Related papers: 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 $\Z_{\ell}$-extensions of number fields. It was shown that for towers of bouquets, the growth of the $\ell$-part of the…

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…

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…

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…

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…

Let $p$ and $\ell$ be prime numbers, and $d\ge1$ an integer. We formulate and prove Iwasawa main conjectures of the Picard groups and Bowen--Franks groups in $\mathbb{Z}_p^d$-towers of digraphs. In particular, we relate the $\ell$ parts of…

For varieties over a finite field $\mathbb F_q$ with "many" automorphisms, we study the $\ell$-adic properties of the eigenvalues of the Frobenius operator on their cohomology. The main goal of this paper is to consider towers such as $y^2…

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…

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…

To extend Iwasawa's classical theorem from ${\mathbb Z}_p$-towers to ${\mathbb Z}_p^d$-towers, Greenberg conjectured that the exponent of $p$ in the $n$-th class number in a ${\mathbb Z}_p^d$-tower of a global field $K$ ramified at finitely…

We study the growth of the Tate-Shafarevich and p-infinity Selmer groups for isogenous abelian varieties in towers of number fields, with an emphasis on elliptic curves. The growth types are usually exponential, as in the setting of…

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…

In this note, I develop a representation-theoretic refinement of the Iwasawa theory of finite Cayley graphs. Building on analogies between graph zeta functions and number-theoretic L-functions, I study $\mathbb{Z}_\ell$-towers of Cayley…

Over all non-prime finite fields, we construct some recursive towers of function fields with many rational places. Thus we obtain a substantial improvement on all known lower bounds for Ihara's quantity $A(\ell)$, for $\ell = p^n$ with $p$…

We consider the deconstruction/reconstruction of extensions in varieties of algebras which are modules expanded by multilinear operators. The parametrization of extensions determined by abelian ideals with unary actions agrees with the…

We describe an explicit `higher rank' Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of general number fields. We then show that this theory leads to a concrete new strategy for proving…

In this paper we develop Host--Kra and inverse Gowers theory for abelian groups of bounded exponent. We show that the Host--Kra factors $Z^{\leq k}(\mathrm{X})$ associated with actions of such groups admit extensions with the structure of…

Let $\ell$ and $p$ be distinct primes, and let $\G$ be an abelian pro-$p$-group. We study the structure of the algebra $\L:=\Z_\ell[[\G]]$ and of $\L$-modules. The algebra $\L$ turns out to be a direct product of copies of ring of integers…

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the…

Let $L$ be a number field and let $\ell$ be a prime number. Rasmussen and Tamagawa conjectured, in a precise sense, that abelian varieties whose field of definition of the $\ell$-power torsion is both a pro-$\ell$ extension of $L(\mu_\ell)$…