Applications of Modular Forms to Elliptic Curves and Representation Theory Open Access

Cotron, Tessa (Fall 2017)

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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 ssp(j) for a given elliptic curve E over a field Fq. We look at these results, which give four modular forms that reduce to the supersingular polynomial ssp(j). We also look at the Atkin orthogonal polynomials which give another way of finding polynomials that reduce modulo p to ssp(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 FK 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 ce 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

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