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

Abstract

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 $\lambda$-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.

25 pages, 1 figure