + = E + X ×G.The directed edge (e, σ) connects (o(e), σ) to (t(e), σ•α(e)) (where we recall that o(e) and t(e) denote the origin and the target of the edge e, respectively), and the inversion map is given by If G 1 is another finite abelian group and f : G → G 1 is a group morphism, then one obtains a morphism of multigraphs If both X(G, S, α) and X(G, S, f • α) are connected and f is surjective, then f * is a cover in the sense of Definition 2.2 and in fact is a Galois cover with group of covering transformations isomorphic to ker(f ).In particular, if f : G → {1} is the group morphism into the trivial group and both X and X(G, S, α) are connected, then we get a Galois cover f * : X(G, S, α) → X with group of covering transformation isomorphic to G.
We now recall the notion of a Z ℓ -tower of multigraphs.
Definition 2.5.Let ℓ be a rational prime, and let X be a connected graph.A Z ℓ -tower above X consists of a sequence of covers of connected graphs such that for all positive integer n, the cover X n /X obtained from composing the covers X n → X n−1 → . . .→ X 1 → X is Galois with Galois group Gal(X n /X) isomorphic to Z/ℓ n Z.
We specialize the above construction to the case when G = Z ℓ .Set α /n to denote the mod-ℓ n reduction of α.We obtain a Z ℓ -tower above X: Set t := |S| and enumerate S = {s 1 , . . ., s t }, and let α i denote α(s i ).We may view α as a vector α = (α 1 , α 2 , . . ., α t ) ∈ Z t ℓ .The following assumption on the multigraphs X(Z/ℓ n Z, S, α /n ) is necessary to relate the growth patterns in Picard groups to Iwasawa invariants.In section 2.3, we introduce conditions under which the following assumption is satisfied.
Assumption 2.6.Let ℓ be a prime (with ℓ = 2 allowed) and let α : S → Z ℓ be a function.We assume that the derived multigraphs X(Z/ℓ n Z, S, α /n ) are connected for all n ≥ 0.
Examples: Let us illustrate the above construction of a Z ℓ -tower through some explicit examples.
By Theorem 2.15 below, all the multigraphs X(Z/3 n Z, S, α /n ) are connected.We obtain a Z 3 -tower above the dumbbell multigraph X which can be pictured as follows: Note that the graphs X(Z/3 n Z, S, α /n ) are precisely the generalized Petersen graphs G(3 n , 11).Let u X be the number of vertices of X and label V X = {v 1 , . . ., v u X }.Let o, t : E + X → V X be the origin and terminus functions taking an edge to its source and target vertex, respectively.Thus, we have that i(e) = (o(e), t(e)).
= v}, and let val X (v) := #E v be the valency of v.The divisor group Div(X) is the free abelian group on the vertices V X of X.It consists of formal sums of the form D = v∈V X n v v, where n v ∈ Z for all v.The degree of D is the sum deg(D) := v n v ∈ Z, and this defines a homomorphism deg : Div(X) → Z, the kernel of which is denoted Div 0 (X).Let M(X) be the abelian group of Z-valued functions on V X .The discrete Laplacian map div : M(X) → Div(X) is defined as follows.For f ∈ M(X), one sets We let Pr(X) be the image of div.It is easy to see that Pr(X) is a subgroup of Div 0 (X), since the subjacent directed graph is eulerian, meaning that every vertex has in-degree equals to its out-degree.The Picard group is taken to be the quotient Pic(X) := Div(X)/ Pr(X), and we set Pic 0 (X) := Div 0 (X)/ Pr(X).Set κ X to be the cardinality of Pic 0 (X), which is a finite group and is equal to the number of spanning trees in X.
The valency matrix is the u X × u X diagonal matrix D = (d i,j ), with The adjacency matrix A = (a i,j ) is given by a i,j = Twice the number of undirected loops at i, i = j; The number of undirected edges connecting the i-th and j-th vertices, i ̸ = j.
The matrix Q := D − A is called the Laplacian matrix.
We let Z ℓ [x] (resp.Z ℓ T ) denote the polynomial ring (resp.formal power series ring) in the indeterminate x (resp.T ) with coefficients in Z ℓ .Denote by Z ℓ [x; Z ℓ ] the ring whose elements are expressions of the form f (x) = a c a x a , where a runs over a finite set in Z ℓ and the coefficient c a is in Z ℓ .Here, multiplication is defined by setting x a x b = x a+b for a, b ∈ Z ℓ .Associate a matrix M (x) to the pair (X, α) as follows Let (1 + T ) b be the formal power series With this convention in place, we define the characteristic series to be which is a formal power series in Z ℓ T .The convention used in this article differs from that used in [MV22b], where −T is used in place of T .This convention simplifies our calculations.
Lemma 2.7.Let X be a multigraph, ℓ be a prime and α : S → Z ℓ a function.The constant term of the characteristic series f (T ) (see (2.2)) is equal to 0.
