3F2-Hypergeometric Series and Supersingular Elliptic Curves Open Access
Pitman, Sarah Christine (2014)
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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