Related papers: On ordinary isogeny graphs with level structure

We enhance an isogeny graph of elliptic curves by incorporating level structures defined by bases of the kernels of iterates of the Verschiebung map. We extend several previous results on isogeny graphs with level structures defined by…

We study $\ell$-isogeny graphs of ordinary elliptic curves defined over $\mathbb{F}_q$ with an added level structure. Given an integer $N$ coprime to $p$ and $\ell,$ we look at the graphs obtained by adding $\Gamma_0(N),$ $\Gamma_1(N),$ and…

Fix a prime number $\ell$. Graphs of isogenies of degree a power of $\ell$ are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a…

Famously, Kohel proved that isogeny graphs of ordinary elliptic curves are beautifully structured objects, now called volcanos. We prove graph structural theorems for abelian varieties of any dimension with commutative endomorphism ring and…

We investigate properties of $\mathbb{Z}_p$-towers of graph coverings that arise from a constant voltage assignment. We prove the existence and uniqueness (up to isomorphisms) of such towers. Furthermore, we study the Iwasawa invariants of…

Let $p,q,l$ be three distinct prime numbers and let $N$ be a positive integer coprime to $pql$. For an integer $n\ge 0$, we define the directed graph $X_l^q(p^nN)$ whose vertices are given by isomorphism classes of elliptic curves over a…

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…

We give a detailed presentation of $\ell$-isogeny graphs associated with ordinary elliptic curves defined over $\mathbb{F}_p$. We then focus on the following inverse problem: given an abstract volcano $V$, do there always exist primes…

We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree p. For elliptic curves over l-adic fields, the…

Let $p$ be an odd prime number. In this article, we study the variation of Iwasawa invariants among $p$-congruent elliptic curves over certain $p$-adic Lie extensions. We investigate both the classical Selmer group as well as the fine…

The remarkable structure and computationally explicit form of isogeny graphs of elliptic curves over a finite field has made them an important tool for computational number theorists and practitioners of elliptic curve cryptography. This…

A graph-theoretic analogue of Iwasawa theory, initiated by Gonet and Valli\`eres, 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…

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…

The Jacobian group (also known as the critical group or sandpile group) is an important invariant of a finite, connected graph $X$; it is a finite abelian group whose cardinality is equal to the number of spanning trees of $X$ (Kirchhoff's…

Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree $\ell$, their structure has been investigated graph-theoretically. We generalise the notion of $\ell$-isogeny graphs to…

It is known that any isogeny graph consisting of ordinary elliptic curves over $\mathbb{F}_q$ with $q = p$ or $p^2$ has a special structure, called a volcano graph. We have a bound $h < \log_2 \sqrt{4q}$ of a height $h$ of the $2$-volcano…

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…

In this paper, we add the information of level structure to supersingular elliptic curves and study these objects with the motivation of isogeny-based cryptography. Supersingular elliptic curves with level structure map to Eichler orders in…

Let $\ell$ be an odd prime, and suppose $E$ is an elliptic curve defined over the rational numbers $\mathbb{Q}$. If $E$ has an $\ell$-torsion point, then there has been significant work done on characterizing the $\ell$-divisibility of the…

Booher, Cais, Kramer-Miller and Upton study a class of $\mathbf{Z}_p$-tower of curves in characteristic $p$ with ramification controlled by an integer $d$. In the special case that $d$ divides $p-1$, they prove a formula for the higher…