Hypergeometric decomposition of symmetric K3 quartic pencils
Charles F. Doran, Tyler L. Kelly, Adriana Salerno, Steven Sperber, John Voight, Ursula Whitcher
aa r X i v : . [ m a t h . N T ] J a n HYPERGEOMETRIC DECOMPOSITION OFSYMMETRIC K3 QUARTIC PENCILS
CHARLES F. DORAN, TYLER L. KELLY, ADRIANA SALERNO, STEVEN SPERBER, JOHN VOIGHT,AND URSULA WHITCHER
Abstract.
We study the hypergeometric functions associated to five one-parameter deformationsof Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchsdifferential equations; we count points using Gauss sums and rewrite this in terms of finite fieldhypergeometric sums; then we match up each differential equation to a factor of the zeta function,and we write this in terms of global L -functions. This computation gives a complete, explicitdescription of the motives for these pencils in terms of hypergeometric motives. Contents
1. Introduction 12. Picard–Fuchs equations 73. Explicit formulas for the number of points 214. Proof of the main theorem and applications 39Appendix A. Remaining Picard–Fuchs equations 48Appendix B. Finite field hypergeometric sums 56References 681.
Introduction
Motivation.
There is a rich history of explicit computation of hypergeometric functions as-sociated to certain pencils of algebraic varieties. Famously, in the 1950s, Igusa [Igu58] studied theLegendre family of elliptic curves and found a spectacular relation between the F -hypergeometricPicard–Fuchs differential equation satisfied by the holomorphic period and the trace of Frobenius.More generally, the link between the study of Picard–Fuchs equations and point counts via hyperge-ometric functions has intrigued many mathematicians. Clemens [Cl03] referred to this phenomenonas “Manin’s unity of mathematics.” Dwork studied the now-eponymous Dwork pencil [Dwo69, §6j,p. 73], and Candelas–de la Ossa–Rodríguez-Villegas considered the factorization of the zeta func-tion for the Dwork pencil of Calabi–Yau threefolds in [CDRV00, CDRV01], linking physical andmathematical approaches. More recently, given a finite-field hypergeometric function defined over Q , Beukers–Cohen–Mellit [BCM15] construct a variety whose trace of Frobenius is equal to thefinite field hypergeometric sum up to certain trivial factors.1.2. Our context.
In this paper, we provide a complete factorization of the zeta function and moregenerally a factorization of the L -series for some pencils of Calabi–Yau varieties, namely, families ofK3 surfaces. We study certain Delsarte quartic pencils in P (also called invertible pencils ) whicharise naturally in the context of mirror symmetry, listed in (1.2.1). Associated to each family wehave a discrete group of symmetries acting symplectically (i.e., fixing the holomorphic form). Ourmain theorem (Theorem 1.4.1 below) shows that hypergeometric functions are naturally associated Date : January 28, 2020. o this collection of Delsarte hypersurface pencils in two ways: as Picard–Fuchs differential equationsand as traces of Frobenius yielding point counts over finite fields.(1.2.1) Pencil Equation Symmetries Bad primes F x + x + x + x − ψx x x x µ × µ F L x + x x + x x + x x − ψx x x x µ , F L x + x + x x + x x − ψx x x x µ L L x x + x x + x x + x x − ψx x x x µ × µ L x x + x x + x x + x x − ψx x x x µ , Here we write µ n for the group of n th roots of unity. The labels F and L stand for “Fermat" and“loop", respectively, as in [DKSSVW17].In previous work [DKSSVW17], we showed that these five pencils share a common factor in theirzeta functions, a polynomial of degree 3 associated to the hypergeometric Picard–Fuchs differentialequation satisfied by the holomorphic form—see also recent work of Kloosterman [Klo17]. Also ofnote is that the pencils are also related in that one can take a finite group quotient of each family andfind that they are then birational to one another [BvGK12]. However, these pencils (and their zetafunctions) are not the same! In this article, we investigate the remaining factors explicitly (againrecovering the common factor). In fact, we show that each pencil is associated with a distinct andbeautiful collection of auxiliary hypergeometric functions.1.3. Notation.
We use the symbol ⋄ ∈ F = { F , F L , F L , L L , L } to signify one of the five K3pencils in (1.2.1). Let ψ ∈ Q r { , } . Let S = S ( ⋄ , ψ ) be the set of bad primes in (1.2.1) togetherwith the primes dividing the numerator or denominator of either ψ or ψ − . Then for p S , theK3 surface X ⋄ ,ψ has good reduction at p , and for q = p r we let(1.3.1) P ⋄ ,ψ,q ( T ) := det(1 − Frob rp T | H ét , prim ( X ⋄ ,ψ , Q ℓ )) ∈ T Z [ T ] be the characteristic polynomial of the q -power Frobenius acting on primitive second degree étalecohomology for ℓ = p , which is independent of ℓ . (Recall that the primitive cohomology of ahypersurface in P n is orthogonal to the hyperplane class.) Accordingly, the zeta function of X ⋄ ,ψ over F q is(1.3.2) Z q ( X ⋄ ,ψ , T ) = 1(1 − T )(1 − qT ) P ⋄ ,ψ,q ( T )(1 − q T ) . The Hodge numbers of X ⋄ ,ψ imply that the polynomial P ⋄ ,ψ,q ( T ) has degree 21. Packaging thesetogether, we define the (incomplete) L -series(1.3.3) L S ( X ⋄ ,ψ , s ) := Y p S P ⋄ ,ψ,p ( p − s ) − convergent for s ∈ C in a right half-plane.Our main theorem explicitly identifies the Dirichlet series L S ( X ⋄ ,ψ , s ) as a product of hypergeo-metric L -series. To state this precisely, we now introduce a bit more notation. Let ααα = { α , . . . , α d } and βββ = { β , . . . , β d } be multisets with α i , β i ∈ Q ≥ that modulo Z are disjoint. We associate afield of definition K ααα,βββ to ααα, βββ , which is an explicitly given finite abelian extension of Q . For certainprime powers q and t ∈ F q , there is a finite field hypergeometric sum H q ( ααα ; βββ | t ) ∈ K ααα,βββ definedby Katz [Kat90] as a finite field analogue of the complex hypergeometric function, normalized byMcCarthy [McC12b], extended by Beukers–Cohen–Mellit [BCM15], and pursued by many authors:see section 3.1 for the definition and further discussion, and section 3.2 for an extension of this def-inition. We package together the exponential generating series associated to these hypergeometricsums into an L -series L S ( H ( ααα ; βββ | t ) , s ) : see section 4.1 for further notation. .4. Results.
Our main theorem is as follows.
Main Theorem 1.4.1.
The following equalities hold with t = ψ − and S = S ( ⋄ , ψ ) . (a) For the Dwork pencil F , L S ( X F ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · L S ( H ( , ; 0 , | t ) , s − , φ − ) · L S ( H ( ; 0 | t ) , Q ( √− , s − , φ √− ) where (1.4.2) φ − ( p ) := (cid:18) − p (cid:19) = ( − ( p − / is associated to Q ( √− | Q , and φ √− ( p ) := (cid:18) √− p (cid:19) = ( − (Nm( p ) − / is associated to Q ( ζ ) | Q ( √− . (b) For the Klein–Mukai pencil F L , L S ( X F L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · L S ( H ( , , ; 0 , , | t − ) , Q ( ζ ) , s − where L S ( H ( , , ; 0 , , | t − ) , s ) = L S ( H ( , , ; 0 , , | t − ) , s ) are defined over K = Q ( √− . (c) For the pencil F L , L S ( X F L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · L S ( Q ( ζ ) | Q , s − L S ( H ( , ; 0 , | t ) , s − , φ − ) · L S ( H ( ; 0 | t ) , Q ( √− , s − , φ √− ) · L S ( H ( , ; 0 , | t − ) , Q ( ζ ) , s − , φ √ ) where L S ( H ( , ; 0 , | t − ) , s ) = L S ( H ( , ; 0 , | t − ) , s ) are defined over K = Q ( √− , (1.4.3) φ √ ( p ) := (cid:18) √ p (cid:19) ≡ (Nm( p ) − / (mod p ) is associated to Q ( ζ , √ | Q ( ζ ) ,and L ( Q ( ζ ) | Q , s ) := ζ Q ( ζ ) ( s ) /ζ Q ( s ) is the ratio of the Dedekind zeta function of Q ( ζ ) andthe Riemann zeta function. (d) For the pencil L L , L S ( X L L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · ζ Q ( √− ( s − L S ( H ( , ; 0 , | t ) , s − , φ − ) · L S ( H ( , , , ; 0 , , , | t ) , Q ( √− , s − , φ √− φ ψ ) where (1.4.4) φ ψ ( p ) := (cid:18) ψp (cid:19) is associated to Q ( p ψ ) | Q . e) For the pencil L , L S ( X L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · ζ ( s − L S ( H ( , , , ; 0 , , , | t − ) , Q ( ζ ) , s − . We summarize Theorem 1.4.1 for each of our five pencils in (1.4.5): we list the degree of the L -factor, the hypergeometric parameters, and the base field indicating when it arises from basechange. A Dedekind (or Riemann) zeta function factor has factors denoted by -.(1.4.5) Pencil Degree ααα βββ Base Field3 , , , , Q F · , , Q · Q ( √− , from Q F L , , , , Q , , , , Q ( ζ ) , from Q ( √− , , , , Q · - - Q ( ζ ) , from Q F L , , Q Q ( √− , from Q , , Q ( ζ ) , from Q ( √− , , , , Q L L · - - Q ( √− , from Q , , Q , , , , , , Q ( √− , from Q , , , , Q L · - - Q , , , , , , Q ( ζ ) , from Q We extensively checked the equality of Euler factors in Main Theorem 1.4.1 in numerical cases(for many primes and values of the parameter ψ ): for K3 surfaces we used code written by Costa[CT14], and for the finite field hypergeometric sums we used code in Pari/GP and Magma [BCP97],the latter available for download [V18]. See also Example 4.8.3.Additionally, each pencil has the common factor L S ( H ( , , ; 0 , , | t ) , s ) , giving another proofof a result in previous work [DKSSVW17]: we have a factorization over Q [ T ] (1.4.6) P ⋄ ,ψ,p ( T ) = Q ⋄ ,ψ,p ( T ) R ψ,p ( T ) with R ψ,p ( T ) of degree independent of ⋄ ∈ F . The common factor R ψ,p ( T ) is given by the actionof Frobenius on the transcendental part in cohomology, and the associated completed L -function L ( H ( , , ; 0 , , | t ) , s ) is automorphic by Elkies–Schütt [ES08] (or see our summary [DKSSVW17,§5.2]): it arises from a family of classical modular forms on GL over Q , and in particular, it hasanalytic continuation and functional equation. See also recent work of Naskręcki [Nas17].The remaining factors in each pencil in Main Theorem 1.4.1 yield a factorization of Q ⋄ ,ψ,p ( T ) ,corresponding to the algebraic part in cohomology (i.e., the Galois action on the Néron–Severigroup). In particular, the polynomial Q ⋄ ,ψ,p ( T ) has reciprocal roots of the form p times a root ofunity. The associated hypergeometric functions are algebraic by the criterion of Beukers–Heckman[BH89], and the associated L -functions can be explicitly identified as Artin L -functions: see section .7. The algebraic L -series can also be explicitly computed when they are defined over Q [Coh, Nas].For example, if we look at the Artin L -series associated to the Dwork pencil F , Cohen has giventhe following L -series relations (see Proposition 4.7.2):(1.4.7) L S ( H ( , ; 0 , | ψ − ) , s, φ − ) = L S ( s, φ − ψ ) L S ( s, φ − − ψ ) L S ( H ( ; 0 | ψ − , Q ( √− , s, φ √− ) = L S ( s, φ − ψ ) ) L S ( s, φ − − ψ ) ) . In particular, it follows that the minimal field of definition of the Néron–Severi group of X F ,ψ is Q ( ζ , p − ψ , p ψ ) . The expressions (1.4.7), combined with Main Theorem 1.4.1(a), resolvea conjecture of Duan [Dua18]. (For geometric constructions of the Néron–Severi group of X F ,ψ , seeBini–Garbagnati [BG14] and Kloosterman [Klo17]; the latter also provides an approach to explicitlyconstruct generators of the Néron–Severi group for four of the five families studied here, with thestubborn case F L still unresolved.) Our theorem yields a explicit factorization of Q ⋄ ,ψ,q ( T ) forthe Dwork pencil over F q for any odd q (see Corollary 4.7.4). As a final application, Corollary 4.8.1shows how the algebraic hypergeometric functions imply the existence of a factorization of Q ⋄ ,ψ,p ( T ) over Q [ T ] depending only on q for all families. Remark . Our main theorem can be rephrased as saying that the motive associated to primitivemiddle-dimensional cohomology for each pencil of K3 surfaces decomposes into the direct sum ofhypergeometric motives as constructed by Katz [Kat90]. These motives then govern both thearithmetic and geometric features of these highly symmetric pencils. Absent a reference, we do notinvoke the theory of hypergeometric motives in our proof.1.5.
Contribution and relation to previous work.
Our main result gives a complete decompo-sition of the cohomology for the five K3 pencils into hypergeometric factors. We provide formulasfor each pencil and for all prime powers q , giving an understanding of the pencil over Q . Ad-dressing these subtleties, and consequently giving a result for the global L -function, are unique toour treatment. Our point of view is computational and explicit; we expect that our methods willgeneralize and perhaps provide an algorithmic approach to the hypergeometric decomposition forother pencils.As mentioned above, the study of the hypergeometricity of periods and point counts enjoys along-standing tradition. Using his p -adic cohomology theory, Dwork [Dwo69, §6j, p. 73] showed forthe family F that middle-dimensional cohomology decomposes into pieces according to three typesof differential equations. Kadir in her Ph.D. thesis [Kad04, §6.1] recorded a factorization of the zetafunction for F , a computation due to de la Ossa. Building on the work of Koblitz [Kob83], Salerno[Sal09, §4.2.1–4.2.2] used Gauss sums in her study of the Dwork pencil in arbitrary dimension; undercertain restrictions on q , she gave a formula for the number of points modulo p in terms of truncatedhypergeometric functions as defined by Katz [Kat90] as well as an explicit formula [Sal13a, §5.4]for the point count for the family F . Goodson [Goo17b, Theorems 1.1–1.3] looked again at F and proved a similar formula for the point counts over F q for all primes q = p and prime powers q ≡ . In [FLRST15], Fuselier et al. define an alternate finite field hypergeometric function(which differs from those by Katz, McCarthy, and Beukers-Cohen-Mellit) that makes it possible toprove identities that are analogous to well-known ones for classical hypergeometric functions. Theythen use these formulas to compute the number of points of certain hypergeometric varieties.Several authors have also studied the role of hypergeometric functions over finite fields for theDwork pencil in arbitrary dimension, which for K3 surfaces is the family F given in Main Theorem1.4.1. McCarthy [McC16] extended the definition of p -adic hypergeometric functions to provide aformula for the number of F p points on the Dwork pencil in arbitrary dimension for all odd primes p , extending his results [McC12a] for the quintic threefold pencil. Goodson [Goo17a, Theorem 1.2]then used McCarthy’s formalism to rewrite the formula for the point count for the Dwork family in rbitrary dimension in terms of hypergeometric functions when ( n + 1) | ( q − and n is even. Seealso Katz [Kat09], who took another look at the Dwork family.Miyatani [Miy15, Theorem 3.2.1] has given a general formula that applies to each of the fivefamilies, but with hypotheses on the congruence class of q . It is not clear that one can derive ourdecomposition from the theorem of Miyatani.A different line of research has been used to describe the factorization structure of the zetafunction for pencils of K3 surfaces or Calabi–Yau varieties that can recover part of Main Theorem1.4.1. Kloosterman [Klo07a, Klo07b] has shown that one can use a group action to describe thedistinct factors of the zeta function for any one-parameter monomial deformation of a diagonalhypersurface in weighted projective space. He then applied this approach [Klo17] to study the K3pencils above and generalize our work on the common factor. His approach is different from boththat work and the present one: he uses the Shioda map [Shi86] to provide a dominant rational mapfrom a monomial deformation of a diagonal (Fermat) hypersurface to the K3 pencils. The Shiodamap has been used in the past [BvGK12] to recover the result of Doran–Greene–Judes matchingPicard–Fuchs equations for the quintic threefold examples, and it was generalized to hypersurfacesof fake weighted-projective spaces and BHK mirrors [Bin11, Kel13]. Kloosterman also providessome information about the other factors in some cases.1.6. Proof strategy and plan of paper.
The proof of Main Theorem 1.4.1 is an involved calcu-lation. Roughly speaking, we use the action of the group of symmetries to calculate hypergeometricperiods and then use this decomposition to guide an explicit decomposition of the point count intofinite field hypergeometric sums.Our proof follows three steps. First, in section 2, we find all Picard–Fuchs equations via thediagrammatic method developed by Candelas–de la Ossa–Rodríguez-Villegas [CDRV00, CDRV01]and Doran–Greene–Judes [DGJ08] for the Dwork pencil of quintic threefolds. For each of our fivefamilies, we give the Picard–Fuchs equations in convenient hypergeometric form.Second, in section 3, we carry out the core calculations by counting points over F q for thecorresponding pencils using Gauss sums. This technique begins with the original method of Weil[Wei49], extended by Delsarte and Furtado Gomida, and fully explained by Koblitz [Kob83]. Wethen take these formulas and, using the hypergeometric equations found in section 2 and carefulmanipulation, link these counts to finite field hypergeometric functions. The equations computedin section 2 do not enter directly into the proof of the theorem, but they give an answer that canthen be verified by some comparatively straightforward manipulations. These calculations confirmthe match predicted by Manin’s “unity” (see [Cl03]).Finally, in section 4, we use the point counts from section 3 to explicitly describe the L -series foreach pencil, and prove Main Theorem 1.4.1. We conclude by relating the L -series to factors of thezeta function for each pencil.1.7. Acknowledgements.
The authors heartily thank Xenia de la Ossa for her input and manydiscussions about this project. They also thank Simon Judes for sharing his expertise, Frits Beukers,David Roberts, Fernando Rodríguez-Villegas, and Mark Watkins for numerous helpful discussions,Edgar Costa for sharing his code for computing zeta functions, and the anonymous referee for helpfulcorrections and comments. The authors would like to thank the American Institute of Mathematics(AIM) and its SQuaRE program, the Banff International Research Station, SageMath, and theMATRIX Institute for facilitating their work together. Doran acknowledges support from NSERCand the hospitality of the ICERM at Brown University and the CMSA at Harvard University. Kellyacknowledges that this material is based upon work supported by the NSF under Award No. DMS-1401446 and the EPSRC under EP/N004922/1. Voight was supported by an NSF CAREER Award(DMS-1151047) and a Simons Collaboration Grant (550029). . Picard–Fuchs equations
In this section, we compute the Picard–Fuchs equations associated to all primitive cohomologyfor our five symmetric pencils of K3 surfaces defined in (1.2.1). Since we are working with pencilsin projective space, we are able to represent 21 of the h ( X ψ ) = 22 dimensions of the second degreecohomology as elements in the Jacobian ring, that is, the primitive cohomology of degree two forthe quartic pencils in P . We employ a more efficient version of the Griffiths–Dwork techniquewhich exploits discrete symmetries. This method was previously used by Candelas–de la Ossa–Rodríguez-Villegas [CDRV00, CDRV01] and Doran-Greene–Judes [DGJ08]. Gährs [Gäh13] used asimilar combinatorial technique to study Picard–Fuchs equations for holomorphic forms on invertiblepencils. After explaining the Griffiths–Dwork technique for symmetric pencils in projective space,we carry out the computation for two examples in thorough detail, and then state the results of thecomputation for three others.2.1. Setup.