Proof.Note that the constant term of f (T ) is det(Q).The vector (1, . . ., 1) t is in the null-space of Q, see [CP18, proof of Corollary 9.10] for further details.Therefore, Q is singular and the result follows.□ The following result shows that f X,α (T ) ̸ = 0 under the assumptions imposed on α.The proof is essentially contained in [Val21,MV22a,MV22b], we recall these details.First, we introduce some further notation.Let Y /X be an abelian cover of multigraphs and set G := Gal(Y /X).Let G be the group of characters ψ : G → C × .Given ψ ∈ G, denote the Artin-Ihara L-function by Here, the product runs over all primes c of X, l(c) is the length of c and Y /X c is the Frobenius automorphism at c. (Recall that a prime in a multigraph is an equivalence class of closed paths that have no backtrack or tail, and that are not multiples of a strictly smaller closed path, where two such paths are identified if they are the same up to their starting vertex.For further details, the reader is referred to [Ter11, Chapter 18].)Let χ(X) denote the Euler characteristic of X, which is defined as follows We shall impose the following assumption without further mention.
That χ(X) ̸ = 0 means that X does not consist of a single cycle, or a "cycle with hair", cf.[Ter11, Figure 2.1].When χ(X) = 0, X is referred to as a "bad graph", cf.P.10 of op.cit.(and references therein), since many arithmetic properties of zeta functions are not satisfied in this case.For instance, the analog of the prime number theorem for such bad graphs does not hold, cf.Theorem 10.1 of op.cit.Definition 2.9.Let Y /X be an abelian cover of multigraphs with automorphism group G, and for each i = 1, . . ., u X , let w i be a fixed vertex in the fiber of v i .
(1) For σ ∈ G, we let the matrix A(σ) to be the u X × u X matrix A(σ) = (a i,j (σ)) defined via Twice the number of undirected loops at the vertex w i , if i = j and σ = 1; The number of undirected edges connecting w i to w σ j , otherwise.
Proof.The relationship between the power series If Y /X is an abelian cover of multigraphs and ψ is a character of G = Gal(Y /X), then the three-term determinant formula [Ter11, Theorem 18.15] for the Artin-Ihara L-function gives where A ψ is the twisted adjacency matrix of Definition 2.9, D is the valency matrix of X, and χ(X) is the Euler characteristic of X. From now on, we let Coming back to the abelian ℓ-tower where Here, it is understood that ζ ℓ n = exp(2πi/ℓ n ) and that an embedding of Q into Q ℓ has been fixed once and for all so that we view ζ ℓ n and the values of the characters where the product is over all non-trivial characters of Z/ℓ n Z, and κ n denotes the number of spanning trees of X(Z/ℓ n Z, S, α /n ).The above formula crucially requires the Assumption 2.8.It follows that h X (1, ψ) ̸ = 0 for all non-trivial characters, and thus Q(T ) ̸ = 0 by (2.3).□ Associated to the characteristic series, are the Iwasawa µ and λ invariants, which we now proceed to define.A polynomial is said to be a distinguished polynomial if it is monic and all non-leading coefficients are divisible by ℓ.
Definition 2.11.By the Weierstrass Preparation theorem, we may factor the characteristic polynomial as follows f (T ) = ℓ µ g(T )u(T ), where g(T ) is a distinguished polynomial in Z ℓ T , and u(T ) is a unit in Z ℓ T .Since u(T ) is a unit in Z ℓ T , the constant term u(0) is a unit in Z ℓ .According to Lemma 2.7, the constant term of f (T ) is equal to 0, and hence T divides g(T ).In particular, we have that deg g ≥ 1.The number µ ∈ Z ≥0 is the µ-invariant and depends on (ℓ, α).Throughout it is assumed that Assumption 2.6 is satisfied for α.Thus, we denote this invariant by µ ℓ (X, α), however, when ℓ, X and α : S → Z ℓ are understood, we simply denote it by µ.On the other hand, the λ-invariant is defined to be the quantity Remark 2.12.
• According to Lemma 2.10, we have that f (T ) ̸ = 0, and thus the µ and λ invariants are defined.• The ν-invariant is not detectable from the characteristic series, and we postpone its definition till Definition 2.14.• In our setting the λ-invariant is not the degree of g(T ), i.e., the conventional definition of the λ-invariant of a power series f (T ).The shift by −1 is necessary in studying the growth of Picard numbers in a Z ℓ -tower.
Let α : S → Z ℓ be a function such that Assumption 2.6 is satisfied.Given n ≥ 0, let X n = X(Z/ℓ n Z, S, α /n ) and write κ n for the number of spanning trees of Let ℓ be a prime and α be as above.Then, there exists n 0 > 0 and ν ∈ Z, such that for all n ≥ n 0 .The invariants µ and λ are given as in Definition 2.11.
Definition 2.14.The ν-invariant is the uniquely determined constant for which the asymptotic formula (2.4) holds.
2.3.Connectedness of multigraphs in abelian ℓ-towers.Let X be a connected multigraph, ℓ be a prime number and let α : S → Z ℓ .In this section, we introduce explicit conditions on X which are sufficient for Assumption 2.6 to hold.Let be an abelian ℓ-tower over X. Set also G n = Gal(X n /X).We have natural group morphisms that are compatible so that we get a group morphism Fix now a spanning tree of X, say T ⊆ E X , a section γ : E X → E + X , and set S := γ(E X ).We let S(T) be the collection of s ∈ S such that π(s) ∈ T. For each s ∈ S, let c s = d s • s be the closed path in X, where d s is the unique geodesic path in T going from t(s) to o(s).Let ⟨c s ⟩ be the corresponding cycle in H 1 (X, Z).Note that if s ∈ S(T), then ⟨c s ⟩ = 0.The collection By successively lifting the tree T to X 1 , then to X 2 and so on, one can construct isomorphisms of multigraphs for which all the squares and the triangle in the diagram commute.In other words, once a spanning tree T for X has been fixed (which also specifies a Z-basis for H 1 (X, Z)), every abelian ℓ-tower over X arises from a function α : S → Z ℓ satisfying α(s) = 0 for all s ∈ S(T).
