Hypergeometric Point Counts for Dwork K3-Surfaces Heidi Goodson

[email protected]

Department of Mathematics, University of Minnesota

math.umn.edu/∼goods052

Objective

Main Results

Example p−1 , d

Recall that the (d − 2)-dimensional Dwork hypersurface is given by d d d x1 + x2 + . . . + xd − dλx1x2 · · · xd = 0.

Throughout, let d be an integer, p ≡ 1 (mod d) be prime, t = T be a generator d for the group F× p , and  be the trivial character. Our first result gives a point count 4 4 4 4 formula for Dwork K3-surfaces (d=4): x1 + x2 + x3 + x4 − 4λx1x2x3x4 = 0.

Our main objective is to express the number of points on Dwork hypersurfaces in terms of Greene’s finite field hypergeometric series. We would like to develop formulas that hold uniformly for all primes p ≡ 1 (mod d).

Theorem (G.): Dwork K3-Surface Point Count

Hypergeometric Series Classical hypergeometric series are defined in terms of the Pochhammer symbol 

n+1Fn



a0 a1 . . . b1 . . .

 an  x bn

∞ X

(a0)k . . . (an)k k x. := k=0 (b1)k . . . (bn)k k!

Greene [1] defined an analogous finite field hypergeometric series in terms of Jacobi sums. For example, let A, B, C be characters d ×. Then in F p 

2 F1

A 

 B  x C







p X AχBχ χ(x). := Cχ p−1 χ χ p

When λ4 ≡ 1 (mod p), the number of points on the Dwork K3-surface is given by   t 2t 3t T T T  2 t 2  NFp(λ) = p + p + 1 + 3pT (−1) + p 3F2 . 1   p More generally, for λ 6≡ 0 (mod p),       3t 3t t t 2t 3t 1 T T T T T T 2 4 2 2     + 3p F NFp(λ) = p + p + 1 + p 3F2 λ 2 1 4 2t T   λ p Tt p t

+ 12pT (−1)1F0



 2t 4 T λ

p

.

The proof of this result starts with a theorem of Koblitz [6] that defines the point count in terms of a sum of Gauss sum expressions. We then use properties of Gauss and Jacobi sums to write the formula in terms of hypergeometric series. The 3F2 series, which we consider to be the “main” term, comes from a particularly nice Gauss sum expression in Koblitz’s formula. The following theorem extends this result to higher dimensional Dwork hypersurfaces.

Theorem (G.): Dwork Hypersurface Point Count Motivation: Legendre Curves For λ 6= 0, 1, we define an elliptic curve in the Legendre family by 2

Eλ : y = x(x − 1)(x − λ). In 1995 Koike [2] showed that the trace of Frobenius, and thus the point count, for curves in this family can be expressed in terms of finite field hypergeometric series:   ψ ψ 2 2 λ , aEλ(p) = −ψ2(−1)p · 2F1   p where ψ2 is a character of order 2. Similar expressions have been developed for other curves [3] and for certain modular Calabi-Yau threefolds [4]. The period associated to the Legendre elliptic curve is given by  1 1 2 2 λ .  F 2 1 1 

π=

Note the similarity between the trace and period expressions. This is to be expected for two reasons. Manin proved in [5] that the period and the trace of Frobenius of any algebraic curve are congruent modulo p. Furthermore, one can show that, in general, “matching” classical and finite field 2F1 hypergeometric series expressions are congruent modulo p.

The “main” term in the point count formula for the (d − 2)-dimensional Dwork hypersurface is the hypergeometric series   t 2t (d−1)t 1 T T . . . T d−2  . p · d−1Fd−2  d  ...  λ p

We compute the point counts in Sage using our finite field hypergeometric expression for some small primes. Prime 5 13 17 29 37 41 # Points 16 352 88 1216 1296 1240 Table 1: Point count for x41 + x42 + x43 + x44 − 24x1x2x3x4 = 0

We have also verified in Sage that the classical and finite field 3F2 hypergeometric series associated to this surface are congruent modulo these primes.

Next Steps Question: When can we expect an algebraic variety to have a hypergeometric point count? Question: What is the relationship between point counts and periods associated to higher dimensional varieties? We would like to show that the classical and finite hypergeometric series associated to higher dimensional Dwork hypersurfaces are congruent modulo p. Ideally this would not just be a neat combinatorial identity, but rather a statement about the geometry of these surfaces. Question: Can we identify modular forms associated to particular Dwork K3-surfaces? This would extend the results of others that give formulas for the coefficients of modular forms in terms of hypergeometric series.

References [1] J. Greene, Hypergeometric functions over finite fields, Trans. Amer. Math. Soc., 301(1) 1987, 77-101.

Connection to Periods Proposition: Dwork Hypersurface Period The period of the Dwork K3-surface is given by   1 2 3 1 4 4 4  π = 3 F2  4 . 1 1 λ Similarly, the period of the (d − 2)-dimensional Dwork hypersurface is given by   1 2 d−1 1 · · · d . π = d−1Fd−2  d d d 1 ··· 1 λ In both cases we have the same “matching” of classical and finite field hypergeometric terms that we saw in the elliptic curve case.

[2] M. Koike, Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields, Hiroshima Math. J., 24(1) 1995, 43-52. [3] C. Lennon, Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves, Proc. Amer. Math. Soc., 139(6) 2011, 1931-1938. [4] D. McCarthy, On a supercongruence conjecture of Rodriguez-Villegas, Proc. Amer. Math. Soc., 140(7) 2012, 2241–2254. [5] J. I. Manin, Hasse-Witt Matrix of an Algebraic Curve, Amer. Math. Soc. Transl. Ser. 2, 45 1965, 245-264. [6] N. Koblitz, The number of points on certain families of hypersurfaces over finite fields, Compositio Math., 48(1) 1983, 3-23.

Hypergeometric Point Counts for Dwork K3-Surfaces

We would like to develop formulas that hold uniformly for all primes p ≡ 1 (mod d). Hypergeometric Series. Classical hypergeometric series are defined in terms ...

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