We briefly review the computational technique of Griffiths–Dwork [CDRV00, CDRV01,DGJ08], and we begin with the setup in some generality.Let X ⊂ P n be a smooth projective hypersurface over C defined by the vanishing of F ( x , . . . , x n ) ∈ C [ x , . . . , x n ] homogeneous of degree d . Let A i ( X ) be the space of rational i -forms on P n with polarlocus contained in X , or equivalently regular i -forms on P n \ X . By Griffiths [Gri69, Corollary 2.1],any ϕ ∈ A n ( X ) can be written as(2.1.1) ϕ = Q ( x , . . . , x n ) F ( x , . . . , x n ) k Ω , where k ≥ and Q ∈ C [ x , . . . , x n ] is homogeneous of degree deg Q = k deg F − ( n + 1) and(2.1.2) Ω := n X i =0 ( − i x i d x ∧ . . . ∧ d x i − ∧ d x i +1 ∧ . . . ∧ d x n . We define the de Rham cohomology groups(2.1.3) H i ( X ) := A i ( X )d A i − ( X ) . There is a residue map
Res : H n ( X ) → H n − ( X, C ) made famous by seminal work of Griffiths [Gri69], mapping into the middle-dimensional Betticohomology of the hypersurface X . Given ϕ ∈ A n ( X ) , we choose an ( n − -cycle γ in X and T ( γ ) a circle bundle over γ with an embedding into the complement P n \ X that encloses γ , and define Res( ϕ ) to be the ( n − -cocycle such that(2.1.4) π √− Z T ( γ ) ϕ = Z γ Res( ϕ ) , well-defined for ϕ ∈ H n ( X ) . Two circle bundles T ( γ ) with small enough radius are homologous in H n ( P n \ X, Z ) , so the class Res( ϕ ) ∈ H n − ( X, C ) is well-defined.There is a filtration on H n ( X ) by an upper bound on the order of the pole along X : H n ( X ) ⊆ H n ( X ) ⊆ . . . ⊆ H nn ( X ) = H n ( X ) . This filtration on H n ( X ) is compatible with the Hodge filtration on H n − ( X, C ) : if we define F k ( X ) := H n − , ( X, C ) ⊕ . . . ⊕ H k,n − k − ( X, C ) , then the residue map restricts to Res : H nk ( X ) → F n − k ( X ) . n certain circumstances, we may be able to reduce the order of the pole [Gri69, Formula 4.5]:we have(2.1.5) Ω F ( x i ) k +1 n X j =0 Q j ( x i ) ∂F ( x i ) ∂x j = 1 k Ω F ( x i ) k n X j =0 ∂Q j ( x i ) ∂x j + ω where ω is an exact rational form. In fact, equation (2.1.5) implies that the order of a form ϕ canbe lowered (up to an exact form) if and only if the polynomial Q is in the Jacobian ideal J ( F ) ,that is, the (homogeneous) ideal generated by all partial derivatives of F . So for k ≥ we have anatural identification(2.1.6) H nk ( X ) H nk − ( X ) ∼ −→ (cid:18) C [ x , . . . , x n ] J ( F ) (cid:19) k deg F − ( n +1) which by the residue map induces an identification(2.1.7) (cid:18) C [ x , . . . , x n ] J ( F ) (cid:19) k deg F − ( n +1) → H n − k,k − ( X ) , whose image is the primitive cohomology group H n − k,k − prim ( X ) , which we know is the cohomologyorthogonal to the hyperplane class since X is a hypersurface in P n . Example . For X a quartic hypersurface in P , the identification (2.1.7) reads(2.1.9) C [ x , x , x , x ] k J ( F ) ≃ H − k,k prim ( X ) . In this case, the Hodge numbers are given by h , = 1 , h , = 35 − · , and h , =165 − ·
56 + 6 ·
10 = 1 .2.2.
Griffiths–Dwork technique.
Now suppose that X ψ is a pencil of hypersurfaces in the pa-rameter ψ , defined by F ψ = 0 . Let { γ j } j be a basis for H n − ( X ψ , C ) with cardinality h n − :=dim C H n − ( X ψ , C ) . Remark . There is a subtle detail about taking a parallel transport using an Ehresmann con-nection to obtain a (locally) unique horizontal family of homology classes [DGJ08, §2.3]. This detaildoes not affect our computations.We then choose a basis of (possibly ψ -dependent) ( n − -forms Ω X ψ ,i ∈ H n − ( X, C ) so that eachof the forms Ω X ψ ,i ∈ H n − ( X, C ) has fixed bidegree ( p, q ) which provides a basis for the Hodgedecomposition H n − ( X, C ) = L p + q = n − H p,q ( X ) for each fixed ψ . We now examine the periodintegrals Z γ j Ω X ψ ,i for ≤ i, j ≤ h n − .We want to understand how these integrals vary with respect to the pencil parameter ψ . To doso, we simply differentiate with respect to ψ , or equivalently integrate on the complement of X ψ in P n as outlined above. Using the residue relation (2.1.4), we rewrite:(2.2.2) Z γ j Ω X ψ ,i = Z T ( γ j ) Q i F kψ Ω , for some Q i ∈ C [ x , . . . , x n ] k deg F ψ − ( n +1) and k ∈ Z ≥ (and circle bundle T ( γ j ) with sufficientlysmall radius as above). By viewing F ψ as a function F : C → C [ x , . . . , x n ] with parameter ψ , we an differentiate F ( ψ ) with respect to ψ and study how this period integral varies:(2.2.3) dd ψ Z T ( γ j ) Q i F ( ψ ) k Ω = − k Z T ( γ j ) Q i F ( ψ ) k +1 d F d ψ Ω Note that the right-hand side of (2.2.3) gives us a new ( n − -form.We know that we will find a linear relation if we differentiate dim C H n − ( X ψ , C ) times, givingus a single-variable ordinary differential equation called the Picard–Fuchs equation for the period R γ j Ω X ψ ,i . In practice, fewer derivatives may be necessary.For simplicity, we suppose that F ψ is linear in the variable ψ . Then the Griffiths–Dwork technique for finding the Picard–Fuchs equation is the following procedure (see [CK99] or [DGJ08] for a moredetailed exposition):1. Differentiate the period b times, ≤ b ≤ h n − . We obtain the equation (cid:18) dd ψ (cid:19) b Z T ( γ j ) Q i F ( ψ ) k Ω = ( k + b − k − Z T ( γ j ) Q i F ( ψ ) k + b (cid:18) − d F d ψ (cid:19) b Ω .
2. Write(2.2.4) Q i (cid:18) − d F d ψ (cid:19) b = h n − X j =1 α j Q j + J where α k ∈ C ( ψ ) and J is in the Jacobian ideal, so we may write J = P i A i ∂F ψ ∂x i with A i ∈ C ( ψ )[ x , . . . , x n ] for all i .3. Use (2.1.5) to reduce the order of the pole of JF ( ψ ) k Ω . We obtain a new numerator polynomialof lower degree.4. Repeat steps 2 and 3 for the new numerator polynomials, until the b th derivative is expressed interms of the chosen basis for cohomology.5. Use linear algebra to find a C ( ψ ) -linear relationship between the derivatives.While algorithmic and assured to work, this method can be quite tedious to perform. Moreover,the structure of the resulting differential equation may not be readily apparent.2.3. A diagrammatic Griffiths–Dwork method.
In this section, we give a computational tech-nique that uses discrete symmetries of pencils of Calabi–Yau hypersurfaces introduced by Candelas–de la Ossa–Rodríguez-Villegas [CDRV00, CDRV01]. To focus on the case at hand, we specialize tothe case of quartic surfaces and explain this method so their diagrammatic and effective adaptationof the Griffiths–Dwork technique can be performed for the five pencils that we want to study.Let x v := x v x v x v x v and let k ( v ) := P i v i ; for a monomial arising from (2.1.9), we have k ( v ) ∈ Z ≥ . Fix a cycle γ , and consider the periods(2.3.1) ( v , v , v , v ) := Z T ( γ ) x v F k ( v )+1 ψ Ω . Consider the relation:(2.3.2) ∂ i x i x v F k ( v )+1 ψ = x v F k ( v )+1 ψ (1 + v i ) − ( k ( v ) + 1) x v F k ( v )+2 ψ x i ∂ i F ψ . e can use (2.3.2) in order to simplify the computation of the Picard–Fuchs equation: integratingover T ( γ ) , the left hand side vanishes, so we can solve for ( v , v , v , v ) :(2.3.3) (1 + v i )( v , v , v , v ) := Z T ( γ ) x v F k ( v )+1 ψ Ω = ( k ( v ) + 1) Z T ( γ ) x v x i ∂ i F ψ F k ( v )+2 ψ Ω . Example . Consider the Dwork pencil F , the pencil defined by the vanishing of F ψ = x + x + x + x − ψx x x x . Simplifying the right-hand side of (2.3.3) gives us the relation of periods: (1 + v i )( v , v , v , v ) = 4( k ( v ) + 1) (( v , . . . , v i + 4 , . . . , v ) − ψ ( v + 1 , v + 1 , v + 1 , v + 1)) or in a more useful form(2.3.5) ( v , . . . , v i + 4 , . . . , v ) = 1 + v i k ( v ) + 1) ( v , v , v , v ) + ψ ( v + 1 , v + 1 , v + 1 , v + 1) for i = 0 , , , .Recall we can also find a relation between various ( v , v , v , v ) by differentiating with respectto ψ . Rewriting (2.2.3) in the current notation, we obtain(2.3.6) dd ψ ( v , v , v , v ) = (4( k ( v ) + 1))( v + 1 , v + 1 , v + 1 , v + 1) , yielding a dependence of the monomials with respect to the successive derivatives with respect to ψ . Using the relations (2.3.3) and (2.3.6), we will compute the Picard–Fuchs equations associatedto periods that come from primitive cohomology. The key observation is that these two operationsrespect the symplectic symmetry group.Restricting now to our situation, let ⋄ ∈ { F , F L , F L , L L , L } signify one of the five K3families in (1.2.1) defined by F ⋄ ,ψ and having symmetry group H = H ⋄ as in (1.2.1). Then H actson the -dimensional C -vector space(2.3.7) V := ( C [ x , x , x , x ] /J ( F ψ )) giving a representation H → GL( V ) . As H is abelian, we may decompose V = L χ W χ where H acts on W χ by a (one-dimensional) character χ : H → C × . Conveniently, each subspace W χ hasa monomial basis. Moreover, the relations from the Jacobian ideal (2.3.3) and (2.3.6) respect theaction of H , so we can apply the Griffiths–Dwork technique to the smaller subspaces W χ .2.4. Hypergeometric differential equations.
In fact, we will find that all of our Picard–Fuchsdifferential equations are hypergeometric. In this section, we briefly recall the definitions [Sla66].
Definition . Let n, m ∈ Z , let α , . . . , α n ∈ Q and β , . . . , β m ∈ Q > , and write ααα = { α j } j and βββ = { β j } j as multisets. The (generalized) hypergeometric function is the formal series(2.4.2) F ( ααα ; βββ | z ) := ∞ X k =0 ( α ) k · · · ( α n ) k ( β ) k · · · ( β m ) k z k ∈ Q [[ z ]] , where ( x ) k is the rising factorial (or Pochhammer symbol ) ( x ) k := x ( x + 1) · · · ( x + k −
1) = Γ( x + k )Γ( x ) and ( x ) := 1 . We call ααα the numerator parameters and βββ the denominator parameters . e consider the differential operator θ := z dd z and define the hypergeometric differential operator (2.4.3) D ( ααα ; βββ | z ) := ( θ + β − · · · ( θ + β m − − z ( θ + α ) · · · ( θ + α n ) . When β = 1 , the hypergeometric function F ( ααα ; βββ | z ) is annihilated by D ( ααα ; βββ | z ) .2.5. The Dwork pencil F . We now proceed to calculate Picard–Fuchs equations for our fivepencils. We begin in this section with the Dwork pencil F , the one-parameter family of projectivehypersurfaces X ψ ⊂ P defined by the vanishing of the polynomial F ψ := x + x + x + x − ψx x x x . The differential equations associated to this pencil were studied by Dwork [Dwo69, §6j]; our approachis a bit more detailed and explicit, and this case is a good warmup as the simplest of the five caseswe will consider.There is a H = ( Z / Z ) symmetry of this family generated by the automorphisms(2.5.1) g ( x : x : x : x ) = ( −√− x : √− x : x : x ) g ( x : x : x : x ) = ( −√− x : x : √− x : x ) . A character χ : H → C × is determined by χ ( g ) , χ ( g ) ∈ h√− i , and we write χ ( a ,a ) for thecharacter with χ ( a ,a ) ( g i ) = √− a i with a i ∈ Z / Z for i = 1 , , totalling characters. We thendecompose V defined in (2.3.7) into irreducible subspaces with a monomial basis. We cluster thesesubspaces into three types up to the permutation action by S on coordinates:(i) ( a , a ) = (0 , (the H -invariant subspace), spanned by x x x x ;(ii) ( a , a ) both even but not both zero, e.g., the subspace with ( a , a ) = (0 , spanned by x x , x x ; and(iii) ( a , a ) not both even, e.g., the subspace with ( a , a ) = (2 , , spanned by x x .Up to permutation of coordinates, there are , , subspaces of types (i),(ii),(iii), respectively.By symmetry, we just need to compute the Picard–Fuchs equations associated to one subspace ofeach of these types. In other words, we only need to find equations satisfied by the monomials x x x x , x x , x x , and x x , corresponding to (1 , , , , (2 , , , , (0 , , , , and (3 , , , ,respectively.The main result for this subsection is as follows. Proposition 2.5.2.
The primitive middle-dimensional cohomology group H prim ( X F ,ψ , C ) has periods whose Picard–Fuchs equations are hypergeometric differential equations as follows: periods are annihilated by D ( , , ; 1 , , | ψ − ) , periods are annihilated by D ( , ; 1 , | ψ − ) , and periods are annihilated by D ( ; 1 | ψ − ) . By the interlacing criterion [BH89, Theorem 4.8], the latter two hypergeometric equations havealgebraic solutions.We state and prove each case of Proposition 2.5.2 with an individual lemma.
Lemma 2.5.3.
The Picard–Fuchs equation associated to the period ψ (0 , , , is the hypergeometricdifferential equation D ( , , ; 1 , , | ψ − ) . roof. We recall the equations (2.3.5) and (2.3.6):(2.5.4) ( v , . . . , v i + 4 , . . . , v ) = 1 + v i k ( v ) + 1) ( v , v , v , v ) + ψ ( v + 1 , v + 1 , v + 1 , v + 1); (2.5.5) dd ψ ( v , v , v , v ) = (4( k ( v ) + 1))( v + 1 , v + 1 , v + 1 , v + 1) . These equations imply a dependence among the terms ( v , v , v , v ) , ( v + 1 , v + 1 , v + 1 , v + 1) , and ( v , . . . , v i + 4 , . . . , v ) denoted in the following diagram: ( v , v , v , v ) / / (cid:15) (cid:15) ( v + 1 , v + 1 , v + 1 , v + 1)( v , . . . , v i + 4 , . . . , v ) In order to use these dependences, we build up a larger diagram:(2.5.6) (0 , , , / / (cid:15) (cid:15) (1 , , , / / (cid:15) (cid:15) (2 , , , / / (cid:15) (cid:15) (3 , , , , , , / / (cid:15) (cid:15) (5 , , , / / (cid:15) (cid:15) (6 , , , , , , / / (cid:15) (cid:15) (5 , , , , , , − / / (cid:15) (cid:15) (4 , , , , , , It may be useful to point out that the same period must appear in two places by simple linear algebra:the vectors (4 , , , , (0 , , , , (0 , , , , (0 , , , and (1 , , , are linearly dependent.Using (2.3.5) and (2.3.6) and letting η := ψ dd ψ , we see that:(2.5.7) (0 , , ,
0) = 14 ( η + 1)(4 , , , , , ,
0) = 18 ( η + 1)(4 , , , , , ,
0) = 112 ( η + 1)(4 , , , ψ (4 , , ,
0) = (3 , , , ow, we can use the fact that ( η − a ) ψ a = ψ a η for a ∈ Z to great effect:(2.5.8) η (0 , , ,
0) = 4 ψ (1 , , , η − η (0 , , ,
0) = 4 ψη (1 , , ,
1) = 8 · ψ (2 , , , η − η − η (0 , , ,
0) = 12 · · ψ η (2 , , ,
2) = 12 · · ψ (3 , , , · · ψ (4 , , , · ψ ( η + 1)(4 , , , ψ ( η + 1) (4 , , , ψ ( η + 1) (0 , , , . We conclude that(2.5.9) (cid:2) ( η − η − η − ψ ( η + 1) (cid:3) (0 , , ,
0) = 0 . We then multiply by ψ to obtain (cid:2) ψ ( η − η − η − ψ ψ ( η + 1) (cid:3) (0 , , ,
0) = 0 (cid:2) ( η − η − η − − ψ ( η ) (cid:3) ψ (0 , , ,
0) = 0 . Finally, substitute t := ψ − and let θ := t dd t = − η/ to see that(2.5.10) (cid:2) ( − θ − − θ − − θ − − t − ( − θ ) (cid:3) ψ (0 , , ,
0) = 0 (cid:2) − t ( θ + )( θ + )( θ + ) + θ (cid:3) ψ (0 , , ,
0) = 0 (cid:2) θ − t ( θ + )( θ + )( θ + ) (cid:3) ψ (0 , , ,
0) = 0 , which is the differential equation D ( , , ; 1 , , | t ) . (cid:3) Lemma 2.5.11.
The Picard–Fuchs equation associated to both ψ (2 , , , and ψ (0 , , , is D ( , ; 1 , | ψ − ) .Proof. By iterating the use of (2.3.5), we can construct a diagram including both (2 , , , and (0 , , , :(2.5.12) (0 , , , (cid:15) (cid:15) / / (1 , , , , − , , (cid:15) (cid:15) / / (4 , , , , , , (cid:15) (cid:15) / / (3 , , , , , , − (cid:15) (cid:15) / / (2 , , , , , , We then obtain the following relations:(2.5.13) η (2 , , ,
0) = ψ ( η + 1)(0 , , , η (0 , , ,
2) = ψ ( η + 1)(2 , , , . e then can use these relations to make a Picard–Fuchs equation associated to the period (2 , , , :(2.5.14) ( η − η (2 , , ,
0) = ψ ( η + 1) η (0 , , , ψ ( η + 1) (cid:0) ψ ( η + 1)(2 , , , (cid:1) = 2 ψ ( η + 1)(2 , , ,
0) + ψ ( η + 1) η (2 , , , ψ ( η + 1)(2 , , , ψ ( η + 4 η + 3)(2 , , , ψ ( η + 1)( η + 3)(2 , , , . By symmetry, we get the same equation for the period (0 , , , , so we have:(2.5.15) (cid:2) ( η − η − ψ ( η + 1)( η + 3) (cid:3) (2 , , ,
0) = 0 (cid:2) ( η − η − ψ ( η + 1)( η + 3) (cid:3) (0 , , ,
2) = 0
Now multiply by ψ and then change variables to t := ψ − with θ := t dd t = − η to obtain: (cid:2) ψ ( η − η − ψψ ( η + 1)( η + 3) (cid:3) (2 , , ,
0) = 0 (cid:2) ( η − η − − ψ η ( η + 2) (cid:3) ψ (2 , , ,
0) = 0 (cid:2) ( − θ − − θ − − t − ( − θ )( − θ + 2) (cid:3) ψ (2 , , ,
0) = 0 (cid:2) t ( θ + )( θ + ) − θ ( θ − ) (cid:3) ψ (2 , , ,
0) = 0 (cid:2) θ ( θ − ) − t ( θ + )( θ + ) (cid:3) ψ (2 , , ,
0) = 0 . This Picard–Fuchs equation is D ( , ; 1 , | ψ − ) . (cid:3) Lemma 2.5.16.
The Picard–Fuchs equation associated to ψ (3 , , , is D ( ; 1 | ψ − ) .Proof. Our strategy again is to use (2.3.5) and (2.3.6) in the order represented by the diagram belowto study the period (3 , , , :(2.5.17) (2 , , − , (cid:15) (cid:15) / / (3 , , , , − , , (cid:15) (cid:15) / / (2 , , , − , , , (cid:15) (cid:15) / / (0 , , , (cid:15) (cid:15) / / (1 , , , , , , (cid:15) (cid:15) / / (4 , , , , , , Using (2.3.5) iteratively in the upper part of the diagram, we see that:(2.5.18) (1 , , ,
2) = ψ (3 , , , . Then using (2.3.6), we then have that(2.5.19) η (0 , , ,
1) = 8 ψ (1 , , ,
2) = 8 ψ (3 , , , . ow, using (2.3.5) again, we have that (3 , , ,
0) = ψ (0 , , , and we can then compute:(2.5.20) ( η − , , ,
0) = ψη (0 , , , ψ (3 , , , ψ (cid:20)
18 (3 , , ,
0) + ψ (4 , , , (cid:21) = 8 ψ (cid:20)
18 (3 , , ,
0) + 18 η (3 , , , (cid:21) = ψ ( η + 1)(3 , , , . We then get the Picard–Fuchs equation associated to the period (3 , , , :(2.5.21) (cid:2) ( η − − ψ ( η + 1) (cid:3) (3 , , ,
0) = 0 . We now will multiply by ψ and then change variables to t = ψ − as in the previous lemma toobtain: (cid:2) ψ ( η − − ψψ ( η + 1) (cid:3) (3 , , ,
0) = 0 (cid:2) ( η − − ψ η (cid:3) ψ (3 , , ,
0) = 0 (cid:2) ( − θ − − t − ( − θ ) (cid:3) ψ (3 , , ,
0) = 0 (cid:2) θ − t ( θ + ) (cid:3) ψ (3 , , ,
0) = 0 , giving rise to the hypergeometric differential equation D ( ; 1 | ψ − ) . (cid:3) We now conclude this section with the proof of the main result.
Proof of Proposition . We combine Lemmas 2.5.3, 2.5.11, and 2.5.16 with the consideration ofthe number of subspaces of each type described above. (cid:3)
The Klein–Mukai pencil F L . We now consider the Klein–Mukai pencil F L , the one-parameter family of hypersurfaces X ψ ⊂ P defined by the vanishing of F ψ := x x + x x + x x + x − ψx x x x . The polynomial F ψ is related to the defining polynomial (1.2.1) by a change in the order of variables.There is a H = Z / Z scaling symmetry of this family generated by the automorphism ( x i ) bythe element g ( x : x : x : x ) = ( ξx : ξ x : ξ x : x ) , where ξ is a seventh root of unity. There are seven characters χ k : H → C × defined by χ k ( g ) = ξ k for k ∈ Z / Z . Note that the monomial bases for the subspaces W χ , W χ , and W χ are cyclicpermutations of one another under the variables x , x , and x . Analogously, so are subspaces W χ , W χ , and W χ . So we have three types of clusters:(i) W χ has the monomial basis { x x x x } ;(ii) W χ has the monomial basis { x x , x x x , x x x } ; and(iii) W χ has the monomial basis { x x , x x x , x x x } .There is one cluster of type (i) and three clusters each of types (ii) and (iii), so h , is decomposedas
19 = 1 + 3 · · . Proposition 2.6.1.
