Distributions of Iwasawa $λ$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866916038202884096 |
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| author | Adachi, Taiga Mizuno, Kosuke Murooka, Ryosuke Tateno, Sohei |
| author_facet | Adachi, Taiga Mizuno, Kosuke Murooka, Ryosuke Tateno, Sohei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23184 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Distributions of Iwasawa $λ$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs Adachi, Taiga Mizuno, Kosuke Murooka, Ryosuke Tateno, Sohei Number Theory Combinatorics 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. |
| title | Distributions of Iwasawa $λ$-invariants of $\mathbf{Z}_p$-towers over supersingular isogeny graphs |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2605.23184 |