On a supercongruence conjecture of Rodriguez-Villegas

In examining the relationship between the number of points over $\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the $p$-th Fourier coefficient of a modular form.


Introduction and Statement of Results
Let F p denote the finite field with p, a prime, elements. In [21] Rodriguez-Villegas examined the relationship between the number of points over F p on certain Calabi-Yau manifolds and truncated generalized hypergeometric series which correspond to a particular period of the manifold. In doing so, he identified numerically 22 possible supercongruences which can be categorized by the dimension, D, of the manifold as outlined below.
We first define the truncated generalized hypergeometric series. For a complex number a and a non-negative integer n let (a) n denote the rising factorial defined by (a) 0 := 1 and (a) n := a(a + 1)(a + 2) · · · (a + n − 1) for n > 0.
Then, for complex numbers a i , b j and z, with none of the b j being negative integers or zero, we define the truncated generalized hypergeometric series r F s a 1 , a 2 , a 3 , . . . , a r b 1 , b 2 , . . . , b s z m := m n=0 (a 1 ) n (a 2 ) n (a 3 ) n · · · (a r ) n (b 1 ) n (b 2 ) n · · · (b s ) n z n n! .
We also let φ(·) denote Euler's totient function and · p the Legendre symbol modulo p.
For D = 1, associated to certain elliptic curves, 4 supercongruences were identified. They were all of the form where φ(d) ≤ 2, 1 ≤ t ≤ 3 and p is a prime not dividing d. These cases have been proven by Mortenson [19], [20].
2010 Mathematics Subject Classification. Primary: 11F33; Secondary: 33C20, 11T24. This work was supported by the UCD Ad Astra Research Scholarship program.
For D = 2 another 4 supercongruences were identified which relate to certain modular K3 surfaces. These were all of the form where φ(d) ≤ 2, p is a prime not dividing d and a(p) is the p-th Fourier coefficient of a weight three modular form on a congruence subgroup of SL(2, Z). The case when d = 2 was originally conjectured by Beukers and Stienstra [5] and was first proven by Van Hamme [23]. Subsequently, proofs were also provided by Ishikawa [10] and Ahlgren [1]. The other D = 2 cases are dealt with by Mortenson [18] where they have been proven for p ≡ 1 (mod d) and up to sign otherwise. The remaining 14 supercongruence conjectures relate to Calabi-Yau threefolds (i.e. D = 3). The threefolds in question are complete intersections of hypersurfaces, of which 13 are discussed by Batyrev and van Straten in [3]. The supercongruences can be expressed as either where φ(d i ) ≤ 2 and p is a prime not dividing d i , or where φ(d) = 4, 1 < r < d − 1 with gcd(r, d) = 1, p is a prime not dividing d and b(p) is the p-th Fourier coefficient of a weight four modular form on a congruence subgroup of SL(2, Z). Only one of these cases has been proven (Kilbourn [11]). It is of the first type with d 1 = d 2 = 2 and is an extension of the Apéry number supercongruence [2]. Let is the Dedekind eta function and q := e 2πiz . Then f is a cusp form of weight four on the congruence subgroup Γ 0 (25). We now list one of the outstanding conjectures of type (1.1). Conjecture 1.1 (Rodriguez-Villegas [21]). If p = 5 is prime and c(p) is as defined in (1.2), then The main result of this paper is the following theorem.
The main approach for proving these types of supercongruences has been to use congruence relations between truncated generalized hypergeometric series and Greene's hypergeometric functions over finite fields [1,2,11,16,18,19,20]. However, as noted in [15], many results using this approach are restricted to primes in certain congruence classes (e.g. p ≡ 1 (mod d) in some of the D = 2 cases described above). In [15], this author extends Greene's hypergeometric functions to the p-adic setting and establishes congruences between this new function and certain truncated generalized hypergeometric series. These congruences cover all 22 hypergeometric series outlined above and are valid for all primes required in each of these cases, thus providing a framework for proving all 22 cases. The proof of Theorem 1.2 relies on one of these congruences along with counting the number of rational points on a modular Calabi-Yau threefold over F p .
Specifically, let p be an odd prime and let n ∈ Z + . For 1 ≤ i ≤ n + 1, let mi di ∈ Q ∩ Z p such that 0 < mi di < 1. Let Γ p (·) denote Morita's p-adic gamma function, ⌊x⌋ denote the greatest integer less than or equal to x and x denote the fractional part of x, i.e. x − ⌊x⌋. Then define An example of one the supercongruence results from [15] is the following theorem.
Taking d = 5 in Theorem 1.3 yields Therefore Theorem 1.2 will be established (after checking the case when p = 2) on proving the following theorem.
. As mentioned above, n+1 G extends Greene's hypergeometric function over finite fields, which was introduced in [7]. Let F * p denote the group of multiplicative characters of F * p . We extend the domain of χ ∈ F * p to F p by defining χ(0) := 0 (including the trivial character ε) and denote B as the inverse of B.
p and x ∈ F p , define the Gaussian hypergeometric series by We recover the Gaussian hypergeometric series from n+1 G via the following result.
where ω is the Teichmüller character. Then Using Proposition 1.5 it is easy to see the following corollary to Theorem 1.4.
The remainder of the paper is organized as follows. Section 2 recalls some properties of Gauss and Jacobi sums, and the p-adic gamma function. The proof of Theorems 1.2 and 1.4 appear in Section 3.