The group H prim ( X F L ,ψ ) has periods whose Picard–Fuchs equations arehypergeometric differential equations, with periods annihilated by D ( , , ; 1 , , | ψ − ) nd periods each annihilated by the following operators: D ( , , ; , , | ψ ) , D ( − , , ; 0 , , | ψ ) , D ( − , − , ; − , , | ψ ) ,D ( , , ; , , | ψ ) , D ( − , , ; 0 , , | ψ ) , D ( − , − , ; − , , | ψ ) . Again, the latter operators have an algebraic solution. To prove Proposition 2.6.1, we againuse the diagrammatic method outlined above, but in this case we have different periods that arerelated. Notice that we have the following differentials ∂ i multiplied by x i :(2.6.2) x ∂ F ψ = 3 x x + x x − ψx x x x x ∂ F ψ = 3 x x + x x − ψx x x x x ∂ F ψ = 3 x x + x x − ψx x x x x ∂ F ψ = 4 x − ψx x x x We can make linear combinations of these equations so that the right hand side is just a linearcombination of two monomials, for example,(2.6.3) (9 x ∂ + x ∂ − x ∂ ) F ψ = 28( x x − ψx x x x ) . Now using (2.3.3), we obtain the following period relations analogous to (2.3.5), written in multi-index notation:(2.6.4) v + (3 , , ,
0) = f ( v )28( k ( v ) + 1) v + ψ ( v + (1 , , , , v + (0 , , ,
0) = f ( v )28( k ( v ) + 1) v + ψ ( v + (1 , , , , v + (0 , , ,
1) = f ( v )28( k ( v ) + 1) v + ψ ( v + (1 , , , , and v + (0 , , ,
4) = 1 + v k ( v ) + 1) v + ψ ( v + (1 , , , where(2.6.5) f ( v ) := 9( v + 1) + ( v + 1) − v + 1) ,f ( v ) := − v + 1) + 9( v + 1) + ( v + 1) , and f ( v ) := ( v + 1) − v + 1) + 9( v + 1) . Lemma 2.6.6.
Let t = ψ − . The Picard–Fuchs equation associated to the period ψ (0 , , , is thehypergeometric differential equation D ( , , ; 1 , , | t ) . roof. We build the following diagram using (2.6.4) and (2.3.6): (0 , , , / / (cid:15) (cid:15) (1 , , , / / (cid:15) (cid:15) (2 , , , / / (cid:15) (cid:15) (3 , , , , , , / / (cid:15) (cid:15) (4 , , , / / (cid:15) (cid:15) (5 , , , , , , / / (cid:15) (cid:15) (4 , , , , , , − / / (cid:15) (cid:15) (4 , , , , , , When one runs through this computation, one can see that we get the same Picard–Fuchs equationfor the invariant period as we did with the Fermat:(2.6.7) (cid:2) ( η − η − η − ψ ( η + 1) (cid:3) (0 , , ,
0) = 0
By multiplying by ψ and changing variables to t = ψ − and θ = t dd t , we can see by followingthrough the computation seen in (2.5.10) that: (cid:2) θ − t ( θ + )( θ + )( θ + ) (cid:3) ψ (0 , , ,
0) = 0 , which is the differential equation D ( , , ; 1 , , | t ) . (cid:3) Lemma 2.6.8.
For the Klein–Mukai family X ψ ,the period (0 , , , is annihilated by D ( , , ; , , | ψ ) ,the period ψ (0 , , , is annihilated by D ( − , , ; 0 , , | ψ ) , andthe period ψ (2 , , , is annihilated by D ( − , − , ; − , , | ψ ) .Proof. For the character χ ( g ) = ξ associated to ∈ Z / Z , we have the following diagram: (0 , , , / / (cid:15) (cid:15) (1 , , , , , , / / (cid:15) (cid:15) (3 , , , , , , − / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (1 , , , , , , e then have the following relations:(2.6.9) η (0 , , ,
2) = 117 ψ + ψ η (2 , , , η (2 , , ,
0) = 27 ψ (0 , , ,
1) + ψη (0 , , , η (0 , , ,
1) = 17 ψ (0 , , ,
2) + ψη (0 , , , . Now we can use these relations to compute the Picard–Fuchs equations associated to (2 , , , , (0 , , , , and (0 , , , . We first do this for the period (0 , , , :(2.6.10) η (0 , , ,
1) = 17 ψ (0 , , , ψη (0 , , , η − η (0 , , ,
1) = 16549 ψ (2 , , ,
0) + 267 ψ η (2 , , ,
0) + ψ η (2 , , , η − η − η (0 , , ,
1) = ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) This gives us the Picard–Fuchs equation for the period (0 , , , :(2.6.11) (cid:2) ( η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (0 , , ,
1) = 0 , Letting u = ψ and σ = u dd u , we get the following hypergeometric form: (cid:2) (4 σ − σ − σ − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (0 , , ,
1) = 0 (cid:2) ( σ − )( σ − ) σ − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (0 , , ,
1) = 0 , which is the hypergeometric differential equation D ( , , ; 1 , , | u ) .We then do the same for (0 , , , :(2.6.12) η (0 , , ,
2) = 117 ψ + ψ η (2 , , , η − η (0 , , ,
2) = 3649 ψ (0 , , ,
1) + 207 ψ η (0 , , ,
1) + ψ η (0 , , , η − η − η (0 , , ,
2) = ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) (0 , , , . This gives us the Picard–Fuchs equation for the period (0 , , , :(2.6.13) (cid:2) ( η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (0 , , ,
2) = 0 . By multiplying by ψ and changing variables to u = ψ and σ = u dd u , we get: ψ (cid:2) ( η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (0 , , ,
2) = 0 (cid:2) ( η − η − η − − ψ (cid:0) η − (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) ψ (0 , , ,
2) = 0 (cid:2) (4 σ − σ − σ − − u (cid:0) σ − (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) ψ (0 , , ,
2) = 0 (cid:2) ( σ − σ − )( σ − ) − u (cid:0) σ − (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) ψ (0 , , ,
2) = 0 , which is the hypergeometric differential equation D ( , , − ; 0 , , | u ) . e finally look at (2 , , , :(2.6.14) η (2 , , ,
0) = 27 ψ (0 , , ,
1) + ψη (0 , , , η − η (2 , , ,
0) = 27 ψη (0 , , ,
1) + ψη (0 , , , ψ (0 , , ,
2) + 107 ψ η (0 , , ,
2) + ψ η (0 , , , η − η − η (2 , , ,
0) = 949 ψ η (0 , , ,
2) + 107 ψ η (0 , , ,
2) + ψ η (0 , , , ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) (2 , , , . This gives us the Picard-Fuchs equation for the period (2 , , , :(2.6.15) (cid:2) ( η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (2 , , ,
0) = 0
By multiplying by ψ and again changing variables we get: ψ (cid:2) ( η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (2 , , ,
0) = 0 (cid:2) ( η − η − η − − ψ (cid:0) η − (cid:1) (cid:0) η − (cid:1) (cid:0) η + (cid:1)(cid:3) ψ (2 , , ,
0) = 0 (cid:2) (4 σ − σ − σ − − u (cid:0) σ − (cid:1) (cid:0) σ − (cid:1) (cid:0) σ + (cid:1)(cid:3) ψ (2 , , ,
0) = 0 (cid:2) ( σ − )( σ − σ − ) − u (cid:0) σ − (cid:1) (cid:0) σ − (cid:1) (cid:0) σ + (cid:1)(cid:3) ψ (2 , , ,
0) = 0; at last, we have the hypergeometric differential equation D ( , − , − ; 0 , − , | u ) . (cid:3) Lemma 2.6.16.
For the Klein–Mukai family X ψ ,the period (0 , , , is annihilated by D ( , , ; , , | ψ ) ,the period ψ (1 , , , is annihilated by D ( − , , ; 0 , , | ψ ) , andthe period ψ (0 , , , is annihilated by D ( − , − , ; − , , | ψ ) .Proof. We use the following diagram: (0 , , , / / (cid:15) (cid:15) (1 , , , , , , − / / (cid:15) (cid:15) (3 , , , , , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (1 , , , , , , nd then compute the following period relations:(2.6.17) η (0 , , ,
0) = 107 ψ (0 , , ,
1) + ψη (0 , , , η (0 , , ,
1) = 57 ψ (1 , , ,
2) + ψη (1 , , , η (1 , , ,
2) = − ψ (0 , , ,
0) + ψ η (0 , , , . By cyclically using these relations, we get the following Picard–Fuchs equations:(2.6.18) (cid:2) ( η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (0 , , ,
1) = 0; (cid:2) ( η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (1 , , ,
2) = 0; (cid:2) ( η − η − η − ψ (cid:0) η − (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (0 , , ,
0) = 0 . We then multiply these equations above by , ψ, and ψ , respectively and then change coordinatesto u = ψ and σ = u dd u to obtain the following: (cid:2) ( σ − )( σ − ) σ − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (0 , , ,
1) = 0; (cid:2) ( σ − σ − )( σ − ) − u (cid:0) σ − (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) ψ (1 , , ,
2) = 0; (cid:2) ( σ − )( σ − σ − ) − u (cid:0) σ − (cid:1) (cid:0) σ − (cid:1) (cid:0) σ + (cid:1)(cid:3) ψ (0 , , ,
0) = 0 . which are D ( , , ; 1 , , | u ) , D ( , , − ; 0 , , | u ) , and D ( − , , − ; 0 , , − | u ) , respec-tively. (cid:3) We conclude this section by combining these results.
Proof of Proposition . Combine Lemmas 2.6.6, 2.6.8, and 2.6.16. (cid:3)
Remaining pencils.
For the remaining three pencils F L , L L , and L , the Picard–Fuchsequations can be derived in a similar manner. The details can be found in Appendix A; we statehere only the results. Proposition 2.7.1.
The group H prim ( X F L ,ψ , C ) has periods whose Picard–Fuchs equations arehypergeometric differential equations as follows: periods are annihilated by D ( , , ; 1 , , | ψ − ) , periods are annihilated by D ( , ; 1 , | ψ − ) , periods are annihilated by D ( ; 1 | ψ ) , periods are annihilated by D ( , ; 1 , | ψ ) , and periods are annihilated by D ( , − ; 0 , | ψ ) .Proof. See Proposition A.1.2. (cid:3)
Proposition 2.7.2.
The group H prim ( X L L ,ψ , C ) has periods whose Picard–Fuchs equations arehypergeometric differential equations as follows: periods are annihilated by D ( , , ; 1 , , | ψ − ) , periods are annihilated by D ( , , , ; 0 , , , | ψ ) , and periods are annihilated by D ( , ; 1 , | ψ ) .Proof. See Proposition A.2.2. (cid:3) roposition 2.7.3. The group H prim ( X L ,ψ , C ) has periods whose Picard–Fuchs equations arehypergeometric differential equations as follows: periods are annihilated by D ( , , ; 1 , , | ψ − ) , periods are annihilated by D ( , , , ; 1 , , , | ψ ) , periods are annihilated by D ( − , , , ; 0 , , , | ψ ) , periods are annihilated by D ( − , − , , ; − , , , | ψ ) , and periods are annihilated by D ( − , − , − , ; 0 , , − , − | ψ ) .Proof. See Proposition A.3.2. (cid:3) Explicit formulas for the number of points
In this section, we derive explicit formulas for the number of points and identify the hyperge-ometric periods according to the action of the group of symmetries, matching the Picard–Fuchsequations computed in section 2.3.1.
Hypergeometric functions over finite fields.
We begin by defining the finite field analogueof the generalized hypergeometric function (defined in section 2.4); we follow Beukers–Cohen–Mellit[BCM15].Let q = p r be a prime power. We use the convenient abbreviation q × := q − . Let ω : F × q → C × be a generator of the character group on F × q . Let Θ : F q → C × be a nontrivial(additive) character, defined as follows: let ζ p ∈ C be a primitive p th root of unity, and define Θ( x ) = ζ Tr F q | F p ( x ) p . For m ∈ Z , we define the Gauss sum (3.1.1) g ( m ) := X x ∈ F × q ω ( x ) m Θ( x ) . We suppress the dependence on q in the notation, and note that g ( m ) depends only on m ∈ Z /q × Z (and the choice of ω and ζ p ). Remark . Every generator of the character group on F × q is of the form ω k ( x ) := ω ( x ) k for k ∈ ( Z /q × Z ) × , and X x ∈ F × q ω k ( x ) m Θ( x ) = g ( km ) . Similarly, every additive character of F q is of the form Θ k ( x ) := ζ k Tr( x ) p for k ∈ ( Z /p Z ) × , and X x ∈ F × q ω ( x ) m Θ k ( x ) = ω ( k ) − m g ( m ) (see e.g. Berndt [BEW98, Theorem 1.1.3]). Accordingly, we will see below that our definition offinite field hypergeometric functions will not depend on these choices.We will need four basic identities for Gauss sums. Lemma 3.1.3.
The following relations hold: (a) g (0) = − . (b) g ( m ) g ( − m ) = ( − m q for every m q × ) , and in particular g ( q × ) = ( − q × / q. c) For every N | q × with N > , we have (3.1.4) g ( N m ) = − ω ( N ) Nm N − Y j =0 g ( m + jq × /N ) g ( jq × /N ) . (d) g ( pm ) = g ( m ) for all m ∈ Z .Proof. For parts (a)–(c), see Cohen [Coh2, Lemma 2.5.8, Proposition 2.5.9, Theorem 3.7.3]. For(d), we replace x by x p in the definition and use the fact that Θ( x p ) = Θ( x ) as it factors throughthe trace. (cid:3) Remark . Lemma 3.1.3(c) is due originally to Hasse and Davenport, and is called the
Hasse–Davenport product relation .We now build our hypergeometric sums. Let ααα = { α , . . . , α d } and βββ = { β , . . . , β d } be multisetsof d rational numbers. Suppose that ααα and βββ are disjoint modulo Z , i.e., α i − β j Z for all i, j = 1 , . . . , d .Based on work of Greene [Gre87], Katz [Kat90, p. 258], but normalized following McCarthy[McC13, Definition 3.2] and Beukers–Cohen–Mellit [BCM15, Definition 1.1], we make the followingdefinition. Definition . Suppose that(3.1.7) q × α i , q × β i ∈ Z for all i = 1 , . . . , d . For t ∈ F × q , we define the finite field hypergeometric sum by(3.1.8) H q ( ααα, βββ | t ) := − q × q − X m =0 ω (( − d t ) m G ( m + αααq × , − m − βββq × ) where(3.1.9) G ( m + αααq × , − m − βββq × ) := d Y i =1 g ( m + α i q × ) g ( − m − β i q × ) g ( α i q × ) g ( − β i q × ) for m ∈ Z .In this definition (and the related ones to follow), the sum H q ( ααα, βββ | t ) only depends on the classesin Q / Z of the elements of ααα and βββ . Moreover, the sum is independent of the choice of characters ω and Θ by a straightforward application of Remark 3.1.2. The hypothesis (3.1.7) is unfortunatelyrather restrictive—but it is necessary for the definition to make sense as written. Fortunately,Beukers–Cohen–Mellit [BCM15] provided an alternate definition that allows all but finitely many q under a different hypothesis, as follows. Definition . The field of definition K ααα,βββ ⊂ C associated to ααα, βββ is the field generated by thecoefficients of the polynomials(3.1.11) d Y j =1 ( x − e π √− α j ) and d Y j =1 ( x − e π √− β j ) . Visibly, the number field K ααα,βββ is an abelian extension of Q .Suppose that ααα, βββ is defined over Q , i.e., K ααα,βββ = Q . Then by a straightforward verification, thereexist p , . . . , p r , q , . . . , q s ∈ Z ≥ such that(3.1.12) d Y j =1 ( x − e π √− α j )( x − e π √− β j ) = Q rj =1 x p j − Q sj =1 x q j − . ecall we require the ααα, βββ to be disjoint, which implies that the sets { p , . . . , p r } and { q , . . . , q s } are also disjoint.Let D ( x ) := gcd( Q rj =1 ( x p j − , Q sj =1 ( x q j − and M := (cid:0)Q rj =1 p p j j (cid:1)(cid:0)Q sj =1 q − q j j (cid:1) . Let ǫ :=( − P sj =1 q j , and let s ( m ) ∈ Z ≥ be the multiplicity of the root e π √− m/q × in D ( x ) . Finally,abbreviate(3.1.13) g ( pppm, − qqqm ) := g ( p m ) · · · g ( p r m ) g ( − q m ) · · · g ( − q s m ) . For brevity, we say that q is good for ααα, βββ if q is coprime to the least common denominator of ααα ∪ βββ . Definition . Suppose that ααα, βββ are defined over Q and q is good for ααα, βββ . For t ∈ F × q , define(3.1.15) H q ( ααα, βββ | t ) = ( − r + s − q q − X m =0 q − s (0)+ s ( m ) g ( pppm, − qqqm ) ω ( ǫM − t ) m . Again, the hypergeometric sum H q ( ααα, βββ | t ) is independent of the choice of characters ω and Θ .The independence on ω is just as with the previous definition, and in this case the independencefrom Θ comes from the fact that every root of unity has its conjugate, and so again any additionalfactors from changing additive characters cancel out. The apparently conflicting notation is justifiedby the following result, showing that Definition 3.1.14 is more general. Proposition 3.1.16 (Beukers–Cohen–Mellit [BCM15, Theorem 1.3]) . Suppose that ααα, βββ are definedover Q and that (3.1.7) holds. Then Definitions and agree. A hybrid sum.
We will need a slightly more general hypothesis than allowed in the previoussection. We do not pursue the most general case as it is rather combinatorially involved, poses someissues of algebraicity, and anyway is not needed here. Instead, we isolate a natural case, where theindices are not defined over Q but neither does (3.1.7) hold, that is sufficient for our purposes. Definition . We say that q is splittable for ααα, βββ if there exist partitions(3.2.2) ααα = ααα ⊔ ααα ′ and βββ = βββ ⊔ βββ ′ where ααα , βββ are defined over Q and q × α ′ i , q × β ′ j ∈ Z for all α ′ i ∈ ααα ′ and all β ′ j ∈ βββ ′ . Example . If (3.1.7) holds, then q is splittable for ααα, βββ taking ααα = ααα ′ and βββ = βββ ′ and ααα = βββ = ∅ . Likewise, if ααα, βββ is defined over Q , then q is splittable for ααα, βββ for all q . Example . A splittable case that arises for us (up to a Galois action) in Proposition 3.5.1below is as follows. Let ααα = { , , } and βββ = { , , } . We cannot use Definition 3.1.14 since ( x − e π √− / )( x − e πi/ )( x − e π √− / ) Q [ x ] . When q ≡ , we may use Definition3.1.6; otherwise we may not. However, when q ≡ is odd, then q is splittable for ααα, βββ : wemay take ααα = ∅ , ααα ′ = ααα and βββ = βββ , βββ ′ = ∅ .It is now a bit notationally painful but otherwise straightforward to generalize the definition forsplittable q , providing a uniform description in all cases we consider. Suppose that q is splittablefor ααα, βββ . Let ααα be the union of all submultisets of ααα that are defined over Q ; then ααα is definedover Q . Repeat this for βββ . Let p , . . . , p r , q , . . . , q s be such that Q α j ∈ ααα ( x − e π √− α j ) Q β j ∈ βββ ( x − e π √− β j ) = Q rj =1 ( x p j − Q sj =1 ( x q j − . s before, let D ( x ) := gcd( Q rj =1 x p j − , Q sj =1 x q j − and M := Q rj =1 p p j j Q sj =1 q q j j and let s ( m ) be the multiplicity of the root e π √− m/q × in D ( x ) . Finally, let δ := deg D ( x ) . Weagain abbreviate(3.2.5) g ( pppm, − qqqm ) := r Y i =1 g ( p i m ) s Y i =1 g ( − q i m ) for m ∈ Z and(3.2.6) G ( m + ααα ′ q × , − m − βββ ′ q × ) := Y α ′ i ∈ ααα ′ g ( m + α ′ i q × ) g ( α i q × ) Y β ′ i ∈ βββ ′ g ( − m − β ′ i q × ) g ( − β i q × ) . Definition . Suppose that q is good and splittable for ααα, βββ . For t ∈ F × q , with the notation abovewe define the finite field hypergeometric sum H q ( ααα, βββ | t ) := ( − r + s − q q − X m =0 q − s (0)+ s ( m ) G ( m + ααα ′ q × , − m − βββ ′ q × ) g ( pppm, − qqqm ) ω (( − d + δ M t ) m . The following proposition then shows that our definition encompasses the previous ones.
Proposition 3.2.8.
