Abstract
Here we explore elliptic curves, specifically supersingular
elliptic curves, and their relationship to hypergeometric
functions. We begin with some background on elliptic curves,
supersingularity, hypergeometric functions, and then use work of
El-Guindy, Ono, Kaneko, Zagier, and Monks to extend results. In
recent work, Monks described the supersingular locus of families of
elliptic curves in terms of 2F1-hypergeometric functions. We "lift"
his work to the level of 3F2-hypergeometric functions by means of
classical transformation laws and a theorem of Clausen.
Table of Contents
Table of Contents
List of Figures 0
1 - Background: Introduction and statement of results 1
1.1 - Elliptic Curves 1
1.2 - Points on an elliptic curve form an Abelian group 3
1.3 - My work 5
1.4 - Hypergeometric functions 6
2 - Supersingular elliptic curves 10
2.1 - Supersingular elliptic curves 10
2.2 - Supersingular locus over Fp 11
2.3 - Work of Monks 11
3 - Outline of proof of Theorem and tools 13
3.1 - Statement of Clausen's Theorem and Transformation Laws
14
3.2 - Elementary Reduction modulo p 14
4 - Proof of Theorem 1.4 15
4.1 - Proof of Theorem 1.4 28
5 - Examples 30
References 31
List of Figures
1 - Solving ECDLP 2
2 - Chord-tangent law 4
3 - Group law 5
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