Theorem 2.15.Let X be a connected multigraph and let T ⊆ E X be a fixed spanning tree of X.Furthermore, let S be the image of a section as above, and let α : S → Z ℓ be a function satisfying α(s) = 0 for all s ∈ S(T).Then X(Z/ℓ n Z, S, α /n ) is connected for all positive integer n if and only if there exists s ∈ S ∖ S(T) such that α(s) ∈ Z × ℓ .Proof.Let us assume first that there exists s ∈ S ∖ S(T) such that (α(s), ℓ) = 1.Let also X and ā ∈ Z/ℓ n Z.The map p n : X n → X defined via p n (v, ā) = v and p n (e, ā) = e is a morphism of multigraphs (recall that the vertices and (directed) edges of X n consist of elements in V X × Z/ℓ n Z and E + X × Z/ℓ n Z, respectively).Furthermore, for all w ∈ V Xn , the restriction p n | E Xn,w induces a bijection It follows that every path c in X has a unique lift c to X n once the starting point in the fiber of o(c) is fixed.Furthermore, since p n is a morphism of multigraphs one has which gives an injective group morphism Thus, by lifting c and using (2.5) above, for any vertex w ∈ p −1 n (v 1 ), there exists a path c in X n going from w to another vertex in p −1 n (v 2 ).It follows that in order to show that X n is connected, it suffices to show that every vertex in a single fiber can be joined via a path in X n .Under our assumption, there exists s 0 ∈ S ∖ S(T) such that (α(s 0 ), ℓ) = 1, and thus α /n (s 0 ) is a generator for Z/ℓ n Z.Let v ∈ V X be such that v = t(s 0 ) and let σ 0 = α /n (s 0 ) ∈ Z/ℓ n Z.The closed path c s 0 = d s 0 • s 0 can be lifted to a path cs 0 in X n starting at (v, 0).Furthermore, since α(s) = 0 for all s ∈ S(T), we have t(c s 0 ) = (v, σ 0 ).The path cs 0 • σ 0 (c s 0 ) will start at (v, 0) and end at (v, 2 • σ 0 ).Keeping going like this, we can find a path in X n going from (v, 0) to (v, m • σ 0 ) for all positive integer m.Since σ 0 generates Z/ℓ n Z, this shows that every vertex in the fiber of v can be joined via a path in X n .It follows that X n is connected.
Conversely, assume that all multigraphs X n are connected.In particular, X 1 is connected.To simplify the notation, write S ∖ S(T) = {s 1 , . . ., s t }.The function α induces a function on C 1 (X, Z) and thus also on H 1 (X, Z) by restriction.Let also v ∈ V X be a fixed vertex.Since we are assuming that X 1 is connected, there exists a path c in X 1 going from (v, 0) to (v, 1).The path c = p 1 (c) is then a closed path in X, and α(⟨c⟩) = 1.Since the ⟨c s i ⟩ for i = 1, . . ., t form a Z-basis of H 1 (X, Z), we have ⟨c⟩ = n 1 ⟨c s 1 ⟩ + . . .+ n t ⟨c st ⟩, for some n i ∈ Z. Applying α /1 to this last equation gives Now, if all the α(s i ) were divisible by ℓ, then we would get a contradiction.□ The following corollary is most likely well known, we list it here for convenience.
Corollary 2.16.Let X be a bouquet and α : S → Z ℓ .Then, Assumption 2.6 is satisfied if and only if there exists s ∈ S for which ℓ ∤ α(s).
Proof.Since X is a bouquet, the only possible spanning tree consists of the single vertex of X (and no edges).The result follows from Theorem 2.15.□ 2.4.Basic properties of Iwasawa invariants.In this subsection, we prove a number of basic results about the Iwasawa µ and λ invariants associated to multigraphs.The graph X is called a bouquet if there is 1 vertex, in which case M (x) is a 1 × 1-matrix.Suppose that there are t loops in total, labelled e 1 , . . ., e t and α = (α 1 , . . ., α t ) sends e i → α i .We denote the corresponding bouquet by X t .We shall set µ := µ ℓ (X t ; α) and We have that f (T ) = t j=1 f α j (T ), and therefore we may write In particular, we find that ord T =0 f (T ) ≥ 2, and thus λ ≥ 1.It is easy to see that µ = 0 and λ = 1 if and only if β 2 is a unit in Z ℓ .These are the minimal values of the Iwasawa invariants attainable by a bouquet.The following result shows that the Iwasawa invariants µ and λ can be arbitrarily large.
