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

We present here explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of ${_{2}}F_1$-hypergeometric functions over $\mathbb{F}_q$ for $q \equiv 1 (\text{mod} 6)$ and $q \equiv 1 (\text{mod} 4)$.


Introduction and statement of results
In this paper, we consider the problem of expressing traces of Frobenius endomorphisms of certain families of elliptic curves in terms of hypergeometric functions over finite fields. In [4], Greene introduced the notion of hypergeometric functions over finite fields or Gaussian hypergeometric series which are analogous to the classical hypergeometric series. Since then, many interesting relations between special values of these functions and the number of F p -points on certain varieties have been obtained. For example, Koike [6] and Ono [10] gave formulas for the number of F p -points on elliptic curves in terms of special values of Gaussian hypergeometric series. Also, in [1,2] the authors studied this problem for certain families of algebraic curves.
Recently in [3], Fuselier gave formulas for the trace of Frobenius of certain families of elliptic curves which involved Gaussian hypergeometric series with characters of order 12 as parameters, under the assumption that p ≡ 1(mod 12). In [8], Lennon provided a general formula expressing the number of F q -points of an elliptic curve E with j(E) = 0, 1728 in terms of values of Gaussian hypergeometric series for q = p e ≡ 1(mod 12). In [9], for q ≡ 1(mod 3), Lennon also gave formulas for certain elliptic curves involving Gaussian hypergeometric series with characters of order 3 as parameters.
We begin with some preliminary definitions needed to state our results. Let q = p e be a power of an odd prime and F q the finite field of q elements. Extend each character χ ∈ F × q to all of F q by setting χ(0) := 0. If A and B are two where J(A, B) denotes the usual Jacobi sum and B is the inverse of B.
Recall the definition of the Gaussian hypergeometric series over F q first defined by Greene in [4]. For any positive integer n and characters A 0 , A 1 , . . . , A n and B 1 , B 2 , . . . , B n ∈ F × q , the Gaussian hypergeometric series n+1 F n is defined to be where the sum is over all characters χ of F × q . Throughout the paper, we consider an elliptic curve E a,b over F q in Weierstrass form as If we denote by a q (E a,b ) the trace of the Frobenius endomorphism on E a,b , then where #E a,b (F q ) denotes the number of F q -points on E a,b including the point at infinity. In the following theorems, we express a q (E a,b ) in terms of Gaussian hypergeometric series. Theorem 1.1. Let q = p e , p > 0 a prime and q ≡ 1 (mod 6). In addition, let a be non-zero and (−a/3) a quadratic residue modulo q. If T ∈ F × q is a generator of the character group, then the trace of the Frobenius on E a,b can be expressed as where ǫ is the trivial character of F q and k ∈ F q satisfies 3k 2 + a = 0.
Theorem 1.2. Let q = p e , p > 0 a prime, q = 9 and q ≡ 1 (mod 4). Also assume that x 3 + ax + b = 0 has a non-zero solution in F q and T ∈ F × q is a generator of the character group. The trace of the Frobenius on E a,b can be expressed as where ǫ is the trivial character of F q and h ∈ F × q satisfies h 3 + ah + b = 0.

Preliminaries
Define the additive character θ : where ζ = e 2πi/p and tr : F q → F q is the trace map given by For A ∈ F × q , the Gauss sum is defined by We let T denote a fixed generator of F × q . We also denote by G m the Gauss sum G(T m ).
The orthogonality relations for multiplicative characters are listed in the following lemma.
Using orthogonality, we have the following lemma.
The following two lemmas on Gauss sum will be useful in the proof of our results.
Lemma 2.4. (Davenport-Hasse Relation [7]). Let m be a positive integer and let q = p e be a prime power such that q ≡ 1(mod m). For multiplicative characters 3. Proof of the results Theorem 1.1 will follow as a consequence of the next theorem. We consider an elliptic curve E 1 over F q in the form where c = 0. The trace of the Frobenius endomorphism on E 1 is given by We express the trace of Frobenius on the curve E 1 as a special value of a hypergeometric function in the following way.
Theorem 3.1. Let q = p e , p > 0 a prime and q ≡ 1 (mod 6). If T ∈ F × q is a generator of the character group, then the trace of the Frobenius on E 1 is given by where ǫ is the trivial character of F q .
Proof. The method of this proof follows similarly to that given in [3]. Let and denote by #E 1 (F q ) the number of points on the curve E 1 over F q including the point at infinity. Then Using the elementary identity from [5] z∈Fq Now using Lemma 2.2 and then applying Lemma 2.1 repeatedly for each term of (3.4), we deduce that Similarly, Expanding the next term, we have The innermost sum of D is nonzero only when k = 0 or k = q−1 2 . Using the fact that G 0 = −1, we obtain which is zero unless m = − 2 3 n and n = −3l − 3(q−1) Using Davenport-Hasse relation (2.4) for m = 2, ψ = T −l and m = 3, ψ = T l+ q−1 6 respectively, we deduce that . Therefore, Now, if T m−n = ǫ, then we have Replacing l by l − q−1 2 and using (3.5), we obtain Plugging the facts that if l = 0 then G l G −l = qT l (−1) and if l = 0 then G l G −l = qT l (−1) − (q − 1) in appropriate identities for each l, we deduce that Replacing l by l + q−1 2 in the first term and simplifying the second term, we obtain Putting the values of A, B, C, D all together in (3.4) gives Since a q (E 1 ) = q + 1 − #E 1 (F q ), we have completed the proof of the Theorem.
Proof of Theorem 1.1. Since a = 0 and (−a/3) is quadratic residue modulo q, we find k ∈ F × q such that 3k 2 + a = 0. A change of variables (x, y) → (x + k, y) takes the elliptic curve E a,b : Clearly a q (E a,b ) = a q (E ′ a,b ). Since 3k = 0, using Theorem 3.1 for the elliptic curve E ′ a,b , we complete the proof. We now prove a result for q ≡ 1(mod 4) similar to Theorem 3.1 and Theorem 1.2 will follow from this result.
Theorem 3.2. Let q = p e , p > 0 a prime and q ≡ 1 (mod 4). Let E 2 be an elliptic curve over F q defined as q is a generator of the character group, then the trace of the Frobenius on E 2 is given by where ǫ is the trivial character of F q .
Proof. We have #E 2 (F q ) − 1 = #{(x, y) ∈ F q × F q : P (x, y) = 0}, where P (x, y) = x 3 + f x 2 + gx − y 2 . Now, following the same procedure as followed in the proof of Theorem (3.1), we deduce that Since a q (E a,b ) = a q (E ′′ a,b ) and 3h = 0, using Theorem 3.2 for the elliptic curve E ′′ a,b , we complete the proof.