Related papers: Isogeny volcanoes
In this note, I study a comparison map between a motivic and \'{e}tale cohomology group of an elliptic curve over $\mathbb{Q}$ just outside the range of Voevodsky's isomorphism theorem. I show that the property of an appropriate version of…
We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions.
Let $p$ be a prime and $K$ a number field of degree $p$. We count the number of elliptic curves, up to $\bar{K}$-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a…
Computers and algorithms play an ever-increasing role in obtaining new results in graph theory. In this survey, we present a broad range of techniques used in computer-assisted graph theory, including the exhaustive generation of all…
There are several formulas for the number of orbits of the projective line under the action of subgroups of $GL_2$. We give an interpretation of two such formulas in terms of the geometry of elliptic curves, and prove a more general formula…
We give an efficient, deterministic algorithm to decide if two abelian varieties over a number field are isogenous. From this, we derive an algorithm to compute the endomorphism ring of an elliptic curve over a number field.
This paper shows that the topological structures of particle orbits generated by a generic class of vector fields on spherical surfaces, called {\it the flow of finite type}, are in one-to-one correspondence with discrete structures such as…
Inspired by the prospect of having discretized spaces emerge from random graphs, we construct a collection of simple and explicit exponential random graph models that enjoy, in an appropriate parameter regime, a roughly constant vertex…
The theory of graphons has proven to be a powerful tool in many areas of graph theory. In this paper, we introduce several foundational aspects of the theory of digraphons -- asymmetric two-variable functions that arise as limits of…
The secrecy graph is a random geometric graph which is intended to model the connectivity of wireless networks under secrecy constraints. Directed edges in the graph are present whenever a node can talk to another node securely in the…
The asymptotic study of percolation on finite transitive graphs is considered. Several questions and very few answers regarding percolation on finite graphs are presented.
An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. Gekeler has given a product formula, in terms of congruence considerations involving that polynomial, for the size of such an isogeny…
We introduce a new tool for the study of isogeny-based cryptography, namely pairings which are sesquilinear (conjugate linear) with respect to the $\mathcal{O}$-module structure of an elliptic curve with CM by an imaginary quadratic order…
A graph theoretic perspective is taken for a range of phenomena in continuum physics in order to develop representations for analysis of large scale, high-fidelity solutions to these problems. Of interest are phenomena described by partial…
Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as p-quotients of particular infinite…
The chapter presents mathematical models intended for creating a topological drawing of a non-separable non-planar graph based on the methods of G. Ringel's vertex rotation theory. The induced system of cycles generates a topological…
We call a simple abelian variety over $\mathbb{F}_p$ super-isolated if its ($\mathbb{F}_p$-rational) isogeny class contains no other varieties. The motivation for considering these varieties comes from concerns about isogeny based attacks…
A visualized graph is a powerful tool for data analysis and synthesis tasks. In this case, the task of visualization constitutes not only in displaying vertices and edges according to the graph representation, but also in ensuring that the…
Graph burning is a discrete-time process on graphs, where vertices are sequentially burned, and burned vertices cause their neighbours to burn over time. We consider extremal properties of this process in the new setting where the…
The formal group law of an elliptic curve has seen recent applications to computational algebraic geometry in the work of Couveignes to compute the order of an elliptic curve over finite fields of small characteristic. The purpose of this…