Lemma 2.17.Let n 1 ∈ Z ≥0 , n 2 ∈ Z ≥1 , and let ℓ be a prime number.There is a bouquet X t consisting of t = ℓ n 1 +1 + ℓ n 1 loops and a choice of vector α ∈ Z t ℓ \(ℓZ ℓ ) t (which depends on both n 1 and n 2 ) such that the associated Iwasawa invariants satisfy As a result, we find that The degree of 1 − (1 + T ) ℓ n 2 2 is 2ℓ n 2 , which is larger than the degree of (1+T ) Furthermore, all non-leading coefficients of 1 − (1 + T ) ℓ n 2 2 are divisible by ℓ.We have Thus, f (T ) = ℓ n 1 u(T )P (T ), where u(T ) = −(1 + T ) −ℓ n 2 is a unit in the Iwasawa algebra and P (T ) is a distinguished polynomial of degree 2ℓ n 2 .Thus, we have shown that µ = n 1 and λ = 2ℓ n 2 − 1. □ Theorem 2.18.Let X be a multigraph, let ℓ be an odd prime and α : S → Z ℓ be a function such that Assumption 2.6 is satisfied.Then, λ ℓ (X, α) is odd.

Equations over finite fields
Let ℓ be a prime (ℓ = 2 is allowed) and q be a power of ℓ.Denote by F q the finite field with q elements, and let f 1 , . . ., f m ∈ F q [x 1 , . . ., x n ].In this section, we catalogue some well known results on the number of solutions to the system of equations 3.1.Systems of equations.We begin with a generalization of the Chevalley-Warning theorem to a system of equations in many variables.Associate to I = (i 1 , . . ., i n ) ∈ Z n ≥0 the monomial x I := x i 1 1 x i 2 2 . . .x in n , then define the degree to be the sum Given f (x 1 , . . ., x n ) = I c I x I , the degree of f is defined as follows The polynomial is homogenous of degree d if deg(x I ) = d for all I such that c I ̸ = 0.The following is often referred to as Warning's second theorem, cf.[War36].
and that for all i, we have that f i (0, 0, . . ., 0) = 0. Let N be the number of solutions of Proof.The result follows from [LN97, Theorem 6.11].□ 3.2.Quadratic forms and their solutions.A quadratic form is a homogenous polynomial of degree 2. Assume throughout that q is odd, let η : F × q → {±1} be the quadratic character of F q , and extend η to a function on F q by setting η(0) = 0. Given a quadratic form Q(x 1 , . . ., x n ) over F q , express Q as with a i,j = a j,i .The rank of Q is the rank of the associated matrix A = (a i,j ) over F q .We let det Q denote the determinant of A. The form Q is non-degenerate if rank A = n, i.e., A is non-singular.On the other hand, Q is diagonal if a i,j = 0 unless i = j.Proposition 3.2.Suppose that ℓ is odd and q is a power of ℓ.Then, any quadratic form over F q is equivalent to a diagonal form.
Proof.The above result is [LN97, Theorem 6.21].□ Let v : F q → Z be the function defined by Theorem 3.3.Let Q(x 1 , . . ., x n ) be a non-zero quadratic form over F q , and assume that q is odd and n is even.For b ∈ F q , let N Q(x 1 , . . ., x n ) = b be the number of solutions to Q(x 1 , . . ., x n ) = b.Then, we have that where η is the quadratic character of F q and ∆ = det Q.
Proof.The result above is [LN97, Theorem 6.26].□ Given a degenerate quadratic form Q(x 1 , . . ., x n ), we let r denote the rank of Q.According to Proposition 3.2, Q is equivalent to the form r i=1 a i x 2 i , where a i are all non-zero.We set Q 0 (x 1 , . . ., x r ) ∈ F q [x 1 , . . ., x r ] to be the nondegenerate form x n ] be a quadratic form.Setting r := rank Q, we denote by N (Q = 0) the number of solutions of Q = 0.Then, we have that Proof.Let r denote the rank of Q, and first consider the case when r is even.According to Proposition 3.2, Q is equivalent to a diagonal form.Thus, without loss of generality, assume that there are non-zero constants a 1 , . . ., a r such that Thus, for every solution to there are q n−r solutions to Q(x 1 , . . ., x r , . . ., x n ) = 0.As a consequence, we find that N (Q = 0) = q n−r N (Q 0 = 0).Since Q 0 is nondegenerate, the result follows from Theorem 3.3, according to which On the other hand, if r is odd, we have that N (Q 0 = 0) = q r−1 , according to [LN97, Theorem 6.27].□

Statistics for the Iwasawa invariants of bouquets
Fix an odd prime number ℓ. Recall that a bouquet is a multigraph with a single vertex.As a multigraph, each undirected edge corresponds to a pair of edges oriented in opposite directions, starting and ending at the unique vertex.Thus in particular, a bouquet is uniquely determined by the number of undirected loops.Let X t be the bouquet with t undirected loops labelled e 1 , . . ., e t .Associated to a vector α = (α 1 , . . ., α t ) ∈ Z t ℓ is a Z ℓ -tower of X t .We say that α is admissible if for some coordinate α i , we have that ℓ ∤ α i .Note that according to Corollary 2.16, α is admissible if and only if Assumption 2.6 is satisfied.Let τ = τ ℓ be the Haar measure on Z t ℓ and note that τ Z t ℓ \ (ℓZ ℓ ) t = 1 − ℓ −t .In this section, we study two related questions: (1) Given a fixed bouquet X t , how do the Iwasawa invariants µ and λ vary, as α = (α 1 , . . ., α t ) varies over the set of admissible vectors in Z t ℓ ?It follows from Theorem 2.18 that the λ-invariant for a bouquet is odd.Thus, in particular, the λ invariant is ≥ 1.We fix constants µ 0 ∈ Z ≥0 and λ 0 ∈ Z ≥1 , let S(µ 0 , λ 0 , ℓ, t) be the set of vectors α ∈ Z t ℓ \(ℓZ ℓ ) t such that µ ℓ (X t , α) = µ 0 and λ ℓ (X t , α) = λ 0 .This probability is computed as follows The above formula does make sense since in the cases of interest, we shall implicitly show that S(µ 0 , λ 0 , ℓ, t) is measurable.Our computations are possible for certain values of µ 0 and λ 0 .The probabilities Prob ℓ (X t ; µ > µ 0 ), Prob ℓ (X t ; µ = µ 0 ), Prob ℓ (X t ; µ = µ 0 , λ ≥ λ 0 ) etc. are defined similarly.