Suppose that q is good and splittable for ααα, βββ . Then the following statementshold. (a) The hypergeometric sum H q ( ααα, βββ | t ) in Definition is independent of the choice of char-acters ω and Θ . (b) If α i q × , β i q × ∈ Z for all i = 1 , . . . , d , then Definitions and agree. (c) If ααα, βββ are defined over Q , then Definitions and agree.Proof. Part (c) follows directly from ααα = ααα and βββ = βββ (and ααα ′ = βββ ′ = ∅ ), so the definitions in factcoincide. Part (a) follows directly from the independence from Θ and ω of each part of the hybridsum.Part (b) follows by the same argument (due to Beukers–Cohen–Mellit) as in Proposition 3.1.16;for completeness, we give a proof in Lemma B.1.1 in the appendix. (cid:3) Suppose that q is good and splittable for ααα, βββ and let t ∈ F × q . Then by construction H q ( ααα, βββ | t ) ∈ Q ( ζ q × , ζ p ) . Since gcd( p, q × ) = 1 , we have Gal( Q ( ζ q × , ζ p ) | Q ) ≃ Gal( Q ( ζ q × ) | Q ) × Gal( Q ( ζ p ) | Q ) . We now descend the hypergeometric sum to its field of definition, in two steps.
Lemma 3.2.9.
We have H q ( ααα, βββ | t ) ∈ Q ( ζ q × ) .Proof. The action of
Gal( Q ( ζ p ) | Q ) ≃ ( Z /p Z ) × by ζ p ζ kp changes only the additive character Θ .By Proposition 3.2.8, the sum is independent of this choice, so it descends by Galois theory. (cid:3) The group
Gal( Q ( ζ q × ) | Q ) ≃ ( Z /q × Z ) × by σ k ( ζ q × ) = ζ ( q × ) k for k ∈ ( Z /q × Z ) × acts on the finitefield hypergeometric sums as follows. Lemma 3.2.10.
The following statements hold. (a)
Let k ∈ Z be coprime to q × . Then σ k ( H q ( ααα, βββ | t )) = H q ( kααα, kβββ | t ) . (b) We have H q ( ααα, βββ | t ) ∈ K ααα,βββ . (c) We have H q ( pααα, pβββ | t ) = H q ( ααα, βββ | t p ) . roof. To prove (a), note σ k ( ω ( x )) = ω k ( x ) since ω takes values in µ q × ; therefore σ k ( g ( m )) = g ( km ) ,and we have both σ k ( g ( pppm, − qqqm )) = g ( pppkm, − qqqkm ) and σ k ( G ( m + ααα ′ q × , − m − βββ ′ q × )) = G ( km + kααα ′ q × , − km − kβββ ′ q × )) . We have s ( km ) = s ( m ) since D ( x ) ∈ Q [ x ] . Moreover, kααα = ααα and the same with βββ , so the values p i , q i remain the same when computed for kααα, kβββ . Now plug these into the definition of H q ( ααα, βββ | t ) and just reindex the sum by km ← m to obtain the result.Part (b) follows from part (a): the field of definition K ααα,βββ is precisely the fixed field under thesubgroup of k ∈ ( Z /q × Z ) × such that kααα, kβββ are equivalent to ααα, βββ as multisets in Q / Z .Finally, part (c). Starting with the left-hand side, we reindex m ← pm then substitute usingLemma 3.1.3(d)’s implication that g ( pm ) = g ( m ) to get G ( pm + pααα ′ q × , − pm − pβββ ′ q × ) = G ( m + ααα ′ q × , − m − βββ ′ q × ) and g ( ppp ( pm ) , qqq ( pm )) = g ( pppm, qqqm ) ,noting that the quantities ppp and qqq do not change, as pααα = ααα and pβββ = βββ modulo Z (as they aredefined over Q ). Noting that (( − d + δ M ) p = ( − d + δ M ∈ F p ⊆ F q , we then obtain the result. (cid:3) Before concluding this primer on finite field hypergeometric functions, we combine the Gauss sumidentities and our hybrid definition to expand one essential example; this gives a flavor of what isto come. First, we prove a new identity.
Lemma 3.2.11.
We have the following identity of Gauss sums: g ( q × ) g ( q × ) g ( q × ) = g ( q × ) = ( − q × / qg ( q × ) . Proof.
Since q is odd, we use the Hasse–Davenport product relation (Lemma 3.1.3(c)) for N = 2 and m = q × , q × , q × , solving for g ( q × ) , g ( q × ) , g ( q × ) , respectively to find: g ( q × ) = g ( q × ) g ( q × ) g ( q × ) ω (2) − q × / g ( q × ) = g ( q × ) g ( q × ) g ( q × ) ω (2) − q × / g ( q × ) = g ( q × ) g ( q × ) g ( q × ) ω (2) − q × / . Multiply all of these together, divide by g ( q × ) , and cancel to obtain:(3.2.12) g ( q × ) g ( q × ) g ( q × ) g ( q × ) = g ( q × ) ω (2) − q × = ( − q × / q applying Lemma 3.1.3(b) in the last step. (cid:3) Next, we consider our example.
Example . Going back to Example 3.2.4, in the case where q ≡ and q odd, we have ααα = { , , } and βββ = { , , } . Then ααα = ∅ and βββ = βββ . Thus Q α j ∈ ααα ( x − e π √− α j ) Q β j ∈ βββ ( x − e π √− β j ) = 1( x − x + 1) = ( x − x − x − . hus D ( x ) = x − and M = 4 ; and s ( m ) = 1 if m = 0 , q × and s ( m ) = 0 otherwise. ThereforeDefinition 3.2.7 and simplification using Lemma 3.1.3(a)–(b) gives H q ( , , ; 0 , , | t )= − − q q − X m =0 q s ( m ) − g ( m + q × ) g ( m + q × ) g ( m + q × ) g ( q × ) g ( q × ) g ( q × ) g (2 m ) g ( − m ) g ( − m ) ω ( − t ) m . When m = 0 , the summand is just ( − − / (1 − q ) = 1 /q × . When m = q × , applying Lemma3.2.11 we obtain − g ( q × ) g ( q × ) g ( q × ) q × g ( q × ) g ( q × ) g ( q × ) g ( q × ) ω ( − t ) q × / = − qq × g ( q × ) g ( q × ) g ( q × ) ω ( t ) q × / . Therefore(3.2.14) H q ( , , ; 0 , , | t )= 1 q × − qq × g ( q × ) g ( q × ) g ( q × ) ω ( t ) q × / + 1 qq × q − X m =1 m = q × / g ( m + q × ) g ( m + q × ) g ( m + q × ) g ( q × ) g ( q × ) g ( q × ) g (2 m ) g ( − m ) g ( − m ) ω ( − t ) m . Counting points.
Following work of Delsarte [Del51] and Furtado Gomida [FG51], Koblitz[Kob83] gave a formula for the number of points on monomial deformations of diagonal hypersurfaces(going back to Weil [Wei49]). In this subsection, we outline their approach for creating closedformulas that compute the number of points for hypersurfaces in projective space in terms of Gausssums.Let X ⊆ P n be the projective hypersurface over F q defined by the vanishing of the nonzeropolynomial r X i =1 a i x ν i · · · x ν in n ∈ F q [ x , . . . , x n ] so that a i ∈ F q and ν ij ∈ Z ≥ for i = 1 , . . . , r and j = 0 , . . . , n . Suppose that q × := q − does notdivide any of the ν ij . Let U be the intersection of X with the torus G n +1 m / G m ⊆ P n , so that thepoints of U are the points of X with all nonzero coordinates.Let S be the set of s = ( s , . . . , s r ) ∈ ( Z /q × Z ) r such that the following condition holds:(3.3.1) P ri =1 s i ≡ mod q × ) and P ri =1 ν ij s i ≡ mod q × ) for all j = 1 , . . . , n. Let µ × q be the group of q × -th roots of unity. Any element s = ( s , . . . , s r ) ∈ S corresponds to amultiplicative character χ s : µ ( q × ) r → C × χ s ( x , . . . , x r ) = ω (Π ri =1 x s i i ) . Given s ∈ S , we define(3.3.2) c s := ( q × ) n − r +1 q r Y i =1 g ( s i ) if s = 0 and c := ( q × ) n − r +1 ( q × ) r − − ( − r − q . ith this notation, we have the following result of Koblitz, rewritten in terms of Gauss sums sothat we can apply it in our context. Theorem 3.3.3 (Koblitz) . We have (3.3.4) U ( F q ) = X s ∈ S ω ( a ) − s c s . where ω ( a ) − s := ω ( a − s · · · a − s r r ) .Proof. We unpack and repack a bit of notation. Koblitz [Kob83, Theorem 1] proves that U ( F q ) = X s ω ( a ) − s c ′ s where the sum is over all characters of µ ( q × ) r / ∆ where ∆ is the diagonal—this set is in naturalbijection with the set S —and where for s = 0 c ′ s = − q ( q × ) n − r +1 J ( s , . . . , s r ) where J ( s , . . . , s r ) is the Jacobi sum and where c ′ = c as in (3.3.2). It only remains to show that c ′ s = c s for s = 0 . If s i = 0 for all i , then [Kob83, (2.5)] J ( s , . . . , s r ) = g ( s ) · · · g ( s r ) g ( s + . . . + s r ) = − g ( s ) · · · g ( s r ) so c ′ s = c s by definition. If r > and s i = 0 for some i , then [Kob83, below (2.5)] J ( s , . . . , s r ) = − J ( s , . . . , s i − , s i +1 , . . . , s r ) , so iterating and using Lemma 3.1.3(a), (cid:3) (3.3.5) J ( s , . . . , s r ) = − Y i =1 ,...,rs i =0 ( − Y i =1 ,...,rs i =0 g ( s i ) = − Y i =1 ,...,rs i =0 g (0) Y i =1 ,...,rs i =0 g ( s i ) = − r Y i =1 g ( s i ) . In the remaining sections, we apply the preceding formulas to each of our five pencils.3.4.
The Dwork pencil F . In this subsection, we will give a closed formula in terms of finite fieldhypergeometric sums for the number of points in a given member of the Dwork family. Throughoutthis section, we suppose that q is odd. Proposition 3.4.1.
For ψ ∈ F × q , the following statements hold. (a) If q ≡ , then X F ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . (b) If q ≡ , then X F ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | ψ − ) + 3 qH q ( , ; 0 , | ψ − )+ 12( − ( q − / qH q ( ; 0 | ψ − ) . Proposition 3.4.1 has several equivalent formulations and has seen many proofs: see section 1.5 inthe introduction for further references. We present another proof for completeness and to illustratethe method we will apply to all five families in this well-studied case. emark . Quite beautifully, the point counts in Proposition 3.4.1 in terms of finite field hy-pergeometric sums match (up to twisting factors) the indices with multiplicity in the Picard–Fuchsequations computed in Proposition 2.5.2. Although we are not able to use this matching directly,it guides the decomposition of the sums by means of lemmas that can be proven in a technical butdirect manner.We prove Proposition 3.4.1 in four steps:1. We compute the relevant characters and cluster them.2. We use Theorem 3.3.3 to count points where no coordinate is zero and rewrite the sums intohypergeometric functions.3. We count points where at least one coordinate is zero.4. We combine steps 2 and 3 to finally prove Proposition 3.4.1.The calculations are somewhat involved, but we know how to cluster and the answer up to thescaling factors in front: indeed, the parameters of the finite field hypergeometric sums are given bythe calculation of the Picard–Fuchs equations for the Dwork pencil given by Proposition 2.5.2.
Step 1: Computing and clustering the characters.
In order to use Theorem 3.3.3 we must computethe subset S ⊂ ( Z /q × Z ) r given by the constraints in (3.3.1). This is equivalent to solving the systemof congruences:(3.4.3) s s s s s ≡ q × ) . If q ≡ , then by linear algebra over Z we obtain S = n (1 , , , , − k + q × (0 , , , − , k + q × (0 , , , − , k : k i ∈ Z /q × Z o . These solutions can be clustered in an analogous way as done in section 2.5:(i) S := { k (1 , , , , −
4) : k ∈ Z /q × Z } ,(ii) S := { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } ,(iii) S := { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } , and(iv) S := { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } .The last three (ii)–(iv) all behave in the same way, due to the evident symmetry.If instead q ≡ , then S = n (1 , , , , − k + q × (0 , , , − , k + q × (0 , , , − , k : k i ∈ Z /q × Z o ; we cluster again, getting the four clusters above but now together with twelve new clusters:(v) three sets of the form S := { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } ,(vi) three sets of the form S := { k (1 , , , , − − q × (0 , , , ,
0) : k ∈ Z /q × Z } , and(vii) six sets of the form S := { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } ,where the number of sets is given by the number of distinct permutations of the middle threecoordinates. tep 2: Counting points on the open subset with nonzero coordinates. We now give a formula for U F ,ψ ( F q ) for the number of points, applying Theorem 3.3.3.We go through each cluster S i , linking each to a hypergeometric function. Lemma 3.4.4.
For all odd q , (3.4.5) X s ∈ S ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) . Proof.
By Definition 3.1.14, H q ( , , ; 0 , , | ψ − ) = 1 q × q − X m =0 q − s (0)+ s ( m ) g (4 m ) g ( − m ) ω (4 ψ ) − m = − q × + − q − g ( q × ) q × + 1 q × q − X m =1 m = q × / q − g (4 m ) g ( − m ) ω (4 ψ ) − m = − q × − g ( q × ) qq × + 1 qq × q − X k =1 k = q × / g ( − k ) g ( k ) ω (4 ψ ) k the latter by substituting k = − m . Now we expand to match terms: X s ∈ S ω ( a ) − s c s = c (0 , , , , + c ( q × / , , , , + q − X k =1 k = q × / ω ( − ψ ) k c ( k,k,k,k, − k ) = ( q × ) − ( − qq × − qq × g ( q × ) + 1 qq × q − X k =1 k = q × / ω (4 ψ ) k g ( k ) g ( − k ) (3.4.6) = q − q + 6 q − q × − g ( q × ) qq × + 1 qq × q − X k =1 ,k = q × ω (4 ψ ) k g ( k ) g ( − k )= q − q + 3 + H q ( , , ; 0 , , | ψ − ) . (cid:3) Lemma 3.4.7.
For i = 2 , , , (3.4.8) X s ∈ S i ω ( a ) − s c s = ( − q × / (cid:0) qH q ( , ; 0 , | ψ − ) (cid:1) . Proof.
By definition,(3.4.9) H q ( , ; 0 , | ψ − ) = 1 q × q − X m =0 q − s (0)+ s ( m ) g (4 m ) g ( − m ) ω (2 ψ ) − m = − q × + 1 q × q − X m =1 m = q × / q − g (4 m ) g ( − m ) ω (2 ψ ) − m using s ( m ) = 1 if m = 0 , q × and s ( m ) = 0 otherwise.By symmetry, X s ∈ S ω ( a ) − s c s = X s ∈ S ω ( a ) − s c s = X s ∈ S ω ( a ) − s c s . o we only need to consider i = 2 . Then:(3.4.10) X s ∈ S ω ( a ) − s c s = c ( q × / , , , , + c ( q × / , , , , + q − X k =1 k = q × / ω ( − ψ ) k c ( k (1 , , , , − q × / , , , , = − g ( q × ) qq × + 1 qq × q − X k =1 k = q × / ω ( − ψ ) k g ( k ) g ( k + q × ) g ( − k ) . Next, we use the Hasse–Davenport product relation (Lemma 3.1.3(c)) with N = 2 | q × to get g (2 k ) = − ω (2) k g ( k ) g (0) g ( k + q × ) g ( q × ) which rearranges using g (0) = − to(3.4.11) g ( k ) g ( k + q × ) = ω (2) − k g (2 k ) g ( q × ) . Using Lemma 3.1.3(b) gives g ( q × ) = ( − q × / q ; substituting this and (3.4.11) into (3.4.10) simpli-fies to X s ∈ S ω ( a ) − s c s = − − q × / q × + 1 qq × q − X k =1 k = q × / ω ( − ψ ) k ( ω (2) − k g (2 k ) g ( q × )) g ( − k )= − − q × / q × + 1 q × q − X k =1 k = q × / ( − q × / ω ( − ψ ) k g (2 k ) g ( − k )= ( − q × / − q × + 1 q × q − X k =1 k = q × / ω ( − ψ ) k g (2 k ) g ( − k ) . Looking back at (3.4.9), we rearrange and insert a factor q to find the hypergeometric sum: ( − q × / X s ∈ S ω ( a ) − s c s = 2 q − q × − qq × + qq × q − X m =1 m = q × / q − ω (2 ψ ) − m g ( − m ) g (4 m )= 2 + qH q ( , ; 0 , | ψ − ) as claimed. (cid:3) Lemma 3.4.12.
Suppose q ≡ . Then X s ∈ S ω ( a ) − s c s = ( − q × / qH q ( ; 0 | ψ − ) + ( − q × / − g ( q × ) + g ( q × ) g ( q × ) . roof. Plugging into the definition of the finite field hypergeometric sum and then pulling out terms m = jq × / with j = 0 , , , , we get(3.4.13) H q ( ; 0 | ψ − ) = 1 q × q − X m =0 ω ( − ψ − ) m g ( m + q × ) g ( − m ) g ( q × )= − q × + ( − q × / q × g ( q × ) (cid:0) g ( q × ) + g ( q × ) (cid:1) + 1 q × q − X m =0 q × ∤ m ω ( − ψ − ) m g ( m + q × ) g ( − m ) g ( q × ) . Hasse–Davenport (Lemma 3.1.3(c)) implies(3.4.14) g (4 m ) = − ω (4) m g ( m ) g ( m + q × ) g ( m + q × ) g ( m + q × ) g (0) g ( q × ) g ( q × ) g ( q × ) For m = j q × , multiplying (3.4.14) by g ( − m − q × ) g ( − m ) and simplifying, we get:(3.4.15) ( − m qg ( − m − q × ) = ω (4) m ( − m g ( m ) g ( m + q × ) g ( m + q × ) g ( − m ) g ( q × ) g ( m ) g ( m + q × ) g ( m + q × ) g ( − m ) = ( − − m ω (4) − m qg ( q × ) g ( − m − q × ) . Now we look at the point count. First, we take the definition:(3.4.16) X s ∈ S ω ( a ) − s c s = 1 qq × q − X k =0 ω ( − ψ ) k g ( k ) g ( k + q × ) g ( k + q × ) g ( − k ) . We then tease out the four terms with k = jq × / . The cases k = 0 , q × give(3.4.17) qq × g ( q × ) g ( q × ) + 1 qq × g ( q × ) g ( q × ) = g ( q × ) + g ( q × ) q × g ( q × ) because g ( q × ) = q as q ≡ . The terms with k = q × , q × are(3.4.18) − qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) = − ( − q × / q + 1 q × = ( − q × / (cid:18) − qq × (cid:19) . using Lemma 3.1.3(b) with m = q × to get g ( q × ) g ( q × ) = ( − q × / q .For the remaining terms in the sum, we plug in (3.4.15) to get(3.4.19) qq × q − X k =0 q × ∤ k ω ( − ψ ) k g ( k ) g ( k + q × ) g ( k + q × ) g ( − k )= 1 qq × q − X k =0 q × ∤ k ω ( − ψ ) k g ( k + q × )( − − k ω (4) − k qg ( q × ) g ( − k − q × )= qq × q − X k =0 q × ∤ k ω ( − ψ ) k g ( k + q × ) g ( − k − q × ) g ( q × ) . ext, we reindex this summation with the substitution m = − k − q × to obtain(3.4.20) qq × q − X m =0 q × ∤ m ω ( − ψ ) − m − q × / g ( − m ) g ( m + q × ) g ( q × ) = ( − q × / qq × q − X m =0 q × ∤ m ω ( − ψ − ) m g ( m + q × ) g ( − m ) g ( q × ) . Taking (3.4.16), expanding and substituting (3.4.17), (3.4.18), and (3.4.20) then gives ( − q × / X s ∈ S ω ( a ) − s c s = ( − q × / g ( q × ) + g ( q × ) q × g ( q × ) + 1 − qq × + qq × q − X m =0 q × ∤ m ω ( − ψ − ) m g ( m + q × ) g ( − m ) g ( q × ) . We are quite close to (3.4.13), but the first term is off by a factor q . Adding and subtracting gives ( − q × / X s ∈ S ω ( a ) − s c s = 1 + qH q ( ; 0 | ψ − ) − ( − q × / g ( q × ) + g ( q × ) g ( q × ) as claimed. (cid:3) Lemma 3.4.21. If q ≡ , then X s ∈ S ω ( a ) − s c s = X s ∈ S ω ( a ) − s c s = X s ∈ S ω ( a ) − s c s . Proof.
We start with (3.4.16) and reindex with m = k + q × : X s ∈ S ω ( a ) − s c s = 1 qq × q − X k =0 ω ( − ψ ) k g ( k ) g ( k + q × ) g ( k + q × ) g ( − k )= 1 qq × q − X m =0 ω ( − ψ ) m g ( m + q × ) g ( m + q × ) g ( m ) g ( − m )= X s ∈ S ω ( a ) − s c s . The equality for S holds reindexing with m = k + q × . (cid:3) We now put these pieces together to give the point count for the toric hypersurface.
Proposition 3.4.22.