Preliminaries
We briefly recall some properties of Gauss and Jacobi sums and the p-adic gamma function, and also develop some preliminary results which we will use in Section 3. Throughout we let Z p , Q p and C p denote the ring of p-adic integers, the field of padic numbers and the p-adic completion of the algebraic closure of Q p , respectively.
2.1. Gauss and Jacobi Sums. We first recall some properties of multiplicative characters. In particular, we note the following orthogonal relations. For and, for x ∈ F p we have We now introduce some properties Gauss and Jacobi sums. For further details see [4] and [9], noting that we have adjusted results to take into account ε(0) = 0. Let ζ p be a fixed primitive p-th root of unity in Q p . We define the additive character θ : F p → Q p (ζ p ) by θ(x) := ζ x p . We note that Q p contains all (p − 1)-th roots of unity and in fact they are all in Z * p . Thus we can consider multiplicative characters of F * p to be maps χ : F * p → Z * p . Recall then that for χ ∈ F * p , the Gauss sum g(χ) is defined by g(χ) := x∈Fp χ(x)θ(x). It easily follows from (2.2) that we can express the additive character as a sum of Gauss sums. Specifically, for x ∈ F * p we have (2. 3) The following important result gives a simple expression for the product of two Gauss sums. For χ ∈ F * p we have Another important product formula for Gauss sums is the Hasse-Davenport formula.

(2.5)
We can relate generalized Jacobi sums to Gauss sums in the following way. For χ 1 , χ 2 , . . . , χ k ∈ F * p not all trivial, (2.6) We now develop some new results which we will use in Section 3. Proof.
Corollary 2.3. Let p ≡ 1 (mod 5) be prime and let ψ be a character of order 5 of F * p . If a, b, c ∈ Z are such that a + c, b + c ≡ 0 (mod 5), then Proof. Using (2.5) and (2.6) we see that Applying Lemma 2.2 then yields the result.
We now recall a formula for counting zeros of polynomials in affine space using the additive character. If f (x 1 , x 2 , . . . x n ) ∈ F p [x 1 , x 2 , . . . x n ], then the number of points, N * p , in A n (F p ) satisfying f (x 1 , x 2 , . . . x n ) = 0 is given by In [13], Koblitz provides a formula for the number of points in P n−1 (F p ) on the hypersurface (2.8) Let T be a fixed generator for the group of characters of F * p and set t := p−1 d . Define an equivalence relation ∼ on W by w ∼ w′ if w − w′ is a multiple modulo d of (1, 1, . . . , 1). ( (2.10) Then we have the following theorem. x i = 0, for some d, λ ∈ F p . Let W , ∼ and N p (0, w) be defined by (2.8), (2.9) and (2.10) respectively. Let T be a fixed generator for the group of characters of F * p , let p ≡ 1 (mod d) and define t := p−1 d . Then 2.2. p-adic preliminaries. We first define the Teichmüller character to be the primitive character ω : F p → Z * p satisfying ω(x) ≡ x (mod p) for all x ∈ {0, 1, . . . , p− 1}. We now recall the p-adic gamma function. For further details, see [12]. Let p be an odd prime. For n ∈ Z + we define the p-adic gamma function as for x = 0, where n runs through any sequence of positive integers p-adically approaching x. This limit exists, is independent of how n approaches x, and determines a continuous function on Z p with values in Z * p . We now state a product formula for the p-adic gamma function. If m ∈ Z + , p ∤ m and x = r p−1 with 0 ≤ r ≤ p − 1 then (2.11) The Gross-Koblitz formula [8] allows us to relate Gauss sums and the p-adic gamma function. Let π ∈ C p be the fixed root of x p−1 + p = 0 which satisfies π ≡ ζ p − 1 (mod (ζ p − 1) 2 ). Then we have the following result.