4.1.The vanishing of the µ-invariant.In this section, we study when the µ-invariant of a Z ℓ -tower of a bouquet X t vanishes.We are interested in the power series (4.1) where we recall that Upon expanding the binomial coefficients, we deduce that the coefficient of where c n,i ∈ Z is independent of a.More precisely, c n,i is the coefficient of x 2i in −x(x − 1) . . .(x − n + 1).Furthermore, if n = 2m is even, then the highest term in a is a 2m with coefficient c 2m,m = −1, whereas if n = 2m + 1 is odd, then the highest term is a 2m with coefficient c 2m+1,m = m(2m + 1).We write f (T ) = n≥2 β n T n , in particular, we have Lemma 4.1.Let ℓ be an odd prime number and f (T ) be the power series associated to the Z ℓ -tower of the bouquet X t given by the admissible vector α = (α 1 , . . ., α t ).Suppose that the associated µ-invariant µ ℓ (X t , α) is positive.Then, for all m ≥ 1, we have that Proof.If µ > 0, then, all coefficients β n are divisible by ℓ µ .Setting p i := k α 2i k , we find that Then, we find that ⌊n/2⌋ i=1 c n,i p i ≡ 0 mod ℓ µ .Set n = 2m, and obtain from the above that Therefore, by induction on m, we find that p m ≡ 0 mod ℓ µ for all m.□ Next, we prove a result on the µ-invariant.Note that since α is assumed to be admissible, we have that #{k : α k ̸ ≡ 0 mod ℓ} > 0.
Proof.Let us write p i := t k=1 α 2i k .By Lemma 4.1, we find that p i ≡ 0 mod ℓ µ for all i ≥ 1.
Let us write N = #{k : α k ̸ ≡ 0 mod ℓ}.For 1 ≤ i ≤ N , let p ′ i denote the power sum k and let e i denote the symmetric sum the same elements.In particular, p ′ i ≡ 0 mod ℓ for all i.Newton's identities (see [Mea92]) say that where e 0 is defined to be 1.In particular, N e N ≡ 0 mod ℓ.By definition, Thus, we deduce that ℓ|N .□ The result below improves upon [Val21, Corollary 5.8].
Proof.Note that since there exists some value α k such that ℓ ∤ α k , we find that #{k : and the result follows.□ The following result is a refinement of Proposition 4.2.
Proof.Recall from the proof Proposition 4.2 that we have This implies that The determinant of the Vandermonde matrix (x i ) x∈Q ℓ ,1≤i≤(ℓ−1)/2 is nonzero mod ℓ.Therefore, we deduce that r x ≡ 0 mod ℓ µ for all x.□ Remark 4.5.The converse to the statement in Proposition 4.4 is not true.For instance, let ℓ = t = 3 and α = (1, 1, 2), we find that In particular, the µ-invariant is zero.This also gives a counterexample for the converse of both Lemma 4.1 and Proposition 4.2.
Proof.If µ > 0, then Proposition 4.2 tells us that #{k : α k ̸ ≡ 0 mod ℓ} is a multiple of ℓ.Since each residue class modulo ℓ in α ∈ Z t ℓ \ (ℓZ ℓ ) t has the same measure, it is enough to count the number of elements in (r 1 , . . ., r t ) ∈ F t ℓ \ {0} such that #{k : r k ̸ = 0} is a multiple of ℓ and t k=1 r 2i k = 0, for all i ≥ 1.For each 1 ≤ i ≤ ⌊t/ℓ⌋, we may choose iℓ indices k 1 , . . ., k iℓ so that r j ̸ = 0 if and only if j ∈ {k b | b = 1, . . ., iℓ}.By Proposition 4.4, amongst these iℓ elements, we have to distribute r 2 j over the m classes in Q ℓ so that the number of times each quadratic residue occurs is a multiple of ℓ.
Let a 1 ℓ, . . ., a m ℓ be the number of times the residue classes in Q ℓ occur.We have There are (iℓ)!/ m i=1 (a j ℓ)! ways to distribute iℓ distinguishable elements in m distinguishable sets so that the number of elements in these sets are given by a 1 ℓ, . . ., a m ℓ.Once we have distributed the iℓ elements amongst the m classes, each element has two possible values modulo ℓ, giving us 2 iℓ choices.Summing up all possible choices of a j 's gives the expression on the right-hand side of the inequality.□ Remark 4.7.Our numerical calculations suggest that for a given ℓ, the right-hand side of the inequality given by Theorem 4.6 tends to In what follows, we prove a necessary and sufficient condition for µ > 0 in the special case where α k ∈ Z for all k.Given an integer a ∈ Z, let us write d k (a) for the coefficient of X k in the shifted Chebyshev polynomial P a (X) used in [MV22a,§2].More explicitly, where the Chebyshev polynomials T a (X) ∈ Z[X] of the first kind are defined by the relation T a (cos(θ)) = cos(aθ) for all θ ∈ R (see [MV22a,§2]).s∈S P α(s) (X) belongs to ℓZ[X] if and only if ℓ | t j for all j = 1, . . ., m. Proof.We have P a (X) = P −a (X) for all a ∈ Z.Thus, (4.2) It is clear that ℓ | t j for all j = 1, . . ., m implies that the polynomial above belongs to ℓZ[X].
