arXiv subjects

Sohei Tateno

Publications and source records attributed to Sohei Tateno.

6 recordsLinked to original sources

arXiv2026

Iwasawa theory for vertex-weighted graphs

Chung-Langlands established a matrix-tree theorem for positive-real valued vertex-weighted graphs, and Wu-Feng-Sato developed a theory of Ihara zeta functions for those graphs. In this paper, generalizing and refining these previous works, we initiate the Iwasawa theory for vertex-weighted graphs, which is a generalization of the Iwasawa theory for graphs initiated by Gonet and Vallières independently. First, we generalize the matrix-tree theorem by Chung-Langlands to arbitrary field-valued vertex-weighted graphs. Second, we refine and prove the so-called decomposition formula for vertex-weighted graphs and edge-weighted graphs without any assumption. Applying these results, we prove the Iwasawa-type formula and a refinement of Kida's formula for $\mathbb{Z}_p^d$-towers of vertex-weighted graphs. Our refinement of the decomposition formulas allows us to estimate the root-wise growth of weighted complexities in $\mathbb{Z}_p^d$-towers. We also provide several numerical examples.

math.CO↗

arXiv2026

Iwasawa Theory for K3 Surfaces over Finite Fields

In this paper, we initiate Iwasawa theory for K3 surfaces over finite fields. First, using the Artin-Tate conjecture, which is known to hold for K3 surfaces, we prove an analogue of Mazur's control theorem for elliptic curves over number fields. Second, we prove an analogue of Iwasawa's class number formula for Brauer groups in two different ways. We also give explicit examples in the case of Kummer surfaces. Finally, we establish an analogue of the Iwasawa main conjecture for Brauer groups.

math.NT↗

arXiv2026

Distributions of Iwasawa $λ$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs

A graph-theoretic analogue of Iwasawa theory, initiated by Gonet and Vallières, has attracted considerable interest in the study of Iwasawa invariants. On the other hand, for a pair of prime numbers $(r,\ell)$, one obtains a graph, called the supersingular $\ell$-isogeny graph (SIG), whose adjacency matrix has eigenvalues given by the $\ell$-th Fourier coefficients of the weight 2 Eisenstein series and newforms of level $r$. In this paper, we fix prime numbers $r$ and $p$, and let $\ell$ vary over infinitely many primes. We then investigate the distribution of the Iwasawa $λ$-invariants of the constant $\mathbf{Z}_p$-towers over the SIGs, thereby revealing connections among graph theory, Iwasawa theory, elliptic curves, and the Galois representations attached to newforms. At the end of this paper, we propose a conjecture concerning the Galois orbits of newforms.

math.NT↗

arXiv2025

Iwasawa theory for weighted graphs

Let $p$ be a prime number and let $d$ be a positive integer. In this paper, we generalize Iwasawa theory for graphs initiated by Gonet and Vallières to weighted graphs. In particular, we prove an analogue of Iwasawa's class number formula and that of Kida's formula for compatible systems of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of weighted graphs. We also provide numerical examples of characteristic elements and Iwasawa invariants. At the end of this paper, we give an application of the ideas of Iwasawa theory to the theory of discrete-time quantum walks in graphs.

math.NT↗

arXiv2025

The Iwasawa invariants of $\mathbb{Z}_p^{\,d}$-covers of links

Let $p$ be a prime number and let $d\in \mathbb{Z}_{>0}$. In this paper, following the analogy between knots and primes, we study the $p$-torsion growth in a compatible system of $(\mathbb{Z}/p^n\mathbb{Z})^d$-covers of 3-manifolds and establish several analogues of Cuoco--Monsky's multivariable versions of Iwasawa's class number formula. Our main goal is to establish the Cuoco--Monsky type formula for branched covers of links in rational homology 3-spheres. In addition, we prove the precise formula over integral homology 3-spheres prompted by Greenberg's conjecture. We also derive results on reduced Alexander polynomials and on the Betti number periodicity. Furthermore, we investigate the twisted Whitehead links in $S^3$ and point out that the Iwasawa $μ$-invariant of a $\mathbb{Z}_p^{\,2}$-cover can be an arbitrary non-negative integer. We also calculate the Iwasawa $μ$ and $λ$-invariants of the Alexander polynomials of all links in Rolfsen's table.

math.GT↗

arXiv2024

On Iwasawa's class number formula for $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions

Let $p$ be a prime number. In this paper, we estimate the variation of the sizes of quotients of certain finitely generated $p$-torsion Iwasawa modules, which are closely related to class numbers. We also construct some $\mathbb{Z}_p\rtimes\mathbb{Z}_p$-extensions whose Iwasawa $μ$-invariant is nonzero. At the end of this paper, we calculate the determinants of some matrices that are related to the groups $\mathbb{Z}_p\rtimes\mathbb{Z}_p$.

math.NT↗