arXiv subjects

Daniel Vallières

Publications and source records attributed to Daniel Vallières.

15 recordsLinked to original sources

arXiv2026

Bowen--Franks groups and minus class groups of cyclotomic number fields with prime conductor

Let $p$ be an odd rational prime and consider the cyclotomic number field $K = \mathbb{Q}(ζ_{p})$ of conductor $p$. We construct a directed graph $Y$ on $p-1$ vertices for which the torsion part of the corresponding Bowen--Franks group is closely related to the minus part of the class group of $K$. In particular, both groups have the same cardinality up to an explicit power of $p$. Furthermore, they are both $\mathrm{Gal}(K/\mathbb{Q})$-modules, and we prove the equality of the cardinalities of their isotypic components after tensoring them with the valuation ring of an appropriate $\ell$-adic field for $\ell \nmid p-1$.

math.NT↗

arXiv2025

Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite graphs and Ihara zeta and $L$-functions

We revisit the theory of Ihara $L$-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study if the Artin formalism is satisfied by these $L$-functions. As an application, we give a proof of the analogue of Iwasawa's asymptotic class number formula for the $p$-part of the number of spanning trees in branched $\mathbb{Z}_{p}$-towers of finite connected graphs using Ihara zeta and $L$-functions. Moreover, we relate a generator for the characteristic ideal of the finitely generated torsion Iwasawa module that governs the growth of the $p$-part of the number of spanning trees in such towers with Ihara $L$-functions.

math.NT↗

arXiv2025

An analogue of the Herbrand-Ribet theorem in graph theory

We study an analogue of the Herbrand-Ribet theorem, and its refinement by Mazur and Wiles, in graph theory. For an odd prime number $p$, we let $\mathbb{F}_{p}$ and $\mathbb{Z}_{p}$ denote the finite field with $p$ elements and the ring of $p$-adic integers, respectively. We consider Galois covers $Y/X$ of finite graphs with Galois group $Δ$ isomorphic to $\mathbb{F}_{p}^{\times}$. Given a $\mathbb{Z}_{p}$-valued character of $Δ$, we relate the cardinality of the corresponding character component of the $p$-primary subgroup of the degree zero Picard group of $Y$ to the $p$-adic absolute value of the special value at $u=1$ of the corresponding Artin-Ihara $L$-function.

math.NT↗

arXiv2025

An analogue of Kida's formula in graph theory

Let $\ell$ be a rational prime and let $p:Y\rightarrow X$ be a Galois cover of finite graphs whose Galois group is a finite $\ell$-group. Consider a $\mathbb{Z}_{\ell}$-tower above $X$ and its pullback along $p$. Assuming that all the graphs in the pullback are connected, one obtains a $\mathbb{Z}_{\ell}$-tower above $Y$. Under the assumption that the Iwasawa $μ$-invariant of the tower above $X$ vanishes, we prove a formula relating the Iwasawa $λ$-invariant of the $\mathbb{Z}_{\ell}$-tower above $X$ to the Iwasawa $λ$-invariant of the pullback. This formula is analogous to Kida's formula in classical Iwasawa theory. We present an application to the study of structural properties of certain noncommutative pro-$\ell$ towers of graphs, based on an analogy with classical results of Cuoco on the growth of Iwasawa invariants in $\mathbb{Z}_\ell^2$-extensions of number fields. Our investigations are illustrated by explicit examples.

math.NT↗

arXiv2024

Spanning trees in $\mathbb{Z}$-covers of a finite graph and Mahler measures

Using the special value at $u=1$ of Artin-Ihara $L$-functions, we associate to every $\mathbb{Z}$-cover of a finite connected graph a polynomial which we call the \emph{Ihara polynomial}. We show that the number of spanning trees for the finite intermediate graphs of such a cover can be expressed in terms of the Pierce-Lehmer sequence associated to a factor of the Ihara polynomial. This allows us to express the asymptotic growth of the number of spanning trees in terms of the Mahler measure of this polynomial. Specializing to the situation where the base graph is a bouquet or the dumbbell graph gives us back previous results in the literature for circulant and $I$-graphs (including the generalized Petersen graphs). We also express the $p$-adic valuation of the number of spanning trees of the finite intermediate graphs in terms of the $p$-adic Mahler measure of the Ihara polynomial. When applied to a particular $\mathbb{Z}$-cover, our result gives us back Lengyel's calculation of the $p$-adic valuations of Fibonacci numbers.

math.NT↗

arXiv2024

Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite graphs

We initiate the study of Iwasawa theory for branched $\mathbb{Z}_{p}$-towers of finite connected graphs. These towers are more general than what have been studied so far, since the morphisms of graphs involved are branched covers, a particular kind of harmonic morphisms of graphs. We prove an analogue of Iwasawa's asymptotic class number formula for the $p$-part of the number of spanning trees in this setting. Moreover, we find an explicit generator for the characteristic ideal of the torsion Iwasawa module governing the growth of the $p$-part of the number of spanning trees in such towers.