Proof.Theorem 2.18 asserts that λ is odd, thus λ ≥ 1.Alternatively, this may be directly shown as follows.Express f (T ) as ℓ µ P(T )u(T ), where P(T ) is a distinguished polynomial and u(T ) is a unit in Λ.Here µ = µ ℓ (X t , a) is the µ-invariant and λ = λ ℓ (X t , a) = deg P − 1.Note that since β i = 0 for i ≤ 2, it follows that T 2 divides P(T ), and thus deg P(T ) ≥ 2 and λ ≥ 1.
Then, the following assertions hold.
Proof.First, assume that µ = 0 and λ = 2k − 1. Setting p i := j α 2i j , we find that According to Lemma 4.11, β n is divisible by ℓ for all values of n < 2k, and β 2k is not divisible by ℓ.The assertion of part (2) follows immediately from this.We now prove (1).In particular, we assume that 2k < ℓ.Then, 2 n! is an ℓ-adic unit for all integers n such that 0 ≤ n ≤ 2k.
Suppose that (b) holds.Then, p i is divisible by ℓ for all i < k, whereas p k is not divisible by ℓ.It follows that ℓ divides β n for all n < 2k, but β 2k is not divisible by ℓ.Thus, (a) holds by Lemma 4.11.

4.3.
Results for small values of t.In this subsection, we study the distribution of the λ-invariant in greater detail for various values of t, starting with t = 1.Throughout, it is assumed that α = (α 1 , . . ., α t ) is admissible.
The bouquet X 1 consists of a single loop.Let ℓ be any prime number (ℓ = 2 is allowed).Even though Assumption 2.8 does not hold, we can compute to find that β 2 = α 2 .Since ℓ ∤ α, we find that β 2 ∈ Z × ℓ , and thus, µ = 0 and λ = 1 for all choices of a ∈ Z × ℓ .In particular, we find for all primes ℓ, The Iwasawa invariants are seen to be µ = 0 and λ = 1 for all choices of α.
Proposition 4.19.Suppose that ℓ ≥ 3 and t = 2.Then, we have that µ We now suppose that ℓ ≥ 5 (the case ℓ = t = 3 seems a lot more subtle; see Remark 4.5 and 4.14).As before, Theorem 4.13 tells us that the µ-invariant is always zero.We study the distribution of λ-invariants in the following proposition.
Theorem 4.21.With respect to notation above, Remark 4.22.We note here that the limit above is equal to the limit of the densities in Theorem 4.18 as t → ∞.This does not follow from formal arguments and further work is required to obtain the result.In fact, it is crucial that the quantity δ defined above lies in the open interval (0, 1).The method used in proving Theorem 4.21 does not extend to δ ≥ 1.
Lemma 4.23.With respect to notation above, for some absolute constant c > 0 which is independent of t and x.
Proof.The number of α such that max The number of tuples such that such that max |α 1 |, |α 2 |, . . ., |α t | ≤ x and ℓ divides all α i is 2⌊x/ℓ⌋ + 1 t .Therefore, we find that We have that for an absolute constant c > 0. In the above inequalities, we have used the mean value theorem to obtain the bound and similarly with x replaced by x/ℓ.□ Proposition 4.24.With respect to the notation above, there is an absolute constant c > 1 such that Proof.Note that Therefore, we deduce from Lemma 4.23 that After replacing c with a larger absolute constant, we find that each of the three terms in the above sum are < c 3 (2x) ⌊x δ ⌋−1 , and thus, their sum is < c(2x) ⌊x δ ⌋−1 .This proves the result.□ Set g(ℓ, t) to be the number of tuples ᾱ = (ᾱ 1 , . . ., ᾱt ) ∈ F t ℓ such that Note that by the proof of Theorem 4.18, we have that Lemma 4.25.With respect to the notation above, Note that #U is equal to ℓ t − 1 − g(ℓ, t).Given ᾱ ∈ U, set T ≤x (ᾱ) to denote the subset of T ≤x = T ≤x (0, 1, ℓ, t) consisting of tuples α that reduce to ᾱ modulo-ℓ.Let α ′ i be the integer such that 0 ≤ α ′ i < ℓ and that α ′ i reduces to ᾱi .We have It then follows that Let This allows us to deduce that Proof.We have the following estimates In the above, the second inequality follows from Lemma 4.25, and the third inequality follows from (4.6).□ Proposition 4.27.With respect to notation above, we have that for some absolute constant c > 0.
