Related papers: Isogeny volcanoes
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
Topology identification and inference of processes evolving over graphs arise in timely applications involving brain, transportation, financial, power, as well as social and information networks. This chapter provides an overview of graph…
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…
Using methods from analytic number theory, for $m > 5$ and for $m = 4$, we obtain asymptotics with power-saving error terms for counts of elliptic curves with a cyclic $m$-isogeny up to quadratic twist over the rational numbers. For $m >…
We count the number of rational elliptic curves of bounded naive height that have a rational $N$-isogeny, for $N \in \{2,3,4,5,6,8,9,12,16,18\}$. For some $N$, this is done by generalizing a method of Harron and Snowden. For the remaining…
The presented material is devoted to the equivalent conversion from the vertex graphs to the edge graphs. We suggest that the proved theorems solve the problem of the isomorphism of graphs, the problem of the graph's enumeration with the…
Intersection graphs are very important in both theoretical as well as application point of view. Depending on the geometrical representation, different type of intersection graphs are defined. Among them interval, circular-arc, permutation,…
We introduce an analogue of the Mertens conjecture for elliptic curves over finite fields. Using a result of Waterhouse, we classify the isogeny classes of elliptic curves for which this conjecture holds in terms the size of the finite…
Elliptic curves are planar curves which can be used to define an abelian group. The efficient computation of discrete logarithms over this group is a longstanding problem relevant to cryptography. It may be possible to efficiently compute…
In this paper we present an iterative construction of irreducible polynomials over finite fields based upon repeated applications of transforms induced by endomorphisms of odd prime degree of ordinary elliptic curves.
Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…
For an orientable surface $S$ of finite topological type with genus $g \geq 3$, we construct a finite set of curves whose union of iterated rigid expansions is the curve graph of $S$. The set constructed, and the method of rigid expansion,…
We define a graph structure associated in a natural way to finite fields that nevertheless distinguishes between different models of isomorphic fields.
Elliptic curves have a well-known and explicit theory for the construction and application of endomorphisms, which can be applied to improve performance in scalar multiplication. Recent work has extended these techniques to hyperelliptic…
The results of computer searches for large graphs with given (small) degree and diameter are presented. The new graphs are Cayley graphs of semidirect products of cyclic groups and related groups. One fundamental use of our ``dense graphs''…
We prove that the path-finding problem in $\ell$-isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised Riemann hypothesis.…
In this talk we discuss mathematical structures associated to Feynman graphs. Feynman graphs are the backbone of calculations in perturbative quantum field theory. The mathematical structures -- apart from being of interest in their own…
This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by…
In this article with study Tamagawa numbers of elliptic curves defined over $\mathbb{Q}$ that have isogenies or torsion points. More precisely, our aim is either to bound the set of primes primes that can divide their Tamagawa numbers or,…
Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general…