# Isogenies of Elliptic Curves and Arithmetical Structures on Graphs Open Access

## Reiter, Tomer (Spring 2021)

Permanent URL: https://etd.library.emory.edu/concern/etds/kk91fm75v?locale=en
Published

## Abstract

In this thesis, we prove two results that come from studying curves. The first is a classification result about elliptic curves. Let Q(2^inf) be the compositum of all quadratic extensions of the rational numbers, Q. Torsion subgroups of rational elliptic curves base changed to Q(2^inf) were classified by Laska, Lorenz, and Fujita. Recently, Daniels, Lozano-Robledo, Najman, and Sutherland classified torsion subgroups of rational elliptic curves base changed to Q(3^inf), the compositum of all cubic extensions of Q. We classify cyclic isogenies of rational elliptic curves base changed to Q(2^inf), for all but finitely many elliptic curves over Q(2^inf).

Next we turn to arithmetical structures, which Lorenzini introduced to model degenerations of curves. Let G be a connected undirected graph on n vertices with no loops but possibly multiedges. Given an arithmetical structure (r, d) on G, we describe a construction which associates to it a graph G' on n-1 vertices and an arithmetical structure (r', d') on G'. By iterating this construction, we derive an upper bound for the number of arithmetical structures on G depending only on the number of vertices and edges of G. In the specific case of complete graphs, possibly with multiedges, we refine and compare our upper bounds to those arising from counting unit fraction representations.

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background - Isogenies of Elliptic Curves 5

2.1 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Galois Representations of Elliptic Curves . . . . . . . . . . . . . . . . 8

3 Isogenies of Elliptic Curves over Q(21) 13

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 The Image of Galois After Base Change . . . . . . . . . . . . . . . . 15

3.3 Large Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 l-adic Images for l = 3; 5; 7; 11; 13 . . . . . . . . . . . . . . . . . . . . 22

3.5 2-adic Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Isogenies of Composite Degrees . . . . . . . . . . . . . . . . . . . . . 28

4 Background - Arithmetical Structures on Graphs 33

4.1 Some Types of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Bounding the Number of Arithmetical Structures on Graphs 37

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2 A Recursive Construction . . . . . . . . . . . . . . . . . . . . . . . . 40

5.3 Upper Bounds on #A(G) . . . . . . . . . . . . . . . . . . . . . . . . 45

5.4 Arithmetical Structures on mKn . . . . . . . . . . . . . . . . . . . . . 51

5.4.1 Specializing to G = mKn . . . . . . . . . . . . . . . . . . . . . 51

5.4.2 Connections to Egyptian fractions . . . . . . . . . . . . . . . . 56

Bibliography 60