Proof.By Lemma 4.26, we have that where c > 0 is an absolute constant.Note that in the above sum, we used the estimate

□
Proof of Theorem 4.21.It follows from Proposition 4.24 that By Proposition 4.27, Recall that δ < 1, and thus ⌊x δ ⌋(2x) −1 goes to 0 as x goes to ∞.We thus find that and the result follows from (4.7).□

Statistics for multigraphs with two vertices
In this section, we study statistical problems associated to multigraphs consisting of 2 vertices v 1 and v 2 .We study an analogue of Question (1) for such graphs.First, let us set up some notation.Let p (resp.q) be the number of undirected loops starting and ending at v 1 (resp.v 2 ).Denote by r the number of undirected edges joining v 1 and v 2 .The Iwasawa invariants we study significantly depend on the choice of the set S. Let e (resp.g) be the number of directed edges in S starting at v 1 (resp.v 2 ) and ending at v 2 (resp.v 1 ).Note that e + g = r.We shall assume that r ≥ 1.If e = 0 and r ≥ 1, then we replace v 1 with v 2 and thus without loss of generality assume that e ≥ 1.Any graph with 2-vertices and choice of S corresponds to the 5-tuple (p, q, r, e, g) and let X e,g p,q,r be the associated datum.It conveniences us to express the vector α as α = (a 1 , . . ., a p , b 1 , . . ., b e , c 1 , . . ., c g , d 1 , . . ., d q ), where a i is the value of α at the i-th loop in S based at v 1 .We set b i to be the value of α at the i-th directed edge in S starting at v 1 and ending at v 2 .Denote by c i the value of α at the i-th directed edge in S starting at v 2 and ending at v 1 .Finally, let d i be the value of α at the i-th loop in S based at v 2 .
We write ⃗ a, ⃗ b, ⃗ c and ⃗ d to denote the vectors (a 1 , . . ., a p ), (b 1 , . . ., b e ), (c 1 , . . ., c g ) and (d 1 , . . ., d q ), respectively.Then, α may be represented as a matrix where the (i, j)-th entry in this matrix represents the values that α takes at the various directed edges that start at v i and end at v j .Thus, the vectors ⃗ a and ⃗ d represent the values taken at loops starting and ending at v 1 and v 2 , respectively.
Recall that Theorem 2.15 gives conditions for Assumption 2.6 to be satisfied.We fix a spanning tree T and assume that α : S → Z ℓ satisfies the conditions of Theorem 2.15.Without loss of generality, assume that T consists of a single edge joining v 1 to v 2 such that S(T) consists of the first directed edge joining v 1 to v 2 .Then the condition that α is equal to 0 on S(T) is equivalent to the condition that b 1 = 0.The condition that α(s) ∈ Z × ℓ for some s ∈ Z × ℓ means that one of the coordinates of α is in Z × ℓ .We say that α is admissible if these conditions on α are satisfied.It follows that if α is admissible, then Assumption 2.6 is satisfied.Set t := p + q + r − 1 and given α, set α 0 ∈ Z t ℓ denote the vector α 0 = (a 1 , . . ., a p , b 2 , . . ., b e , c 1 , . . ., c g , d 1 , . . ., d q ), obtained by dropping the coordinate for b 1 .When α is admissible, at least one of the coordinates of α 0 is not divisible by ℓ.
For example, let ℓ = 5, X the multigraph consisting of two vertices v 1 , v 2 , two loops at v 1 , one loop at v 2 and two undirected edges between v 1 and v 2 , and S = {s 1 , s 2 , s 3 , s 4 , s 5 }, where s 1 , s 2 are the loops at v 1 , s 3 , s 4 are both directed edges going from v 1 to v 2 , and s 5 is a loop at v 2 .Let α : S → Z ℓ be defined via s 1 , s 2 , s 5 → 1, s 3 → 0, and s 4 → 2, then we get the Z 5 -tower The power series f (T ) starts as follows Thus, µ = 0 and λ = 3.Using SageMath, we calculate We have ord 5 (κ n ) = 3n, for all n ≥ 1.
Fixing the choice of S and T, we parametrize the set of admissible vectors by the set of all vectors α 0 ∈ Z t ℓ / (ℓZ ℓ ) t .We denote this set by A(ℓ) and note that the measure equals ℓ t − 1 = (1 − ℓ −t )ℓ t .Let µ α and λ α be the Iwasawa µ and λ-invariants of X e,g p,q,r corresponding to the choice of voltage vector α.Given constants µ 0 and λ 0 , let T (µ = µ 0 , λ = λ 0 , ℓ) be the subset of A(ℓ) such that µ α = µ 0 and λ α = λ 0 .Let Prob ℓ (µ = µ 0 , λ = λ 0 ) denote the probability that µ = µ 0 and λ = λ 0 for X e,g p,q,r .More precisely, the probability is given by where τ is the Haar measure on Z t ℓ as in §4.We are able to compute the above probability in the special case µ 0 = 0 and λ 0 = 1.Since the λ-invariant is odd, these are the minimal values of the Iwasawa invariants.The main result of this section is Theorem 5.3.
Recall that D is the diagonal matrix given by D = diag (2p + r), (2q + r) , and write M (1 + T ) = m i,j (T ) 1≤i,j≤2 (cf.(2.1)).We are able to explicitly calculate the four values of m i,j (T ) as follows We write f (T ) = det M (1 + T ) = ∞ k=0 β k T k and compute the first few values of β k .
Proposition 5.2.Let ℓ be an odd prime number and p, q, r, e, g as above.Then, the following assertions hold.
(1) If ℓ|r, then Q p,e,g,q is a degenerate quadratic form over F ℓ of rank 1.
(2) If ℓ ∤ r, then Q p,e,g,q is a non-degenerate quadratic form over F ℓ of rank t.
