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Special Values of Hypergeometric Series over Finite Fields

Special Values of Hypergeometric Series over Finite Fields. Ron Evans Frank Lam. Some Background. Some Background. Hypergeometric Series. Some Background. Hypergeometric Series. Evaluating over finite fields (GF). Some Background. Hypergeometric Series.

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Special Values of Hypergeometric Series over Finite Fields

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  1. Special Values of Hypergeometric Series over Finite Fields Ron Evans Frank Lam

  2. Some Background

  3. Some Background • Hypergeometric Series

  4. Some Background • Hypergeometric Series • Evaluating over finite fields (GF)

  5. Some Background • Hypergeometric Series • Evaluating over finite fields (GF) • What is meant by ?

  6. Some Background • Hypergeometric Series • Evaluating over finite fields (GF) • What is meant by ? • Notable work on the problem

  7. Known Values of Hypergeometric 3F2 Functions

  8. Known Values of Hypergeometric 3F2 Functions

  9. Some New Values of H(z)

  10. A General Form

  11. A General Form What is r?

  12. A General Form What is r? • r is a function of n • r has to do with the singular modulus of complete elliptic integrals • r is interpreted modulo p

  13. A General Form What is r? • r is a function of n • r has to do with the singular modulus of complete elliptic integrals • r is interpreted modulo p

  14. What does it mean to be interpreted modulo p?

  15. What does it mean to be interpreted modulo p?

  16. What does it mean to be interpreted modulo p?

  17. What does it mean to be interpreted modulo p?

  18. The Proof • What is an elliptic curve?

  19. The Proof • What is an elliptic curve?

  20. The Proof • What is an elliptic curve? • Ono’s elliptic curve theorem

  21. The Proof • What is an elliptic curve? • Ono’s elliptic curve theorem • What are the implications?

  22. The Proof • What is an elliptic curve? • Ono’s elliptic curve theorem • What are the implications? • We can expect the above curve to have points. • We can evaluate H.

  23. What does it mean to count points?

  24. What does it mean to count points?

  25. What does it mean to count points?

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