We discuss recent work of the authors in which we study the translation of classical hypergeometric transformation and evaluation formulas to the finite field setting.

Hypergeometric Functions over Finite Fields

Let λ ∈ ℚ\{0, -1} and l ≥ 2. Denote by Cl, λ the nonsingular projective algebraic curve over ℚ with affine equation given by In this paper, we give a relation between the number of points on Cl, λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [D. McCarthy, 3F2 Hypergeometric series and periods of elliptic curves, Int. J. Number Theory6(3) (2010) 461–470]. We find some special values of 3F2 and 2F1 Gaussian hypergeometric series. Finally we evaluate the value of 3F2(4) which extends a result of [K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(3) (1998) 1205–1223].

Certain values of gaussian hypergeometric series and a family of algebraic curves

Elliptic curves and special values of Gaussian hypergeometric series

We prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.

Hypergeometric type identities in the $p$-adic setting and modular forms

We present here explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of 2F1hypergeometric functions over Fq for q ≡ 1(mod 6) and q ≡ 1(mod 4).

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Hypergeometric functions over $\mathbb {F}_q$ and traces of Frobenius for elliptic curves

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 

$$y^l=x(x-1)(x-\lambda).$$

In this paper, we define Ω(Cl,λ) analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on Cl,λ over a finite field and Gaussian hypergeometric series. Finally, we give an alternate proof of a result of Rouse (Ramanujan J. 12(2):197–205, 2006).

Hypergeometric functions and a family of algebraic curves

Let $\lambda \in \mathbb{Q}\setminus \{0, -1\}$ and $l \geq 2$. Denote by $C_{l,\lambda}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=(x-1)(x^2+\lambda).$$ In this paper we give a relation between the number of points on $C_{l, \lambda}$ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of McCarthy (2010). We find some special values of ${_{3}}F_2$ and ${_{2}}F_1$ Gaussian hypergeometric series. Finally we evaluate the value of ${_{3}}F_2(4)$ which extends a result of Ono (1998).

Rupam Barman

Papers

Certain values of gaussian hypergeometric series and a family of algebraic curves

Elliptic curves and special values of Gaussian hypergeometric series

Hypergeometric functions and a family of algebraic curves

Hypergeometric functions over $\mathbb {F}_q$ and traces of Frobenius for elliptic curves

Certain values of Gaussian hypergeometric series and a family of algebraic curves