Proof.Let Q := Q p,e,g,q and let ⟨•, •⟩ be the associated bilinear form defined by Let u = x 1 , . . ., x p , y 2 , . . ., y e , z 1 , . . ., z g , w 1 , . . ., w q and We note that The assertion (1) is clear since the associated pairing is given by Therefore, in order for ⟨u, •⟩ to be identically 0, it is necessary and sufficient for e i=2 y i − g j=1 z j to equal 0. The rank is therefore equal to t − (t − 1) = 1.Let us prove part (2), i.e., we show that Q is non-degenerate if ℓ ∤ r.Let u ̸ = 0 be such that ⟨u, u ′ ⟩ = 0 for all u ′ .Thus x i and w n are all clearly 0. Consider u ′ such that all coordinates are 0 except for the coordinate y ′ j , which is 1.We find that Since ℓ ∤ r, we find that y j = y k for j ̸ = k.Next, consider u ′ such that all coordinates are 0 except for the coordinate z ′ j which is 1.We find that Setting y := 1 r e i=2 y i − g j=1 z j , it follows from (5.1) and (5.2) that y i = y, z j = −y for all i and j.We find that Therefore, it follows that y must be 0 in order for the form ⟨u, •⟩ to be identically 0.
Hence, we have shown that u = 0, and thus, Q is non-degenerate.□ Theorem 5.3.Let ℓ be an odd prime and fix (p, q, r, e, g) such that e + g = r; set t := p + q + r − 1.Then, the following assertions hold.
(1) Suppose that ℓ | r.Then, (2) Suppose that ℓ ∤ r.Then, there is a number υ ∈ {+1, −1} for which Proof.First, we prove (1), and thus assume that ℓ | r.Let Q be the quadratic form defined in Proposition 5.2.Note that by part (1) of Proposition 5.2, the rank of Q is equal to 1. Let ᾱ0 be the reduction of α 0 modulo ℓ.It follows from Lemma 5.1 that µ α = 0 and λ α = 1 if and only if Q(ᾱ 0 ) ̸ = 0. Therefore, we find that According to Proposition 3.4, N (Q = 0) = ℓ t−1 .Therefore, we find that For part (2), it follows from Proposition 5.2 part (2) that Q is non-degenerate, and thus the rank is t.Setting υ to denote η (−1) t/2 ∆(Q) , it follows from Proposition 3.4 that Therefore, we find that
Computations for values of u up to 7 suggest that the value of β 2 may be given by the following general formula: In order to write down an explicit formula for the first non-zero coefficient of f (T ), we will make simplifying assumptions.First, let us suppose the case where all the terms a i,j are zero except for one.This is referred to as a single voltage assignment and we note that computations in this case have been carried out in [Gon21a].For simplicity, let a := a 12 be the non-zero value and assume that ℓ ∤ a.Then, the derived multigraphs K u (Z/ℓ n Z, S, α /n ) are connected for all n, which follows from Theorem 2.15 by taking T to be any spanning tree that does not contain an edge joining v 1 to v 2 .Under these assumptions, only the coefficient β 2 will be needed to compute the distribution of the Iwasawa invariants.If one wanted to treat the case of more general voltage assignments, then one would need to compute β n for n ≥ 3. Lemma 6.2.Let α be a single voltage assignment described above.The µ-invariant of f (T ) is ord ℓ ((u − 2)u u−3 ) and its λ-invariant is 1.
Since (1 + T ) −a is a unit in the Iwasawa algebra, multiplying f (T ) by (1 + T ) a does not affect its Iwasawa invariants.Thus, µ(f (T )) = ord ℓ (−(u − 2)u u−3 ) + 2µ((1 + T ) a − 1) = ord ℓ (−(u − 2)u u−3 ) because ℓ ∤ a by assumption.This also shows that λ(f (T )) = 1 since the ℓ-adic valuation of β 2 is exactly ord ℓ (−(u − 2)u u−3 ) and (1 + T ) a − 1 = T u(T ), where u(T ) is a unit in the Iwasawa algebra.□ There is another special case where the determinant of M (1 + T ) can be computed explicitly.Suppose that α ′ is a voltage assignment with a 1,j = a for all 2 ≤ j ≤ u − 1 and zero for all other a j,i (0 ≤ i ≤ j ≤ u).The derived multigraphs are again seen to be connected for all n by choosing a suitable spanning tree T in Theorem 2.15.One can choose the spanning tree T to pass through all vertices in the following order v 1 , v t , v t−1 , . . ., v 2 .Such a tree must avoid all edges joining v 1 to v j for 2 ≤ j ≤ t − 1. Lemma 6.3.Let α ′ be the voltage assignment described above.The µ-invariant of f (T ) is ord ℓ ((u − 2)u u−3 ) and its λ-invariant is 1.
Proof.We set t := (1 + T ) a in order to ease notation.With respect to the voltage assignment α ′ , we find that M (1 + T ) is given by .
Define the two 1 × u vectors w and v and the u × u matrix C by .
We get the stated proportion of the theorem by letting x goes to infinity.□ Remark 6.5.The sum of densities in the statement of Theorem 6.4 for all values of (µ, λ) is equal to 1 − 1/ℓ and not 1.This is indeed possible for a sequence of numbers.For instance, consider the sequence g i = i.Then for any value of j > 0, the density of values i for which g i = j is equal to 0, and thus the sum of densities over all values of j is 0, not 1.