Let ψ ∈ F × q . (a) If q ≡ , then (3.4.23) U F ,ψ ( F q ) = q − q − H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − )) . (b) If q ≡ , then (3.4.24) U F ,ψ ( F q ) = q − q + 9 + H q ( , , ; 0 , , | ψ − ) + 3 qH q ( , ; 0 , | ψ − ))+ 12 ( − q × / qH q ( ; 0 | ψ − ) + ( − q × / − g ( q × ) + g ( q × ) g ( q × ) ! . roof. For q ≡ , we have from Lemmas 3.4.4 and 3.4.7:(3.4.25) U F ,ψ ( F q ) = X i =1 X s ∈ S i ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + 3( − − qH q ( , ; 0 , | ψ − ))= q − q − H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − )) For q ≡ , we have from Lemmas 3.4.4, 3.4.7, 3.4.12, and 3.4.21, we have that:(3.4.26) U F ,ψ ( F q ) = X i =1 X s ∈ S i ω ( a ) − s c s + 12 X s ∈ S ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + 3(2 + qH q ( , ; 0 , | ψ − ))+ 12 ( − q × / qH q ( ; 0 | ψ − ) + ( − q × / − g ( q × ) + g ( q × ) g ( q × ) ! which simplifies to the result. (cid:3) Step 3: Count points when at least one coordinate is zero.
Lemma 3.4.27. If q ≡ , then X F ,ψ ( F q ) − U F ,ψ ( F q ) = 4 q + 4 . Proof.
First we compute the number of points when x = 0 and x x x = 0 , i.e., count points onthe Fermat quartic curve V : x + x + x = 0 with coordinates in the torus. All points in V ( F q ) lieon the torus: if e.g. x = 0 and x = 0 , then − x /x ) , but − F × q since q ≡ .We claim that V ( F q ) = q + 1 ; this can be proven in many ways. First, we sketch an elementaryargument, working affinely on x + x = − . The map ( x , x ) ( x , x ) gives a map to theaffine curve defined by C : w + w = − . The number of points on this curve over F q is q + 1 (the projective closure is a smooth conic with no points at infinity), and again all such solutionshave w , w ∈ F × q . Since q ≡ , the squaring map F × q → F × q is bijective. Therefore,for the four points ( ± w , ± w ) with w + w = − , there are exactly four points ( ± x , ± x ) with x i = w i for i = 0 , . Thus V ( F q ) = C ( F q ) = q + 1 . (Alternatively, the map ( x , x ) ( x , x ) is bijective, with image a supersingular genus curve over F q .)Second, and for consistency, we again apply the formula of Koblitz! For the characters, we solve(3.4.28) s s s ≡ q × ) . There are exactly four solutions when q ≡ : S = { (0 , , , q × (1 , , , q × (1 , , , q × (0 , , } . Then by Theorem 3.3.3,(3.4.29) V ( F q ) = c (0 , , + c ( q × / , , + c ( q × / , , + c ( q × / , , = ( q − − ( − q + 3( − q g ( q × ) g ( q × ) g (0)= q − qq + 3 = q + 1 . y symmetry, repeating in each of the four coordinate hyperplanes, we obtain X F ,ψ ( F q ) − U F ,ψ ( F q ) = 4( q + 1) = 4 q + 4 . (cid:3) Lemma 3.4.30. If q ≡ , then X F ,ψ ( F q ) − U F ,ψ ( F q ) = 4 q − − − q × / + 12 g ( q × ) + g ( q × ) g ( q × ) . Proof.
We repeat the argument in the preceding lemma. We cluster solutions to (3.4.28) and countthe number of solutions in the following way:(3.4.31) V ( F q ) = c (0 , , + 6 c ( q × / , , + 3 c ( q × / , , + 3 c ( q × / , , + 3 c ( q × / , , = q − − − q × / − q g ( q × ) g ( q × ) + 3 q g ( q × ) g ( q × )= q − − − q × / + 3 g ( q × ) + g ( q × ) g ( q × ) . There are solutions to x + x = 0 with x x = 0 if q ≡ and zero otherwise, so − q × / solutions in either case. Adding up, we get X F ,ψ ( F q ) − U F ,ψ ( F q ) = 4 V ( F q ) + 6(2 + 2( − q × / )= 4 q − − − q × / + 12 1 q g ( q × ) g ( q × ) + 12 g ( q × ) + g ( q × ) g ( q × ) . (cid:3) Step 4: Conclude.
We now conclude the proof.
Proof of Proposition . We combine Proposition 3.4.22 with Lemmas 3.4.27 and 3.4.30. If q ≡ , then X F ,ψ ( F q ) = U F ,ψ ( F q ) + ( X F ,ψ ( F q ) − U F ,ψ ( F q ))= ( q − q − H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − )) + (4 q + 4)= q + q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . If q ≡ , then the ugly terms cancel, and we have simply X F ,ψ ( F q ) = U F ,ψ ( F q ) + ( X F ,ψ ( F q ) − U F ,ψ ( F q ))= ( q − q + 9 + H q ( , , ; 0 , , | ψ − ) + 3 qH q ( , ; 0 , | ψ − )+ 12( − q × / qH q ( ; 0 | ψ − )) + (4 q − q + q + 1 + H q ( , , ; 0 , , | ψ − ) + 3 qH q ( , ; 0 , | ψ − )+ 12( − q × / qH q ( ; 0 | ψ − ) . (cid:3) The Klein–Mukai pencil F L . In this section, we repeat the steps of the previous sectionbut for the Klein–Mukai pencil F L . We suppose throughout this section that q is coprime to .Our main result is as follows. Proposition 3.5.1.
For q coprime to and ψ ∈ F × q , the following statements hold. (a) If q , then X F L ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | ψ − ) . b) If q ≡ , then X F L ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | ψ − )+ 3 qH q ( , , ; 0 , , | ψ ) + 3 qH q ( , , ; 0 , , | ψ ) . Remark . The new parameters , , ; 0 , , and , , ; 0 , , match the Picard–Fuchsequations in Proposition 2.6.1 as elements of Q / Z , with the same multiplicity. Step 1: Computing and clustering the characters.
As before, we first have to compute the solutionsto the system of congruences: w w w w w ≡ (mod q × ) . By linear algebra over Z , we compute that if q , then the set of solutions is S = { (1 , , , , − w : w ∈ Z /q × Z } . On the other hand if q ≡ , then the set splits into three classes:(i) the set S = { k (1 , , , , −
4) : k ∈ Z /q × Z } ,(ii) three sets of the form S = n k (1 , , , , −
4) + q × (1 , , , ,
0) : k ∈ Z /q × Z o , and(iii) three sets of the form S = n k (1 , , , , −
4) + q × (3 , , , ,
0) : k ∈ Z /q × Z o .The multiplicity of the latter two sets corresponds to cyclic permutations yielding the same productof Gauss sums. Step 2: Counting points on the open subset with nonzero coordinates.
As in the previous section,the hard work is in counting points in the toric hypersurface. We now proceed with each cluster.
Lemma 3.5.3. If q ≡ , then (3.5.4) X s ∈ S ω ( a ) − s c s = qH q ( , , ; 0 , , | ψ ) − q g ( q × ) g ( q × ) g ( q × ) . Proof.
Recall our hybrid hypergeometric sum (3.2.14) from Example 3.2.13, plugging in t = ψ :(3.5.5) H q ( , , ; 0 , , | ψ )= 1 q × − qq × g ( q × ) g ( q × ) g ( q × )+ 1 qq × q − X m =1 m = q × / g ( m + q × ) g ( m + q × ) g ( m + q × ) g ( q × ) g ( q × ) g ( q × ) g (2 m ) g ( − m ) g ( − m ) ω ( − ψ ) m . Our point count formula expands to X s ∈ S ω ( a ) − s c s = 1 qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × )+ 1 qq × q − X k =0 q × ∤ k ω ( − ψ ) k g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( k ) g ( − k ) . e work on the sum. Changing indices to m = k + q × , using the identity g ( m + q × ) = ω (4) − m ( − m q − g ( q × ) g ( − m ) g (2 m ) found by using Hasse–Davenport for N = 2 , and applying Lemma 3.2.11, gives us the summand ω ( − ψ ) m g ( m + q × ) g ( m + q × ) g ( m + q × ) g ( m + q × ) g ( − m )= ω ( − ψ ) m g ( m + q × ) g ( m + q × ) g ( m + q × ) ω (4) − m ( − m q − g ( q × ) g ( − m ) g (2 m ) g ( − m )= ω ( − ψ ) m g ( m + q × ) g ( m + q × ) g ( m + q × ) g ( q × ) g ( q × ) g ( q × ) q − g ( q × ) g (2 m ) g ( − m ) g ( − m )= qω ( − ψ ) m g ( m + q × ) g ( m + q × ) g ( m + q × ) qg ( q × ) g ( q × ) g ( q × ) g (2 m ) g ( − m ) g ( − m ) . Plugging back in, we can relate this to the hypergeometric function (3.5.5): X s ∈ S ω ( a ) − s c s = 1 qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × )+ 1 qq × q − X m =0 q × ∤ m qω ( − ψ ) m g ( m + q × ) g ( m + q × ) g ( m + q × ) g ( q × ) g ( q × ) g ( q × ) g (2 m ) g ( − m ) g ( − m )= 1 qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × )+ qH q ( , , ; 0 , , | ψ ) + qq × − q × g ( q × ) g ( q × ) g ( q × )= qH q ( , , ; 0 , , | ψ ) − q g ( q × ) g ( q × ) g ( q × ) . (cid:3) Lemma 3.5.6. If q ≡ then (3.5.7) X s ∈ S ω ( a ) − s c s = qH q ( , , ; 0 , , | ψ ) − q g ( q × ) g ( q × ) g ( q × ) . Proof.
Apply complex conjugation to Lemma 3.5.3; the effect is to negate indices, as in Proposition3.2.8. (cid:3)
We now put the pieces together to prove the main result in this step.
Proposition 3.5.8.
Suppose ψ ∈ F × q . (a) If q then (3.5.9) U F L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − ) . (b) If q ≡ then (3.5.10) U F L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − )+ 3 qH q ( , , ; 0 , , | ψ ) + 3 qH q ( , , ; 0 , , | ψ ) − q ( g ( q × ) g ( q × ) g ( q × ) + g ( q × ) g ( q × ) g ( q × )) . roof. When q , there is only one cluster of characters, S . By Lemma 3.4.4, we knowthat(3.5.11) U F L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) . When q ≡ we have three clusters of characters, the latter two ( S and S ) with multi-plicity . By Lemmas 3.4.4, 3.5.3, and 3.5.6, these sum to the result. (cid:3) Step 3: Count points when at least one coordinate is zero.
Recall that q is coprime to . Lemma 3.5.12.
Let ψ ∈ F × q . (a) If q , then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 4 q − . (b) If q ≡ , then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 4 q − q ( g ( q × ) g ( q × ) g ( q × ) + g ( q × g ( q × ) g ( q × )) . Proof.
We count solutions with at least one coordinate zero. If x = 0 but x x x = 0 , we countpoints on x + x x = 0 : solving for x , we see there are q − solutions; repeating this for the cases x = 0 or x = 0 , we get q − points.Now suppose x = 0 but x x x = 0 , we look at the equation x x + x x + x x = 0 definingthe Klein quartic. Applying Theorem 3.3.3 again, we find that(3.5.13) s s s ≡ q × ) . If q , then only (0 , , is a solution and c (0 , , = q − . If q ≡ then thesolutions are n k ( q × , q × , q × ) : k ∈ Z / Z o which gives the point count q − q g ( q × ) g ( q × ) g ( q × ) + 3 q g ( q × ) g ( q × ) g ( q × ) . If now at least two of the variables among { x , x , x } are zero, then the equation is just x = 0 hence the last one is also zero and there is only one such point. If x = x = 0 , then the equationis x x = 0 hence another of the first three variables is zero. Consequently there are exactly 3 suchpoints. Totalling up gives the result. (cid:3) Step 4: Conclude.
We now prove Proposition 3.5.1.
Proof of Proposition . By Proposition 3.5.8 and Lemma 3.5.12, if q , then(3.5.14) X F L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + (4 q − q + q + 1 + H q ( , , ; 0 , , | ψ − ) . f q ≡ , then the ugly terms cancel and we get(3.5.15) X F L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − )++ 3 qH q ( , , ; 0 , , | ψ ) + 3 qH q ( , , ; 0 , , | ψ ) − q g ( q × ) g ( q × ) g ( q × )+ (4 q − q + q + 1 + H q ( , , ; 0 , , | ψ − )+ 3 qH q ( , , ; 0 , , | ψ ) + 3 qH q ( , , ; 0 , , | ψ ) as desired. (cid:3) Remaining pencils.
For the remaining three pencils F L , L L , and L , the formula for thepoint counts can be derived in a similar manner. The details can be found in Appendix B; we statehere only the results. Proposition 3.6.1.
For q odd and ψ ∈ F × q , the following statements hold. (a) If q ≡ , then X F L ,ψ ( F q ) = q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . (b) If q ≡ , then X F L ,ψ ( F q ) = q − q + 1 + H q ( , , ; 0 , , | ψ − )+ qH q ( , ; 0 , | ψ − ) − qH q ( ; 0 | ψ − ) . (c) If q ≡ , then X F L ,ψ ( F q ) = q + 7 q + 1 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2 qH q ( ; 0 | ψ − ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) . Proof.
See Proposition B.2.1. (cid:3)
Proposition 3.6.2.
For q odd and ψ ∈ F × q , the following statements hold. (a) If q ≡ , then X L L ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . (b) If q ≡ , then X L L ,ψ ( F q ) = q + 9 q + 1 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) . Proof.
See Proposition B.3.1. (cid:3)
Proposition 3.6.3.
For q coprime to and ψ ∈ F × q , the following statements hold. (a) If q , then X L ,ψ ( F q ) = q + 3 q + 1 + H q ( , , ; 0 , , | ψ − ) . (b) If q ≡ , then X L ,ψ ( F q ) = q + 3 q + 1 + H q ( , , ; 0 , , | ψ − ) + 4 qH q ( , , , ; 0 , , , | ψ ) . Proof.
See Proposition B.4.1. (cid:3) . Proof of the main theorem and applications
In this section, we prove Main Theorem 1.4.1 by converting the hypergeometric point countformulas in the previous section into a global L -series. We conclude with some discussion andapplications.4.1. From point counts to L -series. In this section, we define L -series of K3 surfaces and hy-pergeometric functions, setting up the notation we will use in the proof of our main theorem.We begin with L -series of K3 surfaces. Let ψ ∈ Q r { , } . Let ⋄ ∈ { F , F L , F L , L L , L } signify one of the five K3 families in (1.2.1). Let S = S ( ⋄ , ψ ) be the set of bad primes in (1.2.1)together with the primes dividing the numerator or denominator of either ψ or ψ − . Lemma 4.1.1.
For p S ( ⋄ , ψ ) , the surface X ⋄ ,ψ has good reduction at p .Proof. Straightforward calculation. (cid:3)
Let p S ( ⋄ , ψ ) . The zeta function of X ⋄ ,ψ over F p is of the form(4.1.2) Z p ( X ⋄ ,ψ , T ) = 1(1 − T )(1 − pT ) P ⋄ ,ψ,p ( T )(1 − p T ) where P ⋄ ,ψ,p ( T ) ∈ T Z [ T ] . The Hodge numbers of X ⋄ ,ψ imply that the polynomial P ⋄ ,ψ,p ( T ) hasdegree 21. Equivalently, we have that(4.1.3) P ⋄ ,ψ,p ( T ) = det(1 − Frob − p T | H ét , prim ( X ⋄ ,ψ , Q ℓ )) is the characteristic polynomial of the Frobenius automorphism acting on primitive second degreeétale cohomology for ℓ = p (and independent of ℓ ). We then define the (incomplete) L -series(4.1.4) L S ( X ⋄ ,ψ , s ) := Y p S P ⋄ ,ψ,p ( p − s ) − , convergent for s ∈ C in a right half-plane by elementary estimates.We now turn to hypergeometric L -series, recalling the definitions made in section 3.1–3.2. Let ααα, βββ be multisets of rational numbers that are disjoint modulo Z . Let t ∈ Q r { , } , and let S ( ααα, βββ, t ) be the set of primes dividing a denominator in ααα ∪ βββ together with the primes dividingthe numerator or denominator of either t or t − .Recall the Definition 3.2.7 of the finite field hypergeometric sums H q ( ααα ; βββ | t ) ∈ K ααα,βββ ⊆ C . For aprime power q such that ααα, βββ is splittable, we define the formal series(4.1.5) L q ( H ( ααα, βββ | t ) , T ) := exp (cid:18) − ∞ X r =1 H q r ( ααα ; βββ | t ) T r r (cid:19) ∈ T K ααα,βββ [[ T ]] using Lemma 3.2.10(b). (Note the negative sign; below, this normalization will yield polynomialsinstead of inverse polynomials.)For a number field M , a prime of M is a nonzero prime ideal of the ring of integers Z M of M .We call a prime p of M good (with respect to ααα, βββ, ψ ) if p lies above a prime p S ( ααα, βββ, ψ ) . Nowlet M be an abelian extension of Q containing the field of definition K := K ααα,βββ with the followingproperty:(4.1.6) for all good primes p of M , we have q = Nm( p ) splittable for ααα, βββ .For example, if m is the least common multiple of all denominators in ααα ∪ βββ , then we may take M = Q ( ζ m ) . We will soon see that we will need to take M to be nontrivial extensions of K inProposition 4.3.1 to deal with the splittable hypergeometric function given in Example 3.2.4. Let m be the conductor of M , i.e., the minimal positive integer such that M ⊆ Q ( ζ m ) . Under the canonical dentification ( Z /m Z ) × ∼ −→ Gal( Q ( ζ m ) | Q ) where k σ k and σ k ( ζ m ) = ζ km , let H M ≤ ( Z /m Z ) × besuch that Gal( M | Q ) ≃ ( Z /m Z ) × /H M .Now let p S ( ααα, βββ, ψ ) . Let p , . . . , p r be the primes above p in M , and let q = p f = Nm( p i ) forany i . Recall (by class field theory for Q ) that f is the order of p in ( Z /m Z ) × /H M , and rf = [ M : Q ] .Moreover, the set of primes { p i } i arise as p i = σ k i ( p ) where k i ∈ Z are representatives of thequotient ( Z /m Z ) × / h H M , p i of ( Z /m Z ) × by the subgroup generated by H M and p . We then define(4.1.7) L p ( H ( ααα, βββ | t ) , M, T ) := r Y i =1 L q ( H ( k i ααα, k i βββ | t ) , T f )= Y k i ∈ ( Z /m Z ) × / h H M ,p i L q ( H ( k i ααα, k i βββ | t ) , T f ) ∈ T K [[ T ]] . This product is well-defined up to choice of representatives k i of the cosets in ( Z /m Z ) × / h H M , p i .Indeed, by Lemma 3.2.10: part (c) gives(4.1.8) L q ( H ( pkααα, pkβββ | t ) , T f ) = L q ( H ( kααα, kβββ | t p ) , T f ) = L q ( H ( kααα, kβββ | t ) , T f ) for all k ∈ ( Z /m Z ) × and all good primes p , since t p = t ∈ F p ⊆ F q ; and similarly part (a) impliesit is well-defined for k ∈ ( Z /m Z ) × /H M as H M ≤ H K . Lemma 4.1.9.
The following statements hold. (a)
We have (4.1.10) L p ( H ( ααα ; βββ | t ) , M, T ) ∈ T Q [[ T ]] and (4.1.11) L p ( H ( ααα ; βββ | t ) , M, T ) = L p ( H ( kααα ; kβββ | t ) , M, T ) for all k ∈ Z coprime to p and m . (b) Let H K ≤ ( Z /m Z ) × correspond to Gal( K | Q ) as above, and let (4.1.12) r ( M | K, p ) := [ h H K , p i : h H M , p i ] . Then r ( M | K, p ) is the number of primes in M above a prime in K above p , and (4.1.13) L p ( H ( ααα, βββ | t ) , M, T ) = Y k i ∈ ( Z /m Z ) × / h H K ,p i L q ( H ( k i ααα, k i βββ | t ) , T f ) r ( M | K,p ) . Proof.
For part (a), the descent to Q follows from Galois theory and Lemma 3.2.10; the equality(4.1.11) follows as multiplication by k permutes the indices k i in ( Z /m Z ) × / h H M , p i . For part (b),the fact that r ( M | K, p ) counts the number of primes follows again from class field theory; to get(4.1.13), use Lemma 3.2.10 and the fact that the field of definition of ααα, βββ is K = K ααα,βββ . (cid:3) We again package these together in an L -series:(4.1.14) L S ( H ( ααα ; βββ | t ) , M, s ) := Y p S L p ( H ( ααα ; βββ | t ) , M, p − s ) − . We may expand (4.1.14) as a Dirichlet series(4.1.15) L S ( H ( ααα ; βββ | t ) , M, s ) = X n ⊆ Z M n =(0) a n Nm( n ) s with a n ∈ K = K ααα,βββ ⊂ C , and again the series converges for s ∈ C in a right half-plane. If M = K ααα,βββ , we suppress the notation M and write just L S ( H ( ααα ; βββ | t ) , s ) , etc. inally, for a finite order Dirichlet character χ over M , we let L S ( H ( ααα ; βββ | t ) , M, s, χ ) denote thetwist by χ , defined by(4.1.16) L S ( H ( ααα ; βββ | t ) , M, s, χ ) := X n ⊆ Z M n =(0) χ ( n ) a n Nm( n ) s . The Dwork pencil F . In the remaining sections, we continue with the same notation: let t = ψ − and let S = S ( ⋄ , ψ ) be the set of bad primes in (1.2.1) together with the set of primesdividing the numerator or denominator of t or t − . We now prove Main Theorem 1.4.1(a). Proposition 4.2.1.