Proofs
Proof of Theorem 1.4. Let N p be the number of points in P 4 (F p ) on Therefore, noting that it suffices to prove We will express both sides of (3.1) in terms of Gauss sums, starting with 4 G 1 5 , 2 5 , 3 5 , 4 5 p . Let m 0 := −1, m 5 := p − 2, d 0 = d 5 := p − 1 and m i := i, d i := 5 for 1 ≤ i ≤ 4. Also, let r i = p−1 di for 0 ≤ i ≤ 5. Consequently, from definition (1.3) we get that For a given k, we easily see that k p−1 5 ≤ j ≤ (k + 1) p−1

5
, with equality on the left when k = j = 0. Therefore 0 ≤ (k + 1)(p − 1) − 5j ≤ p − 1 and we can apply (2.11) with m = 5 and x = k + 1 − 5j p−1 to get We now use Theorem 2.5 to convert this to an expression involving Gauss sums.
To satisfy the conditions of the theorem for all arguments of the p-adic gamma functions above, we split off the term when j = 0. For all other values of j, we have 0 ≤ (p − 1) − j, (k + 1)(p − 1) − 5j ≤ p − 2. We then get Now applying (2.4) yields We will now evaluate N p . We first consider the case when p ≡ 1 (mod 5). By Theorem 2.4 with d = 5, λ = 1 we see that where W , ∼ and N p (0, w) are as defined in (2.8), (2.9) and (2.10) respectively, T is a fixed generator for the group of characters of F p and t = p−1 5 . We now describe the elements of both W and W/ ∼ as they apply to our setting. We first note that the contribution of any (w 1 , w 2 , . . . , w 5 ) to the above formula is the same as any permutation of it. We therefore list the elements of these sets up to permutation. We will however indicate, using a superscript, the multiplicity with which it contributes to the formula, i.e., its total number of distinct permutations. As N p (0, w) vanishes if some but not all of the w i are zero we will just list the elements of W for which all w i are non-zero and call it W * . Therefore We now use W * to evaluate N p (0) := w∈W N p (0, w). We note that many of the elements of W * are multiples modulo 5 of each other. Thus
Taking T to be ω −j gives us We now convert the two sums on the right above, which we call S 1 and S 2 respectively, to expressions involving Gauss sums using (2.3). Then, starting with S 2 , we have We now apply (2.1) to the last summation on the right, which yields (p − 1) if e = −a − b − c − d and zero otherwise. So We again apply (2.1) to the last summation on the right and continue in this manner isolating the sum for each x i in turn and applying (2.1). This leads to Taking T to be ω j gives us N p = p 3 + p 2 + p + 1 − 4 G 1 5 , 2 5 , 3 5 , 4 5 p as required.
Proof of Theorem 1.2. One easily checks the result for p = 2. Combining Theorem 1.3 with d = 5 and Theorem 1.4 yields the result.

Remark
Using (2.7) to count points on certain algebraic varieties is by no means new. This author first observed the technique in [6]. In the proof above, we have applied this technique in the case p ≡ 1 (mod 5). We could have also applied this method to the case p ≡ 1 (mod 5), choosing instead to use Theorem 2.4 for reasons of brevity. The two methods are essentially the same with Theorem 2.4 encapsulating much of the work which must be done if (2.7) is used. For a detailed account of how the case p ≡ 1 (mod 5) is established using (2.7), please see [14].