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#
Applications of Modular Forms to Elliptic Curves and Representation
Theory

Cotron, Tessa
(2017)
**Honors Thesis**(92 pages)

**Committee Chair / Thesis Adviser:**Ono, Ken

**Committee Members:**Duncan, John F ; Srivastava, Ajit

**Research Fields:**Mathematics

**Keywords:**Number Theory; Partitions; Elliptic Curves; Modular Forms; Congruences

**Program:**College Honors Program, Mathematics

**Permanent url:**http://pid.emory.edu/ark:/25593/s298z

## Abstract

The theory of modular forms has many
applications throughout number theory. In a recent paper [3],
Bacher and de la Harpe study finitary permutation groups and the
relations between their conjugacy growth series and *p(n)*,
the partition function, and *p(n)*e*,* a generalized partition
function. The authors in [3] conjecture over 200 congruences for
*p(n)*e which
are analogous to the Ramanujan congruences for *p(n).* Along
with this, the study of asymptotics for these formulas is motivated
by the group theory of [3]. We prove all of the conjectured
congruences from [3] and give asymptotic formulas for all of the
*p(n)*e*.* Modular form congruences also play a role in the
theory of elliptic curves. In [11], the authors look at modular
forms and other polynomials which reduce modulo *p* to the
supersingular polynomial *ss*p*(j)* for a given elliptic curve
*E* over a field **F**q*.* We look at these results,
which give four modular forms that reduce to the supersingular
polynomial *ss*p*(j).* We also look at the Atkin orthogonal polynomials
which give another way of finding polynomials that reduce modulo
*p* to *ss*p*(j),* and we examine the hypergeometric properties of
these polynomials and modular forms.

## Table of Contents

1. Introduction 1

1.1. Elliptic Curves 1

1.2. Partition Functions 5

2. Modular Forms 15

2.1. Eisenstein Series 17

2.2 Eta Functions 23

2.3 Operators on Modular Forms 24

2.4 Divisor Polynomials 26

3. Elliptic Curves 29

3.1. Supersingular Elliptic Curves 33

4. Polynomials that Reduce to the Supersingular Polynomial 36

4.1 Proof of Theorem 1.1 38

5. The Atkin Orthogonal Polynomials 43

5.1. Orthogonal Polynomials 43

5.2 The Atkin Polynomials 47

5.3 Hypergeometric Properties of the Atkin Polynomials 51

6. Hypergeometric Properties of
*F*K 57

7. An Asymptotic for
*p(n)*e 61

8. Generalized Ramanujan Congruences 64

8.1. Sturm's Theorem 64

8.2. An Algorithm for the Vector
**c**e 65

8.3. Proof of Theorem 1.14 68

8.4. Proof of Theorem 1.15 69

8.5. Examples of Congruences 71

9. Proof of Theorem 1.19 72

10. Congruences for
*p*2*(n)*74

10.1. Proof of Theorem 1.21 80

11. Appendix 81

11.1 Some Examples of the Form
*p(3n+B)*e *= 0 (mod 3)*81

11.2 Some Examples of the Form
*p(5n+B)*e *= 0 (mod 5)*81

11.3 Some Examples of the Form
*p(7n+B)*e *= 0 (mod 7)*82

11.4 Some Examples of the Form
*p(11n+B)*e *= 0 (mod 11)*82

11.5 Some Examples of the Form
*p(13n+B)*e *= 0 (mod 13)*83

References 83

List of Figures and Tables

1. *y*^{2}=x^{3}*+1...*3

2. Ratio of
*p(n)*e and
*P(n)*e... 64