Let ψ ∈ Q r { , } and let t = ψ − . Then L S ( X F ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · L S ( H ( , ; 0 , | t ) , s − , φ − ) · L S ( H ( ; 0 | t ) , Q ( √− , s − , φ √− ) where (4.2.2) φ − ( p ) = (cid:18) − p (cid:19) = ( − ( p − / is associated to Q ( √− | Q , and φ √− ( p ) = (cid:18) √− p (cid:19) = ( − (Nm( p ) − / is associated to Q ( ζ ) | Q ( √− .Proof. Recall Proposition 3.4.1, where we wrote the number of F q points on F in terms of finitefield hypergeometric functions. We rewrite these for convenience:(4.2.3) X F ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | t ) + φ − ( q )3 qH q ( , ; 0 , | t )+ δ [ q ≡ mod φ √− ( q ) qH q ( ; 0 | t ) where δ [ P ] = 1 , according as if P holds or not.Each summand in (4.2.3) corresponds to a multiplicative term in the exponential generatingseries. The summand q + q + 1 gives the factor (1 − T )(1 − qT )(1 − q T ) in the denominator of(4.1.2), so L S ( X F ,ψ , s ) represents the rest of the sum. The summand H q ( , , ; 0 , , | t ) yields L S ( H ( , , ; 0 , , | t ) , s ) by definition.Next we consider the summand φ − ( q )3 qH q ( , ; 0 , | t ) : for each p S , we have(4.2.4) exp − ∞ X r =1 φ − ( p r )3 p r H p r (cid:0) , ; 0 , | t (cid:1) ( p − s ) r r ! = exp − ∞ X r =1 φ − ( p r ) H p r ( , ; 0 , | t ) p (1 − s ) r r ! = L p ( H p ( , ; 0 , | t ) , p − s , φ − ) ; Combining these for all p S then gives the L -series L S ( H ( , ; 0 , | t ) , s − , φ − ) .We conclude with the final term φ √− ( q ) qH q ( ; 0 | t ) which exists only when q ≡ .We accordingly consider two cases. First, if p ≡ , then in Z [ √− , the two primes p , p bove p have norm p . We compute(4.2.5) exp − ∞ X r =1 φ √− ( p r ) p r H p r ( ; 0 | t ) ( p − s ) r r ! = L p ( H ( ; 0 | t ) , p − s , φ √− ) = L p ( H ( ; 0 | t ) , p − s , φ √− ) L p ( H ( − ; 0 | t ) , p − s , φ √− ) = L p ( H ( ; 0 | t ) , Q ( √− , p − s , φ √− ) where the second equality holds because the definition of the hypergeometric sum only depends onparameters modulo Z and the final equality is the definition (4.1.7) using that Gal( M | Q ) when M = Q ( √− is generated by complex conjugation.Second, if p ≡ , then there is a unique prime ideal p above p with norm Nm( p ) = p ,and(4.2.6) exp − ∞ X r =1 φ √− ( p r ) p r H p r ( ; 0 | t ) ( p − s ) r r ! = L p ( H ( ; 0 | t ) , p − s ) , φ √− ) = L p ( H ( ; 0 | t ) , Q ( √− , p − s , φ √− ) . Taking the product of (4.2.5) and (4.2.6) over all prescribed p , we obtain the last L -series factor. (cid:3) The Klein–Mukai pencil F L . We now prove Theorem 1.4.1(b).
Proposition 4.3.1.
For the Klein–Mukai pencil F L , L S ( X F L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · L S ( H ( , , ; 0 , , | t − ) , Q ( ζ ) , s − where for ααα, βββ = { , , } , { , , } we have field of definition K ααα,βββ = Q ( √− .Remark . By Lemma 4.1.9, we have L S ( H ( , , ; 0 , , | t − ) , s ) = L S ( H ( , , ; 0 , , | t − ) , s ) . Proof.
Recall that by Proposition 3.5.1, we have X F L ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | t )+ 3 qδ [ q ≡ mod (cid:0) H q ( , , ; 0 , , | t − ) + H q ( , , ; 0 , , | t − ) (cid:1) We compute that the field of definition (see Definition 3.1.10) associated to the parameters ααα, βββ = { , , } , { , , } and to ααα, βββ = { , , } , { , , } is Q ( √− . We take M = Q ( ζ ) andconsider a prime p of M , and let q = Nm( p ) . Then q ≡ , and in Example 3.2.4, we haveseen that q is splittable for ααα, βββ .We proceed in two cases, according to the splitting behavior of p in K = K ααα,βββ = Q ( √− .First, suppose that p ≡ , , , or equivalently p splits in K . We have r ( M | K, p ) f = 6 so r ( M | K, p ) = 3 . We then apply Lemma 4.1.9(b), with ( Z /m Z ) × / h H K , p i = {± } , to obtain exp − ∞ X r =1 p fr H p fr (cid:0) , , ; 0 , , | t (cid:1) ( p − s ) fr f r − p fr H p fr (cid:0) , , ; 0 , , | t (cid:1) ( p − s ) fr f r ! = L p f ( H ( , , ; 0 , , | t ) , p − s ) /f L p f ( H ( , , ; 0 , , | t ) , p − s ) /f (4.3.3) = Y k i ∈ ( Z /m Z ) × / h H K ,p i L p ( H ( k i , k i , k i ; 0 , k i , k i | t ) , p − s ) r ( M | K,p ) L p ( H ( , , ; 0 , , | t ) , Q ( ζ ) , p − s ) . To conclude, suppose p ≡ , , , i.e., p is inert in K . Now ( Z /m Z ) × / h H K , p i = { } and r ( M | K, p ) f . By Lemma 3.2.10(c), for all q ≡ , we have that(4.3.4) H q (cid:0) , , ; 0 , , | t (cid:1) = H q (cid:0) p, p, p ; 0 , − , − | t p (cid:1) = H q (cid:0) , , ; 0 , , | t (cid:1) . Using the previous line and Lemma 4.1.9(b), exp − ∞ X r =1 p fr H p fr (cid:0) , , ; 0 , , | t (cid:1) ( p − s ) fr f r +3 p fr H p fr (cid:0) , , ; 0 , , | t (cid:1) ( p − s ) fr f r (cid:19) = exp − ∞ X r =1 H p fr (cid:0) , , ; 0 , , | t (cid:1) ( p − s ) fr f r ! = L p f ( H ( , , ; 0 , , | t ) , p − s ) /f (4.3.5) = L p f ( H ( , , ; 0 , , | t ) , p − s ) r ( M | K,p ) = L p ( H ( , , ; 0 , , | t ) , Q ( ζ ) , p − s ) . (cid:3) The pencil F L . We now prove Theorem 1.4.1(c):
Proposition 4.4.1.
For the pencil F L , L S ( X F L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · L S ( Q ( ζ ) | Q , s − · L S ( H ( , ; 0 , | t ) , s − , φ − ) · L S ( H ( ; 0 | t ) , Q ( √− , s − , φ √− ) · L S ( H ( , ; 0 , | t − ) , Q ( ζ ) , s − , φ √ ) where • the character φ √− is defined in (4.2.2) , • for ααα, βββ = { , } , { , } we have the field of definition K ααα,βββ = Q ( √− , M = Q ( ζ ) , and φ √ ( p ) := (cid:18) √ p (cid:19) ≡ (Nm( p ) − / ( mod p ) is associated to Q ( ζ , √ | Q ( ζ ) , and • L ( Q ( ζ ) | Q , s ) = ζ Q ( ζ ) ( s ) /ζ ( s ) , where ζ Q ( ζ ) ( s ) is the Dedekind zeta function of Q ( ζ ) and ζ ( s ) = ζ Q ( s ) the Riemann zeta function.Proof. We now appeal to Proposition 3.6.1, which we summarize as:(4.4.2) X F L ,ψ ( F q ) = q + q + 1 + 2 q · ( if q ≡ − if q H q ( , , ; 0 , , | t ) + φ − ( q ) qH q ( , ; 0 , | t )+ 2 φ √− ( q ) qδ [ q ≡ mod H q ( ; 0 | t )+ 2 φ √ ( q ) qδ [ q ≡ mod (cid:0) H q ( , ; 0 , | t − ) + H q ( , ; 0 , | t − ) (cid:1) . For the new sum with parameters ααα, βββ = { , } , { , } , we have field of definition K ααα,βββ = Q ( √− because the subgroup of ( Z / Z ) × preserving these subsets is generated by . he term q + q + 1 in (4.4.2) is handled as before. For the next term, by splitting behavior inthe biquadratic field Q ( ζ ) = Q ( √− , √ we obtain(4.4.3) L p ( Q ( ζ ) | Q , pT ) = (1 − pT ) , if p ≡ mod ; (1 − ( pT ) ) − pT = (1 − pT )(1 + pT ) , if p mod . For q ≡ the contribution to the exponential generating series is q , otherwise the contri-bution is q − q = − q .All remaining terms except for the term φ √ ( q ) qδ [ q ≡ mod H q ( , ; 0 , | t − ) + H q ( , ; 0 , | t − )) are handled in the proof of Proposition 4.2.1. We choose M = Q ( ζ ) which has conductor m = 8 .Then H K = h i ≤ ( Z / Z ) × . Let ǫ = (cid:18) √ p (cid:19) for a prime p above p in Q ( ζ ) (and independent ofthis choice).Suppose that p ≡ . We compute that f = 1 and r ( M | K, p ) = 2 . By applyingLemma 4.1.9(b), we have: exp − ∞ X r =1 ǫ r p r H p r (cid:0) , ; 0 , | t − (cid:1) ( p − s ) r r − ∞ X r =1 ǫ r p r H p r (cid:0) , ; 0 , | t − (cid:1) ( p − s ) r r ! = L p ( H (cid:0) , ; 0 , | t − (cid:1) , p − s , φ √ ) L p ( H p r (cid:0) , ; 0 , | t − (cid:1) , p − s , φ √ ) (4.4.4) = Y k ∈ ( Z / Z ) × / h H K ,p i L p ( H (cid:0) k, k ; 0 , k | t − (cid:1) , p − s , φ √ ) = L p ( H (cid:0) , ; 0 , | t − (cid:1) , Q ( ζ ) , p − s , φ √ ) . Suppose now that p ≡ , . Then f = 2 and r ( M | K, p ) = 2 . By Lemma 3.2.10(c), forall q a power of p so that q ≡ , we have that(4.4.5) H q (cid:0) , ; 0 , | t − (cid:1) = H q (cid:0) p, p ; 0 , p | t − p (cid:1) = H q (cid:0) , ; 0 , | t − (cid:1) . Again applying Lemma 4.1.9(b), we have: exp − ∞ X r =1 ǫ r p r H p r (cid:0) , ; 0 , | t − (cid:1) ( p − s ) r r − ∞ X r =1 ǫ r p r H p r (cid:0) , ; 0 , | t − (cid:1) ( p − s ) r r ! = exp − ∞ X r =1 ǫ r p r H p r (cid:0) , ; 0 , | t − (cid:1) ( p − s ) r r ! = L p ( H (cid:0) , ; 0 , | t − (cid:1) , p − s , φ √ ) (4.4.6) = L p ( H (cid:0) , ; 0 , | t − (cid:1) , p − s , φ √ ) r ( M | K,p ) = L p ( H (cid:0) , ; 0 , | t − (cid:1) , Q ( ζ ) , p − s , φ √ ) . Finally, suppose that p ≡ . Then f = 2 and r ( M | K, p ) = 1 , and now exp − ∞ X r =1 ǫ r p r H p r (cid:0) , ; 0 , | t − (cid:1) ( p − s ) r r − ∞ X r =1 ǫ r p r H p r (cid:0) , ; 0 , | t − (cid:1) ( p − s ) r r ! = L p ( H (cid:0) , ; 0 , | t − (cid:1) , p − s , φ √ ) L p ( H (cid:0) , ; 0 , | t − (cid:1) , p − s , φ √ ) (4.4.7) = Y k ∈ ( Z / Z ) × / h H K ,p i L p ( H (cid:0) k, k ; 0 , k | t − (cid:1) , p − s , φ √ ) L p ( H (cid:0) , ; 0 , | t − (cid:1) , Q ( ζ ) , p − s , φ √ ) . (cid:3) The pencil L L . We now prove Theorem 1.4.1(d).
Proposition 4.5.1.
For the pencil L L , we have L S ( X L L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) · ζ Q ( √− ( s − · L S ( H ( , ; 0 , | t ) , s − , φ − ) · L S ( H ( , , , ; 0 , , , | t ) , Q ( i ) , s − , φ √− φ ψ ) where the characters φ − , φ √− are defined in (4.2.2) and (4.5.2) φ ψ ( p ) = (cid:18) ψp (cid:19) is associated to Q ( p ψ ) | Q and ζ Q ( √− ( s ) is the Dedekind zeta function of Q ( √− .Proof. By Proposition 3.6.2, we have the point counts(4.5.3) X L L ,ψ ( F q ) = q + q + 1 + 4 q · ( if q ≡ if q ≡ H q ( , , ; 0 , , | t ) + ( − q × / qH q ( , ; 0 , | t )+ 2( − q × / ω ( ψ ) q × / qδ [ q ≡ mod H q ( , , , ; 0 , , , | t ) . Again by splitting behavior, we have(4.5.4) ζ Q ( i ) ,p ( pT ) = ( (1 − pT ) , if p ≡ mod − p T , if p ≡ mod . For q ≡ , the contribution to the exponential generating series is q , otherwise the con-tribution is .All but the last summand have been identified in the previous propositions, and this one followsin a similar but easier manner (because it has Q as field of definition) applying Lemma 4.1.9(b),with M = Q ( √− and f r ( M | Q , p ) = 2 : exp − ∞ X r =1 − ( p fr ) × / ω ( ψ ) ( p fr ) × / p fr H q ( , , , ; 0 , , , | t ) ( p − s ) fr f r ! = L p f ( H ( , , , ; 0 , , , | t ) , p − s , φ ψ φ − ) /f (4.5.5) = L p f ( H ( , , , ; 0 , , , | t ) , p − s , φ ψ φ − ) r ( M | Q ,p ) = L p ( H ( , , , ; 0 , , , | t ) , Q ( √− , p − s , φ ψ φ − ) . (cid:3) The pencil L . Here we prove Theorem 1.4.1(e).
Proposition 4.6.1.
For the pencil L , L S ( X L ,ψ , s ) = L S ( H ( , , ; 0 , , | t ) , s ) ζ ( s − · L S ( H ( , , , ; 0 , , , | t − ) , Q ( ζ ) , s − , roof. By Proposition 3.6.3, we have that X L ( ψ ) = q + 3 q + 1 + H q ( , , ; 0 , , | t )+ 4 qδ [ q ≡ mod H q ( , , , ; 0 , , , | t − ) . The extra two summands of q in the point count correspond to the L -series factor ζ ( s − . Wenow focus on the remaining new summand qH q ( , , , ; 0 , , , | ψ ) that occurs exactly when q ≡ . Let f be the order of p in ( Z / Z ) × , which divides 4. Take M = Q ( ζ ) . We knowthat K = Q for all possible p , hence r ( M | K, p ) = 4 f − . By Lemma 4.1.9, we have exp − ∞ X r =1 p rf H p rf ( , , , ; 0 , , , | t − ) ( p − s ) rf f r ! = L p f ( H ( , , , ; 0 , , , | t − ) , p − s ) /f (4.6.2) = L p f ( H ( , , , ; 0 , , , | t − ) , p − s ) r ( M | Q ,p ) = L p ( H ( , , , ; 0 , , , | t − ) , Q ( ζ ) , p − s ) . (cid:3) Algebraic hypergeometric functions.
We now turn to some applications of our main the-orem. We begin in this section by setting up a discussion of explicit identification of the algebraichypergometric functions that arise in our decomposition, following foundational work of Beukers–Heckman [BH89].Recall the hypergeometric function F ( z ) = F ( ααα ; βββ | z ) (Definition 2.4.1) for parameters ααα, βββ . Forcertain special parameters, this function may be algebraic over C ( z ) , i.e., the field C ( z, F ( z )) isa finite extension of C ( z ) . By a criterion of Beukers–Heckman, F ( z ) is algebraic if and only ifthe parameters interlace (ordering the parameters, they alternate between elements of ααα and βββ )[BH89, Theorem 4.8]; moreover, all sets of interlacing parameters are classified [BH89, Theorem7.1]. We see in Main Theorem 1.4.1 that for all but the common factor L S ( H ( , , ; 0 , , | t ) , s ) ,the parameters interlace, so this theory applies. Conjecture 4.7.1.
Let t ∈ Q , let ααα, βββ with ααα = βββ = d be such that the hypergeometric function F ( ααα ; βββ | z ) is algebraic. Let M satisfy (4.1.6) . Then L S ( H ( ααα, βββ | t ) , M, s ) is an Artin L -series ofdegree d [ M : K ααα,βββ ] ; in particular, for all good primes p , we have L p ( H ( ααα, βββ | t ) , M, T ) ∈ T Q [ T ] a polynomial of degree d [ M : K ααα,βββ ] . Conjecture 4.7.1 is implicit in work of Katz [Kat90, Chapter 8], and there is current, ongoingwork on the theory of hypergeometric motives that is expected to prove this conjecture, at least forcertain choices of M . An explicit version of Conjecture 4.7.1 could be established in each case forthe short list of parameters that arise in our Main Theorem. For example, we can use the followingproposition about L -series and apply it for the family F , proving a conjecture of Duan [Dua18]. Proposition 4.7.2 (Cohen) . We have the following L -series relations: (4.7.3) L S ( H ( , ; 0 , | ψ − ) , s, φ − ) = L S ( s, φ − ψ ) L S ( s, φ − − ψ ) L S ( ; 0 | ψ − , Q ( √− , s, φ √− ) = L S ( s, φ − ψ ) ) L S ( s, φ − − ψ ) ) , where φ a = (cid:18) ap (cid:19) is the Legendre symbol.Proof. The hypergeometric L -series were computed explicitly by Cohen [Coh, Propositions 6.4 and7.32], and the above formulation follows directly from this computation. (cid:3) More generally, Naskręcki [Nas] has given an explicit description for algebraic hypergeometric L -series of low degree defined over Q using the variety defined by Beukers–Cohen–Mellit [BCM15]. roposition 4.7.2, plugged into our hypergeometric decomposition, gives an explicit decompositionof the polynomial Q F ,ψ,q as follows. Corollary 4.7.4.
We have: Q F ,ψ,q ( T )= (cid:18) − (cid:18) − ψ q (cid:19) qT (cid:19) (cid:18) − (cid:18) − − ψ q (cid:19) qT (cid:19) (cid:18) − (cid:18) − ψ q (cid:19) qT (cid:19) , if q ≡ mod ; (cid:18) − (cid:18) − ψ q (cid:19) qT (cid:19) (1 − qT ) (1 + qT ) , if q ≡ mod where (cid:18) aq (cid:19) denotes the Jacobi symbol. Corollary 4.7.4 explains why the field of definition of the Picard group involves square roots of ψ and − ψ .4.8. Applications to zeta functions.
To conclude, we give an application to zeta functions. InSections 2 and 3, we established a relationship between the periods and the point counts for ourcollection of invertible K3 polynomial families and hypergeometric functions. In particular, both theperiods and the point counts decompose naturally in terms of the group action into hypergeometriccomponents.It is easy to see that the zeta function is the characteristic polynomial of Frobenius acting onour cohomology (i.e., the collection of periods). In this sense, both sections 2 and 3 suggest that,as long as the group action and the action of Frobenius commute, the splitting of Frobenius by thegroup action translates into factors, each corresponding to the Frobenius acting only on a givenisotypical component of the action. However, a priori we only know that this factorization over Q (see e.g. work of Miyatani [Miy15]).Thus, we have the following corollary of Main Theorem 1.4.1. Corollary 4.8.1.
Assuming Conjecture , for smooth X ⋄ ,ψ,q , the polynomials Q ⋄ ,ψ,q ( T ) factorover Q [ T ] under the given hypothesis as follows: (4.8.2) Family Factorization Hypothesis F (deg 2) (deg 1) q ≡ mod (deg 2) q ≡ mod F L (deg 3) (deg 3) q ≡ mod q ≡ mod q ≡ , mod q ≡ , mod F L (1 − qT ) (deg 2)(deg 1) (deg 2) (deg 2) q ≡ mod − qT ) (1 + qT ) (deg 2)(deg 1) (deg 4) q ≡ mod − qT ) (1 + qT ) (deg 2)(deg 2)(deg 4)(deg 4) q ≡ , mod L L (1 − qT ) (deg 2)(deg 4) q ≡ mod − q T ) (deg 2)(deg 8) q ≡ mod L (1 − qT ) (deg 4) q ≡ mod − qT ) (deg 8) q ≡ mod − qT ) (deg 16) q ≡ , mod The factorization in Corollary 4.8.1 is to be read as follows: for the family L L when q ≡ , we have Q ⋄ ,ψ,q ( T ) = (1 − qT ) Q ( T ) Q ( T ) where deg Q ( T ) = 2 and deg Q ( T ) = 4 , but e do not claim that Q , Q are irreducible. A complete factorization into irreducibles depends on ψ ∈ F × q and can instead be computed from the explicit Artin L -series. Proof.
