# Hypergeometric functions and a family of algebraic curves

## Rupam Barman

### 2012

We let Fp denote the finite field with p elements and we extend all characters χ of F×p to Fp by setting χ(0) = 0

## Highlights

• Hypergeometric functions and their relations with algebraic curves have been studied by many mathematicians
• For a0, a1, . . . , ar, b1, b2, . . . , br ∈ C, the ordinary hypergeometric series r+1Fr is defined as r+1Fr a0, a1, b1
• The interplay between ordinary hypergeometric series and Gaussian hypergeometric series has played an important role in character sum evaluation, the representation theory of SL(2, R) and finding the number of points on an algebraic curve over finite fields
• We let Fp denote the finite field with p elements and we extend all characters χ of F×p to Fp by setting χ(0) = 0
• Let λ ∈ Q \ {0, 1} and l ≥ 2, and denote by Cl,λ the nonsingular projective algebraic curve over Q with affine equation given by yl = x(x − 1)(x − λ)
• 1.8, we obtain the following result which generalizes the case l = 2, p ≡ 1 treated in Theorem 1.9