math.NT↗

arXiv2023

On the distribution of Iwasawa invariants associated to multigraphs

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 analogues of Iwasawa invariants, which depend on the prime $\ell$ and the abelian $\ell$-tower of multigraphs. We formulate and study statistical questions about the behaviour of the Iwasawa $μ$ and $λ$ invariants.

math.CO↗

arXiv2023

The non-$\ell$-part of the number of spanning trees in abelian $\ell$-towers of multigraphs

Let $\ell$ and $p$ be two distinct primes. We study the $p$-adic valuation of the number of spanning trees in an abelian $\ell$-tower of connected multigraphs. This is analogous to the classical theorem of Washington--Sinnott on the growth of the $p$-part of the class group in a cyclotomic $\mathbb{Z}_\ell$-extension of abelian extensions of $\mathbb{Q}$. Furthermore, we show that under certain hypotheses, the number of primes dividing the number of spanning trees is unbounded in such a tower.

math.CO↗

arXiv2022

On $\mathbb{Z}_{\ell}^{d}$-towers of graphs

Let $\ell$ be a rational prime. We show that an analogue of a conjecture of Greenberg in graph theory holds true. More precisely, we show that when $n$ is sufficiently large, the $\ell$-adic valuation of the number of spanning trees at the $n$th layer of a $\mathbb{Z}_{\ell}^{d}$-tower of graphs is given by a polynomial in $\ell^{n}$ and $n$ with rational coefficients of total degree at most $d$ and of degree in $n$ at most one.

math.CO↗

arXiv2021

On abelian $\ell$-towers of multigraphs III

Let $\ell$ be a rational prime. Previously, abelian $\ell$-towers of multigraphs were introduced which are analogous to $\Z_{\ell}$-extensions of number fields. It was shown that for towers of bouquets, the growth of the $\ell$-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $\Z_{\ell}$-extensions of number fields). In this paper, we extend this result to abelian $\ell$-towers over an arbitrary connected multigraph (not necessarily simple and not necessarily regular). In order to carry this out, we employ integer-valued polynomials to construct power series with coefficients in $\Z_\ell$ arising from cyclotomic number fields, different than the power series appearing in the prequel. This allows us to study the special value at $u=1$ of the Artin--Ihara $L$-function, when the base multigraph is not necessarily a bouquet.

math.CO↗

arXiv2021

On abelian $\ell$-towers of multigraphs II

Let $\ell$ be a rational prime. Previously, abelian $\ell$-towers of multigraphs were introduced which are analogous to $\mathbb{Z}_{\ell}$-extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the $\ell$-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for $\mathbb{Z}_{\ell}$-extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian $\ell$-towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in $\mathbb{Z}_{\ell}$ and then study the special value at $s=1$ of the Artin-Ihara $L$-function $\ell$-adically.

math.CO↗

arXiv2020

On abelian $\ell$-towers of multigraphs

We study how the $\ell$-adic valuation of the number of spanning trees varies in regular abelian $\ell$-towers of multigraphs. We show that for an infinite family of regular abelian $\ell$-towers of bouquets, the behavior of the $\ell$-adic valuation of the number of spanning trees behave similarly to the $\ell$-adic valuation of the class numbers in $\mathbb{Z}_{\ell}$-extensions of number fields.

math.CO↗

arXiv2019

The special value $u=1$ of Artin-Ihara $L$-functions

We study the special value $u=1$ of Artin-Ihara $L$-functions associated to characters of the automorphism group of abelian covers of multigraphs. In particular, we show an annihilation statement analogous to a classical conjecture of Brumer on annihilation of class groups for abelian extensions of number fields and we also calculate the index of an ideal analogous to the classical Stickelberger ideal in algebraic number theory. Along the way, we make some observations about the number of spanning trees in abelian multigraph coverings that may be of independent interest.

math.NT↗

arXiv2017

Numerical evidence for higher order Stark-type conjectures

We give a systematic method of providing numerical evidence for higher order Stark-type conjectures such as (in chronological order) Stark's conjecture over $\mathbb{Q}$, Rubin's conjecture, Popescu's conjecture, and a conjecture due to Burns that constitutes a generalization of Brumer's classical conjecture on annihilation of class groups. Our approach is general and could be used for any abelian extension of number fields, independent of the signature and type of places (finite or infinite) that split completely in the extension. We then employ our techniques in the situation where $K$ is a totally real, abelian, ramified cubic extension of a real quadratic field. We numerically verify the conjectures listed above for all fields $K$ of this type with absolute discriminant less than $10^{12}$, for a total of $19197$ examples. The places that split completely in these extensions are always taken to be the two real archimedean places of $k$ and we are in a situation where all the $S$-truncated $L$-functions have order of vanishing at least two.

math.NT↗

arXiv2017

Weyl groups of some hyperbolic Kac-Moody algebras

We use the theory of Clifford algebras and Vahlen groups to study Weyl groups of hyperbolic Kac-Moody algebras T_n^{++}, obtained by a process of double extension from a Cartan matrix of finite type T_n, whose corresponding generalized Cartan matrices are symmetric.

math.GR↗