For each case, we need to identify the field of definition for the terms associated to hypergeo-metric functions other than H ( , , ; 0 , , | t ) and check the degrees of the resulting zeta functionfactors using Lemma 4.1.9 and Conjecture 4.7.1. F . The case where q ≡ is straightforward from the statement of Proposition 4.2.1.In the case where q ≡ , we see in the proof that the L -series L S ( H q ( ; 0 | t ) , Q ( √− , s − , φ √− ) factors into a square (see Equations (4.2.5) and (4.2.6)). F L . In the case where q ≡ , , , we see in Equation (4.3.3) that the L -series associatedto H ( , , ; 0 , , | t ) factorizes into two terms with multiplicity /f where f is theorder of q in ( Z / Z ) × . The analogous argument holds for when q ≡ , , usingEquation (4.3.5) to see that the L -series factors into one term with multiplicity /f . F L . The explicit factors follow directly from Equation (4.4.3). The next two factors comefrom the L -series L S ( H ( , ; 0 , | t ) , s − , φ − ) and L S ( H ( ; 0 | t ) , Q ( √− , s − , φ √− ) were dealt with in the F case. The zeta function factorization implied by the L -series L S ( H ( , ; 0 , | t − ) , Q ( ζ ) , s − follows by using Equations (4.4.4), (4.4.6), and (4.4.7). L L . The explicit factors follow directly from Equation (4.5.4). The next factor has been dealtwith above. The final factor is implied by Equation (4.5.5). L . The term ζ ( s − gives the (1 − qT ) factor. The last factor is direct from Equation (4.6.2). (cid:3) Example . Because the reciprocal roots of Q ⋄ ,ψ,q ( T ) are of the form q times a root of 1, thefactors of Q ⋄ ,ψ,q ( T ) over Z are of the form Φ( qT ) , where Φ is a cyclotomic polynomial. We nowgive the explicit zeta functions for the case where q = 281 and ψ = 18 in the table below. We usea SageMath interface to C code written by Costa, which is described in a paper of Costa–Tschinkel[CT14]. Note that the factorizations in Corollary 4.8.1 are sharp for the families F L and L .(4.8.4) Family Q ⋄ ,ψ,q ( T ) F (1 − qT ) (1 + qT ) F L (1 − qT ) (1 + qT ) (1 + q T ) F L (1 + qT + q T + q T + q T + q T + q T ) L L (1 − qT ) (1 + qT ) L (1 − qT ) (1 + qT + q T + q T + q T ) Appendix A. Remaining Picard–Fuchs equations
In this appendix, we provide the details in the computation of the remaining three pencils F L , L L , and L . We follow the same strategy as in sections 2.5–2.6.A.1. The F L pencil. Take the pencil F ψ := x + x + x x + x x − ψx x x x that defines the pencil of projective hypersurfaces X ψ = Z ( F ψ ) ⊂ P . There is a Z / Z scalingsymmetry of this family generated by the element g ( x : x : x : x ) = ( ξ x : x : ξx : ξ x ) where ξ is a primitive eighth root of unity. There are eight characters χ k : H → C × , where χ k ( g ) = ξ k . We can again decompose V into subspaces W χ k and write their monomial bases. ote that the monomial bases for W χ , W χ , W χ , and W χ are the same up to transpositions of x and x or x and x which leave the polynomial invariant; thus, they have the same Picard–Fuchsequations. The monomial bases for W χ and W χ are related by transposing x and x , so they alsohave the same Picard–Fuchs equations. So we are left with four types of monomial bases:(i) W χ has monomial basis { x x x x } ;(ii) W χ has monomial basis { x x x , x x x } ;(iii) W χ has monomial basis { x x x , x x , x x } ; and(iv) W χ has monomial basis { x x , x x , x x x , x x x } .Using (2.3.3), we compute the following period relations:(A.1.1) v + (4 , , ,
0) = 1 + v ω + 1) v + ψ ( v + (1 , , , v + (0 , , ,
0) = 1 + v ω + 1) v + ψ ( v + (1 , , , v + (0 , , ,
1) = 3( v + 1) − ( v + 1)8( ω + 1) v + ψ ( v + (1 , , , v + (0 , , ,
3) = − ( v + 1) + 3( v + 1)8( ω + 1) v + ψ ( v + (1 , , , We can now use the diagram method to prove the following proposition.
Proposition A.1.2.
For the F L family, the primitive cohomology group H prim ( X F L ,ψ , C ) has periods whose Picard–Fuchs equations are hypergeometric differential equations as follows: periods are annihilated by D ( , , ; 1 , , | ψ − ) , periods are annihilated by D ( , ; 1 , | ψ − ) , periods are annihilated by D ( ; 1 | ψ ) , periods are annihilated by D ( , ; 1 , | ψ ) , and periods are annihilated by D ( − , ; 0 , | ψ ) . We consider each of these in turn.
Lemma A.1.3.
The Picard–Fuchs equation associated to the periods ψ (2 , , , and ψ (0 , , , isthe hypergeometric differential equation D ( , ; 1 , | ψ − ) .Proof. For the periods (2 , , , and (0 , , , , corresponding to the quartic monomials x x and x x , we use the diagram , , , / / (cid:15) (cid:15) (3 , , , , , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (1 , , , , − , , / / (cid:15) (cid:15) (4 , , , , , , and get the relations(A.1.4) η (2 , , ,
0) = ψ ( η + 1)(0 , , , η (0 , , ,
2) = ψ ( η + 1)(2 , , , . For the periods (2 , , , and (0 , , , , corresponding to the quartic monomials x x and x x ,we get the same Picard–Fuchs equation(A.1.5) (cid:2) ( η − η − ψ ( η + 3)( η + 1) (cid:3) By multiplying by ψ and substituting t = ψ − and θ = t dd t = − η/ , we get: ψ (cid:2) ( η − η − ψ ( η + 3)( η + 1) (cid:3) (2 , , ,
0) = 0 (cid:2) ( η − η − − ψ ( η + 2) η (cid:3) ψ (2 , , ,
0) = 0 (cid:2) ( θ + )( θ + ) − t − ( θ − ) θ (cid:3) ψ (2 , , ,
0) = 0 (cid:2) ( θ − ) θ − t ( θ + )( θ + ) (cid:3) ψ (2 , , ,
0) = 0 which is the hypergeometric differential equation D ( , ; 1 , | ψ − ) . (cid:3) Lemma A.1.6.
The Picard–Fuchs equation associated to the periods (2 , , , is the hypergeometricdifferential equation D ( ; 1 | ψ ) .Proof. For the period (2 , , , , corresponding to the quartic monomial x x x , we use the diagram , , , / / (cid:15) (cid:15) (3 , , , , − , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (1 , , , − , , , / / (cid:15) (cid:15) (0 , , , , , , One can see that (2 , , ,
2) = (2 + η )(2 , , , , which one can then use to show that:(A.1.7) η (2 , , ,
1) = 8 ψ (0 , , , ψ (1 , , , ψ (2 , , , ψ ( η + 2)(2 , , , . Thus the period (2 , , , corresponding to the quartic monomial x x x satisfies the differentialequation: (cid:2) η − ψ ( η + 2) (cid:3) (2 , , ,
1) = 0
By substituting u = ψ and σ = u dd u = 4 η , we get: (cid:2) η − ψ ( η + 2) (cid:3) (2 , , ,
1) = 0 (cid:2) σ − u ( σ + ) (cid:3) (2 , , ,
1) = 0 . (cid:3) Lemma A.1.8.
The Picard–Fuchs equations associated to the periods (2 , , , , ψ (1 , , , arethe hypergeometric differential equations D ( , ; 1 , | ψ ) , D ( , − ; 0 , | ψ ) , respectively.Proof. For the period (2 , , , , corresponding to the quartic monomial x x x , we use the diagram (0 , − , , / / D (cid:15) (cid:15) (1 , , , − , , , / / D (cid:15) (cid:15) (0 , , , , , , / / D (cid:15) (cid:15) (3 , , , , , , / / D (cid:15) (cid:15) (2 , , , , , , ote that:(A.1.9) η (2 , , ,
1) = 8 ψ (1 , , , , , ,
4) = ( η + )(1 , , , η (1 , , ,
1) = ψ ( η + )(2 , , , . We can compute the two periods that satisfy each of the following Picard–Fuchs equations forthe four sets of pairs:(A.1.10) (cid:2) ( η − η − ψ ( η + )( η + ) (cid:3) (2 , , ,
1) = 0; (cid:2) ( η − η − ψ ( η + )( η + ) (cid:3) (1 , , ,
1) = 0 . With the first Picard–Fuchs equation, we can substitute u = ψ , σ = u dd u = 4 η , and yield theequation: (cid:2) ( η − η − ψ ( η + )( η + ) (cid:3) (2 , , ,
1) = 0; (cid:2) ( σ − ) σ − u ( σ + )( σ + ) (cid:3) (2 , , ,
1) = 0 . which is the hypergeometric differential equation D ( , ; 1 , | u ) . For the second Picard–Fuchsequation, we can multiply by ψ and then substitute to find: (cid:2) ( η − η − ψ ( η + )( η + ) (cid:3) (1 , , ,
1) = 0 ψ (cid:2) ( η − η − ψ ( η + )( η + ) (cid:3) (1 , , ,
1) = 0 (cid:2) ( η − η − − ψ ( η + )( η − ) (cid:3) ψ (1 , , ,
1) = 0 (cid:2) ( σ − σ − ) − u ( σ + )( σ − ) (cid:3) ψ (1 , , ,
1) = 0 , which is the hypergeometric function D ( , − ; 0 , | ψ ) . (cid:3) Proof of Proposition . The periods annihilated by D ( , , ; 1 , , | ψ − ) are those correspond-ing to the holomorphic form. The periods are annihilated by D ( , ; 1 , | ψ − ) are provided byLemma A.1.3. The period annihilated by D ( ; 1 | ψ ) corresponds to a monomial in the basis for W χ , which we compute in Lemma A.1.6. Since W χ and W χ are related by a transposition of x and x , there are two periods annihilated by the hypergeometric differential equation computed here.The periods annihilated by D ( , ; 1 , | ψ ) and the periods annihilated by D ( − , ; 0 , | ψ ) correspond to the monomial bases for W χ , W χ , W χ , and W χ , which are the same up to transpo-sitions. We compute in Lemma A.1.8 the Picard–Fuchs equations for W χ which then give us thateach of those hypergeometric differential equations annihilate 4 periods. (cid:3) A.2.
The L L pencil. Now consider F ψ := x x + x x + x x + x x − ψx x x x that defines the pencil of projective hypersurfaces X ψ = Z ( F ψ ) ⊂ P . There is a Z / Z symmetrywith generator g ( x : x : x : x ) = ( ξx : ξ x : ξ x : ξ x ) where ξ is a primitive eighth root of unity. There are four characters χ a : H → G m , where χ a ( g ) = √− a . We can again decompose V into subspaces W χ a . Out of the eight, the subspaces W χ , W χ , W χ , and W χ are empty. The monomial bases for W χ and W χ are related by atransposition of the variables x and x , so their Picard–Fuchs equations are the same. We havethree types of monomial bases:(i) W χ has monomial basis { x x x x , x x , x x , x x , x x } ;(ii) W χ has monomial basis { x x x , x x x , x x x , x x x } ; and iii) W χ has monomial basis { x x , x x , x x x , x x x , x x x , x x x } .Using (2.3.3), we compute the following period relations:(A.2.1) v + (3 , , ,
0) = 3( v + 1) − ( v + 1)8( ω + 1) v + ψ ( v + (1 , , , v + (1 , , ,
0) = − ( v + 1) + 3( v + 1)8( ω + 1) v + ψ ( v + (1 , , , and the two symmetric relations replacing , with , . Proposition A.2.2.
For the L L family, the primitive cohomology group H prim ( X L L ,ψ , C ) has periods whose Picard–Fuchs equations are hypergeometric differential equations as follows: periods are annihilated by D ( , , ; 1 , , | ψ − ) , periods are annihilated by D ( , , , ; 0 , , , | ψ ) , and periods are annihilated by D ( , ; 1 , | ψ ) . To prove Proposition A.2.2, we use the diagram method above in a few cases and then usesymmetry. We first do two calculations.
Lemma A.2.3.
The Picard–Fuchs equation associated to the periods (2 , , , , (1 , , , , (1 , , , , and (0 , , , is the hypergeometric differential equation D ( , , , ; 0 , , , | ψ ) .Proof. To find the Picard–Fuchs equations corresponding to all these cohomology pieces, we use thediagram (2 , , , / / (cid:15) (cid:15) (3 , , , , , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (1 , , , , , , We obtain the following relations:(A.2.4) η (1 , , ,
2) = ψ ( η + )(2 , , , η (1 , , ,
1) = ψ ( η + )(1 , , , η (0 , , ,
1) = ψ ( η + )(1 , , , η (2 , , ,
0) = ψ ( η + )(0 , , , Using these relations, we can get the Picard–Fuchs equation:(A.2.5) (cid:2) ( η − η − η − η − ψ (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1)(cid:3) (1 , , ,
2) = 0
By substituting u = ψ and σ = u dd u = η , we obtain: (cid:2) (4 σ − σ − σ − σ − u (cid:0) σ + (cid:1) (cid:0) η + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (1 , , ,
2) = 0; ( σ − )( σ − )( σ − ) σ − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (1 , , ,
2) = 0 , which is the hypergeometric differential equation D ( , , , ; 0 , , , | ψ ) . The other three Picard–Fuchs equations are the same due to the symmetry in (A.2.4). (cid:3) Lemma A.2.6.
The Picard–Fuchs equations associated to the periods (2 , , , and (0 , , , isthe hypergeometric differential equation D ( , ; 1 , | ψ ) .Proof. We use the diagram (2 , , , / / (cid:15) (cid:15) (3 , , , , , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (1 , , , , , , / / (cid:15) (cid:15) (3 , , , , , , We obtain the following relations:(A.2.7) η (0 , , ,
2) = ψ ( η + 1)(2 , , , η (2 , , ,
0) = ψ ( η + 1)(0 , , , , giving the following Picard–Fuchs equations:(A.2.8) (cid:2) ( η − η − ψ ( η + 3) ( η + 1) (cid:3) (2 , , ,
0) = 0; (cid:2) ( η − η − ψ ( η + 3) ( η + 1) (cid:3) (0 , , ,
2) = 0;
By substituting u = ψ and σ = u dd u = η , we obtain the hypergeometric form: (cid:2) ( σ − ) σ − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (2 , , ,
0) = 0; (cid:2) ( σ − ) σ − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (0 , , ,
2) = 0 , which is the hypergeometric differential equation D ( , ; 1 , | ψ ) . (cid:3) Proof of
Proposition A.2.2 . The first three periods are the same for each family. Next, by Lemma A.2.3,all the monomial basis elements in W χ are annihilated by the hypergeometric differential equation D ( , , , ; 0 , , , | ψ ) . Since W χ and W χ are related by a transposition, we get 8 periods an-nihilated by it. The Picard–Fuchs equations for the last two periods are given by Lemma A.2.6. (cid:3) A.3.
The L pencil. Finally we consider F ψ := x x + x x + x x + x x − ψx x x x that defines the pencil of projective hypersurfaces X ψ = Z ( F ψ ) ⊂ P . There is a H = Z / Z scalingsymmetry on X ψ generated by the element g ( x : x : x : x ) = ( ξx : ξ x : ξ x : ξ x ) where ξ is a fifth root of unity. There are five characters χ k : H → C × given by χ k ( g ) = ξ k . Wedecompose V into five subspaces W χ k . The monomial bases for W χ , W χ , W χ , and W χ are related y a rotation of the variables x , x , x , and x , so their corresponding Picard–Fuchs equations arethe same. We are then left with two types of monomial bases:(i) W χ has monomial basis { x x x x , x x , x x } ; and(ii) W χ has monomial basis { x x , x x x , x x x , x x x } .For this family, we can compute the period relations:(A.3.1) v + (3 , , ,
0) = 27(1 + v ) − (1 + v ) + 3(1 + v ) − v )80( ω + 1) v + ψ ( v + (1 , , , and its cyclic permutations. Proposition A.3.2.
For the family L , the primitive cohomology group H prim ( X L ,ψ , C ) has periods whose Picard–Fuchs equations are hypergeometric differential equations as follows: periods are annihilated by D ( , , ; 1 , , | ψ − ) , periods are annihilated by D ( , , , ; 1 , , , | ψ ) , periods are annihilated by D ( − , , , ; 0 , , , | ψ ) , periods are annihilated by D ( − , − , , ; − , , , | ψ ) , and periods are annihilated by D ( − , − , − , ; − , − , , | ψ ) .Proof. The period associated to the holomorphic form is found by the same strategy as before.Lastly, we use (A.3.1) to construct the diagram: (2 , , , / / (cid:15) (cid:15) (3 , , , , , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (2 , , , , , , / / (cid:15) (cid:15) (1 , , , , , , and consequently obtain the following relations:(A.3.3) η (1 , , ,
0) = ψ (cid:0) η + (cid:1) (2 , , , η (1 , , ,
2) = ψ (cid:0) η + (cid:1) (1 , , , η (0 , , ,
1) = ψ (cid:0) η + (cid:1) (1 , , , η (2 , , ,
0) = ψ (cid:0) η + (cid:1) (0 , , , . e then cyclically use these relations to find a recursion which yields the following Picard–Fuchsequations:(A.3.4) h(cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) − ψ ( η − η − η − η i (1 , , ,
2) = 0 h(cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) − ψ ( η − η − η − η i (1 , , ,
0) = 0 h(cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) − ψ ( η − η − η − η i (2 , , ,
0) = 0 h(cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) (cid:0) η + (cid:1) − ψ ( η − η − η − η i (0 , , ,
1) = 0 . By multiplying these lines by 1, ψ , ψ , and ψ , respectively, substituting u = ψ , σ = u dd u , andthen multiplying by − u , we obtain the following equations: (cid:2) ( σ − )( σ − )( σ − ) σ − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1)(cid:3) (1 , , ,
2) = 0 (cid:2) ( σ − σ − )( σ − )( σ − ) − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ − (cid:1)(cid:3) ψ (1 , , ,
0) = 0 (cid:2) ( σ − )( σ − σ − )( σ − ) − u (cid:0) σ + (cid:1) (cid:0) σ + (cid:1) (cid:0) σ − (cid:1) (cid:0) σ − (cid:1)(cid:3) ψ (2 , , ,
0) = 0 (cid:2) ( σ − )( σ − )( σ − σ − ) − u (cid:0) σ + (cid:1) (cid:0) σ − (cid:1) (cid:0) σ − (cid:1) (cid:0) σ − (cid:1)(cid:3) ψ (0 , , ,
1) = 0 . These are the claimed hypergeometric differential equations. (cid:3)
Appendix B. Finite field hypergeometric sums
In this part of the appendix, we write down the details of manipulations of hypergeometric sums.B.1.
Hybrid definition.
In this section, we apply the argument of Beukers–Cohen–Mellit to showthat the hybrid definition of the finite field hypergeometric sum reduces to the classical one. Weretain the notation from sections 3.1–3.2.
Lemma B.1.1.
Suppose that q is good and splittable for ααα, βββ . If α i q × , β i q × ∈ Z for all i = 1 , . . . , d ,then Definitions and agree.Proof. Our proof follows Beukers–Cohen–Mellit [BCM15, Theorem 1.3]. We consider G ( m + ααα ′ q × , − m − βββ ′ q × ) = Y α ′ i ∈ ααα ′ g ( m + α ′ i q × ) g ( α i q × ) Y β ′ i ∈ βββ ′ g ( − m − β ′ i q × ) g ( − β i q × ) . We massage this expression, and for simplicity drop the subscripts . First, D ( x ) Y α j ∈ ˆ ααα ( x − e π √− α j ) = r Y j =1 ( x p j − and D ( x ) Y β j ∈ ˜ βββ ( x − e π √− β j ) = s Y j =1 ( x q j − . Write D ( x ) = Q δj =1 ( x − e π √− c j /q × ) . Then(B.1.2) G ( m + ααα ′ q × , − m − βββ ′ q × )= r Y i =1 p i − Y j =0 g ( m + jq × /p i ) g ( jq × /p i ) s Y i =1 q i − Y j =0 g ( − m − jq × /q i ) g ( − jq × /q i ) δ Y j =1 g ( c j ) g ( − c j ) g ( m + c j ) g ( − m − c j ) . Since p i divides q × , by the Hasse–Davenport relation (Lemma 3.1.3(c)) we have that(B.1.3) p i − Y j =0 g ( m + jq × /p i ) g ( jq × /p i ) = − g ( p i m ) ω ( p i ) p i m . nalogously, since q i divides q × , we use Hasse–Davenport to find that(B.1.4) q i − Y j =0 g ( − m − jq × /q i ) g ( − jq × /q i ) = − g ( − q i m ) ω ( q i ) q i m Note that if c j = 0 then g ( c j ) g ( − c j ) = ( − c j q and otherwise, hence δ Y i =1 g ( c j ) g ( − c j ) = ( − P c j q δ − s (0) , where s (0) is the multiplicity of in D ( x ) , or, equivalently, the number of times c j is . Nownote the number of times that m + c j = 0 is the multiplicity of the root e − π √− m/q × in D ( x ) ,which, equivalently, is the multiplicity of e π √− m/q × in D ( x ) as D ( x ) is a product of cyclotomicpolynomials. This implies that δ Y j =1 g ( m + c j ) g ( − m − c j ) = ( − m + c j q δ − λ ( m ) . We then have that(B.1.5) δ Y j =1 g ( c j ) g ( − c j ) g ( m + c j ) g ( − m − c j ) = ( − P j c j q δ − s (0) ( − P j ( m + c j ) q δ − s ( m ) = ( − − δm q s ( m ) − s (0) . Combining Equations (B.1.3), (B.1.4), and (B.1.5), we then have G ( m + ααα ′ q × , − m − βββ ′ q × )= r Y i =1 − g ( p i m ) ω ( p i ) p i m ! s Y i =1 − g ( − q i m ) ω ( q i ) q i m ! (cid:16) ( − − δm q s ( m ) − s (0) (cid:17) = ( − r + s q s ( m ) − s (0) g ( p m ) · · · g ( p r m ) g ( − q m ) · · · g ( − q s m ) ω (( − δ p − p · · · p − p r r q q · · · q q s s ) m = ( − r + s q s ( m ) − s (0) g ( p m ) · · · g ( p r m ) g ( − q m ) · · · g ( − q s m ) ω (( − δ M ) m . By plugging this equation into Definition 3.1.6 for the appropriate factors we obtain the quantitygiven in Definition 3.2.7. (cid:3)
B.2.
The pencil F L .Proposition B.2.1. The number of F q -points on F L can be written in terms of hypergeometricfunctions, as follows: (a) If q ≡ , then X F L ,ψ ( F q ) = q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . (b) If q ≡ , then X F L ,ψ ( F q ) = q − q + 1 + H q ( , , ; 0 , , | ψ − )+ qH q ( , ; 0 , | ψ − ) − qH q ( ; 0 | ψ − ) . (c) If q ≡ , then X F L ,ψ ( F q ) = q + 7 q + 1 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2 qH q ( ; 0 | ψ − ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) . emark B.2.2 . Notice that the hypergeometric functions appearing in the point count correspondto the Picard–Fuchs equations in Proposition 2.7.1. We also see the appearance of six additionaltrivial factors.
Step 1: Computing and clustering the characters.
To use Theorem 3.3.3 we compute the subset S ⊂ ( Z /q × Z ) r given by the constraints in (3.3.1).(a) If q ≡ then S can be clustered in the following way:(i) the set S = { k (1 , , , , −
4) : k ∈ Z /q × Z } and(ii) the set S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } .(b) If q ≡ then S contains the two sets above and:(i) the set S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } and(ii) the set S = { k (1 , , , , −
4) + 3 q × (0 , , , ,
0) : k ∈ Z /q × Z } .(b) If q ≡ then S contains the four sets above and(i) two sets of the form S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } and(ii) two sets of the form S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } . Step 2: Counting points on the open subset with nonzero coordinates.
Lemma B.2.3.
Suppose ψ ∈ F × q . (a) If q ≡ then (B.2.4) U F L ,ψ ( F q ) = q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . (b) If q ≡ then (B.2.5) U F L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − ) − qH q ( ; 0 | ψ − ) − g ( q × ) + g ( q × ) g ( q × ) ! . (c) If q ≡ then (B.2.6) U F L ,ψ ( F q ) = q − q + 7 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2 qH q ( ; 0 | ψ − ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) − g ( q × ) + g ( q × ) g ( q × ) ! − q g ( q × ) g ( q × ) g ( q × ) − q g ( q × ) g ( q × ) g ( q × ) . Proof. If q ≡ then by Lemmas 3.4.4 and 3.4.7(B.2.7) U F L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s = ( q − q + 3 + H q ( , , ; 0 , , | ψ − ))+ − − qH q ( , ; 0 , | ψ − )= q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . If q ≡ then q ≡ but q , so by Lemmas 3.4.4, 3.4.7, 3.4.12,and 3.4.21 B.2.8) U F L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s = ( q − q + 3 + H q ( , , ; 0 , , | ψ − )) + (2 + qH q ( , ; 0 , | ψ − ))+ ( − q × / qH q ( ; 0 | ψ − ) + ( − q × / − g ( q × ) + g ( q × ) g ( q × ) ! + ( − q × / qH q ( ; 0 | ψ − ) + ( − q × / − g ( q × ) + g ( q × ) g ( q × ) ! = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )) − qH q ( ; 0 | ψ − ) − g ( q × ) + g ( q × ) g ( q × ) ! . If q ≡ , then by Lemmas 3.4.4, 3.4.7, 3.4.12, 3.4.21, B.2.12, and B.2.17, we have:(B.2.9) U F L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s = ( q − q + 3 + H q ( , , ; 0 , , | ψ − )) + (2 + qH q ( , ; 0 , | ψ − ))+ 2 qH q ( ; 0 | ψ − ) + 2 − g ( q × ) + g ( q × ) g ( q × ) ! + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) − q g ( q × ) g ( q × ) g ( q × ) − q g ( q × ) g ( q × ) g ( q × )= q − q + 7 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2 qH q ( ; 0 | ψ − ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ )+ 2 ω (2) q × / qH q ( , ; 0 , | ψ ) − g ( q × ) + g ( q × ) g ( q × ) ! − q g ( q × ) g ( q × ) g ( q × ) − q g ( q × ) g ( q × ) g ( q × ) as claimed. (cid:3) Before proving the lemmas that associate the quantities P s ∈ S ω ( a ) − s c s and P s ∈ S ω ( a ) − s c s to hypergeometric sums, we need the following lemma. Lemma B.2.10.
Suppose q ≡ and q = p r for some natural number r and prime p . Then g ( q × ) g ( q × ) g ( q × ) g ( q × ) = ω (2) q × / q. Proof.
Since q ≡ , we can use Hasse–Davenport with N = 2 and m = q × to get that(B.2.11) g ( q × ) = ω (2) q × / g ( q × ) g ( q × ) g ( q × ) . y multiplying both sides by g ( q × ) , and dividing by ω (2) q × / , we have ω (2) − q × / g ( q × ) g ( q × ) = g ( q × ) g ( q × ) g ( q × ) g ( q × ) . We obtain the identity above after noting that g ( q × ) g ( q × ) = ( − q × / q = q , since q ≡ and that ω (2) q × / = ± when q ≡ hence ω (2) q × / = ω (2) − q × / . (cid:3) Lemma B.2.12.
Suppose q ≡ . Then for S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } we have X s ∈ S ω ( a ) − s c s = ω (2) q × / qH q ( , ; 0 , | ψ ) − q g ( q × ) g ( q × ) g ( q × ) − q g ( q × ) g ( q × ) g ( q × ) . Proof.
First, we take the definition of the sum and take out all terms that are of the form ℓ q × toobtain the equality:(B.2.13) X s ∈ S ω ( a ) − s c s = 1 qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) + 1 qq × g ( q × ) g ( q × ) g ( q × )+ 1 qq × q − X k =0 k mod q × ) ω (4 ψ ) k g ( k ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( − k ) . Next, we use the Hasse-Davenport relationship to expand g ( − k ) and then use relation from 3.1.3(b)to cancel out the g ( k + q × ) factor in the summation. Through this, we obtain:(B.2.14) X s ∈ S ω ( a ) − s c s = 1 qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) + 1 qq × g ( q × ) g ( q × ) g ( q × )+ 1 q × q − X k =0 k mod q × ) ω ( ψ ) k g ( k + q × ) g ( k + q × ) g ( − k + q × ) g ( − k + q × ) g ( q × ) . Here, we re-index the summation by m = k + q × to obtain:(B.2.15) X s ∈ S ω ( a ) − s c s = 1 qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) + 1 qq × g ( q × ) g ( q × ) g ( q × )+ 1 q − q − X m =0 m mod q × ) ω ( ψ ) m g ( m + q × ) g ( m + q × ) g ( − m − q × ) g ( − m ) g ( q × ) . e now multiply the final form by the expression g ( q × ) g ( q × ) g ( q × ) g ( q × ) g ( q × ) g ( q × ) = 1 . We put the denominator of this factor into the summation to relate the summation to a hyper-geometric function but factor out the numerator along with a factor of g ( q × ) . We then applyLemma B.2.10 to this factor ahead of the summation. We thus obtain:(B.2.16) X s ∈ S ω ( a ) − s c s = 1 qq × g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) − qq × g ( q × ) g ( q × ) g ( q × ) g ( q × ) + 1 qq × g ( q × ) g ( q × ) g ( q × ) − ω (2) q × / qq × q − X m =0 m mod q × ) ω ( ψ ) m g ( m + q × ) g ( m + q × ) g ( − m − q × ) g ( − m ) g ( q × ) g ( q × ) g ( − q × ) g (0) . By comparing terms of the summations above and the hypergeometric function itself, we obtain thedesired result. (cid:3)
Lemma B.2.17.
Suppose q ≡ . Then for S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } we have X s ∈ S ω ( a ) − s c s = ω (2) q × / qH q ( , ; 0 , | ψ ) − q g ( q × ) g ( q × ) g ( q × ) − q g ( q × ) g ( q × ) g ( q × ) . Proof.
The proof is analogous to the proof in Lemma B.2.12 except we substitute m = k + q × .Alternatively, apply complex conjugation to the conclusion of Lemma B.2.12, negating indices as inthe proof of Lemma 3.5.6. (cid:3) Step 3: Count points when at least one coordinate is zero.
Lemma B.2.18.
The following statements hold. (a) If q ≡ , then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 2 q. (b) If q ≡ , then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 2 q − g ( q × ) + g ( q × ) g ( q × ) ! . (c) If q ≡ , then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 10 q − g ( q × ) + g ( q × ) g ( q × ) ! + q g ( q × ) g ( q × ) g ( q × )+ q g ( q × ) g ( q × ) g ( q × ) . roof. We do this case by case. If x is the only variable equalling zero, then we must count thenumber of solutions in the open torus for the hypersurface Z ( x + y z + z y ) ⊂ P . We can see byusing Theorem 3.3.3 that this depends on q . Here, in case (a) we get q − points, in case (b) weget q − g ( q × ) + g ( q × ) ) g ( q × ) − , and in case (c) we get q − g ( q × ) + g ( q × ) ) g ( q × ) − +2 q − ( g ( q × ) g ( q × ) g ( q × ) + g ( q × ) g ( q × ) g ( q × )) . There are two such cases, when either x or x isthe only variable equalling 0.Next is when both x and x are zero and the other two variables are nonzero. Here the numberof solutions is − q × / .Next is when x is zero but the rest are nonzero. Here this is ( q − times the number of solutionsof Z ( x + y ) in the open torus of P . We then get that the number of solutions is if q and if q ≡ , hence q − points. There are two such cases, when x or x are uniquelyzero.The next case is when x and x are both zero. Then the number of nonzero solutions is exactlythe number of solutions of Z ( x + y ) in the open torus of P , i.e., if q and if q ≡ .There are no rational points where x , and x are both zero and x nonzero and the same whenyou swap x with x or x with x . Finally, there are two more solutions: (0 : 0 : 1 : 0) and (0 : 0 : 0 : 1) . We now count.(a) If q and q is odd, then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 2( q −
1) + 0 + 0 + 0 + 2 = 2 q. (b) If q ≡ , then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 2 q − g ( q × ) + g ( q × ) g ( q × ) ! + 2 + 0 + 0 + 2= 2 q − g ( q × ) + g ( q × ) g ( q × ) ! . (c) If q ≡ , then X F L ,ψ ( F q ) − U F L ,ψ ( F q ) = 2 q − g ( q × ) + g ( q × ) g ( q × ) + 2 q g ( q × ) g ( q × ) g ( q × )+ 2 q g ( q × ) g ( q × ) g ( q × ) (cid:19) + 2 + 2(4( q − q − g ( q × ) + g ( q × ) g ( q × ) + q g ( q × ) g ( q × ) g ( q × )+ q g ( q × ) g ( q × ) g ( q × ) . (cid:3) Step 4: Combine Steps 2 and 3 to reach the conclusion.Proof of Proposition 3.6.1.
We now combine Lemmas B.2.3 and B.2.18 as follows. For (a), for q ≡ , X F L ,ψ ( F q ) = q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) + 2 q = q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . For (b), for q ≡ , X F L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − ) qH q ( ; 0 | ψ − ) − g ( q × ) + g ( q × ) g ( q × ) ! + 2 q − g ( q × ) + g ( q × ) g ( q × ) ! = q − q + 1 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − ) − qH q ( ; 0 | ψ − ) Finally, for (c) with q ≡ , X F L ,ψ ( F q ) = q − q + 5 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2 qH q ( ; 0 | ψ − ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) − g ( q × ) + g ( q × ) g ( q × ) ! − q g ( q × ) g ( q × ) g ( q × ) − q g ( q × ) g ( q × ) g ( q × )+ 10 q − g ( q × ) + g ( q × ) g ( q × ) ! + q g ( q × ) g ( q × ) g ( q × )+ q g ( q × ) g ( q × ) g ( q × )= q + 7 q + 1 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2 qH q ( ; 0 | ψ − ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) + 2 ω (2) q × / qH q ( , ; 0 , | ψ ) . (cid:3) B.3.
The pencil L L .Proposition B.3.1. The number of F q -points on L L can be written in terms of hypergeometricfunctions, as follows: (a) If q ≡ , then (B.3.2) X L L ,ψ ( F q ) = q + q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . (b) If q ≡ , then (B.3.3) X L L ,ψ ( F q ) = q + 9 q + 1 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) . Remark
B.3.4 . Again, notice that the hypergeometric functions appearing in the point count corre-spond to exactly one of the Picard–Fuchs equations in Proposition A.2.2. We also see the appearanceof eight additional trivial factors.
Step 1: Computing and clustering the characters.
As with all the previous families, we first computethe set S of solutions to the system of congruences given by Theorem 3.3.3:(a) If q , and q is odd, then S consists of(i) the set S = { k (1 , , , , −
4) : k ∈ Z /q × Z } and(ii) the set S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } . (b) If q ≡ , then(i) the set S = { k (1 , , , , −
4) : k ∈ Z /q × Z } , (ii) the set S = { k (1 , , , , −
4) + q × (0 , , , ,
0) : k ∈ Z /q × Z } , and(iii) two sets of the form S = { k (1 , , , , −
4) + q × (0 , , , ,
2) : k ∈ Z /q × Z } . Step 2: Counting points on the open subset with nonzero coordinates.
Lemma B.3.5.
Suppose ψ ∈ F × q . For q , we have: (a) If q ≡ , then (B.3.6) U L L ,ψ ( F q ) = q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . b) If q ≡ , then (B.3.7) U L L ,ψ ( F q ) = q − q + 5 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) . Proof.
We do this by cases. For (a), where q ≡ , then by using Theorem 3.3.3 withLemmas 3.4.4 and 3.4.7 we have that(B.3.8) U L L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) − − qH q ( , ; 0 , | ψ − )= q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . For (b) with q ≡ , by using Theorem 3.3.3 with Lemmas 3.4.4, 3.4.7, and B.3.10 below,we have that U L L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s + X s ∈ S ω ( a ) − s c s + 2 X s ∈ S ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + 2 + qH q ( , ; 0 , | ψ − )+ 2( − q × / ω ( ψ ) q × / qH q ( 18 , , ,
78 ; 0 , , , | ψ − ) (B.3.9) = q − q + 5 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) . (cid:3) Lemma B.3.10.
Suppose that q ≡ . Then X s ∈ S ω ( a ) − s c s = ( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) . Proof.
We start with the definition, factor out ω ( ψ ) q × / , and then use the Hasse–Davenport rela-tion (3.1.4) with N = 4 with respect to m = k to obtain:(B.3.11) X s ∈ S ω ( a ) − s c s = 1 qq × q − X k =0 ω ( − ψ ) k + q × g ( k ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( − k + q × )= ω ( ψ ) q × / qq × q − X k =0 ω ( − ψ ) k g (4 k ) ω (4) − k g ( q × ) g ( q × ) g ( q × ) g ( − k + q × ) . Simplify with Lemma 3.1.3(b) to get(B.3.12) X s ∈ S ω ( a ) − s c s = ( − q × / ω ( ψ ) q × / q − q − X k =0 ω ( − ψ ) k g (4 k ) g ( q × ) g ( − k + q × ) . Now we use the Hasse–Davenport relation again with N = 2 and m = − k to find:(B.3.13) X s ∈ S ω ( a ) − s c s = ( − q × / ω ( ψ ) q × / q − q − X k =0 ω ( − ψ ) k g (4 k ) g ( q × ) g ( − k ) g ( q × ) ω (2) k g ( − k ) ! . e now simplify using Lemma 3.1.3(b) again and then expand the summation to get:(B.3.14) X s ∈ S ω ( a ) − s c s = ( − q × / ω ( ψ ) q × / q − q × + 1 q × q − X k =0 k mod q × ) ω (4 ψ ) k q − g (4 k ) g ( − k ) . We finally reindex the sum with m = − k , yielding(B.3.15) X s ∈ S ω ( a ) − s c s = ( − q × / ω ( ψ ) q × / q − q × + 1 q × q − X m =0 m mod q × ) ω (4 ψ ) − m q − g ( − m ) g (8 m ) = ( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) relating back to the finite field hypergeometric sum. (cid:3) Step 3: Count points when at least one coordinate is zero.
Lemma B.3.16.
Let q be an odd prime that is not . Then (a) If q ≡ , then X L L ,ψ ( F q ) − U L L ,ψ ( F q ) = 4 q. (b) If q ≡ , then X L L ,ψ ( F q ) − U L L ,ψ ( F q ) = 12 q − . Proof.
Suppose that x = 0 and the rest are nonzero. Then, by using Theorem 3.3.3, we can seethat there are ( q − − q × / + 1) such points. Since there are four choices of one coordinate beingzero, this counts q − − q × / + 1) points.Suppose now that x = x = 0 and the rest nonzero, then by Theorem 3.3.3 again, we have (( − q × / + 1) points. By symmetry, this is the same as the case where x = x = 0 and the restnonzero, so we now count − q × / + 1) .Next, suppose x = x = 0 and the rest nonzero. Automatically, the polynomial vanishes, hencethere are q − such points. There are 4 such cases from choosing one of x and x and anotherfrom x and x to equal zero, hence we count q − points. Finally, the four points where threecoordinates are zero are all solutions, hence we count 4 more points. Thus X L L ,ψ ( F q ) − U L L ,ψ ( F q ) = 4( q − − q × / + 1) + 2(( − q × / + 1) + 4( q −
1) + 4 . If q ≡ , then ( − q × / = − , so X L L ,ψ ( F q ) − U L L ,ψ ( F q ) = 4 q . If q ≡ ,then ( − q × / = 1 , so X L L ,ψ ( F q ) − U L L ,ψ ( F q ) = 12 q − . (cid:3) Step 4: Combine Steps 2 and 3 to reach the conclusion.Proof of Proposition 3.6.2. If q then, by Lemmas B.3.5 and B.3.16, we have that(B.3.17) X L L ,ψ ( F q ) = ( q − q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − )) + 4 q = q + q + 1 + H q ( , , ; 0 , , | ψ − ) − qH q ( , ; 0 , | ψ − ) . If q ≡ then, by Lemmas B.3.5 and B.3.16, we have that X L L ,ψ ( F q ) = q − q + 5 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − )+ 2( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) + 12 q − q + 9 q + 1 + H q ( , , ; 0 , , | ψ − ) + qH q ( , ; 0 , | ψ − ) (B.3.18) + 2( − q × / ω ( ψ ) q × / qH q ( , , , ; 0 , , , | ψ − ) . (cid:3) B.4.
The pencil L .Proposition B.4.1. The number of F q points on L for q odd is given in terms of hypergeometricfunctions as follows. (a) If q , then X L ,ψ ( F q ) = q + 3 q + 1 + H q ( , , ; 0 , , | ψ − ) . (b) If q ≡ , then X L ,ψ ( F q ) = q + 3 q + 1 + H q ( , , ; 0 , , | ψ − ) + 4 qH q ( , , , ; 0 , , , | ψ ) . Remark
B.4.2 . As before, we can identify the parameters of the hypergeometric function H q ( , , , ; 0 , , , | ψ ) with the parameters of the second Picard–Fuchs equation in Proposition A.3.2. If we use Theorem3.4 of [BCM15] again to shift parameters, then we see that in fact all of the Picard–Fuchs equationssatisfied by the non-holomorphic periods correspond to this same hypergeometric motive over Q .Also notice that in the discussion following Proposition A.3.2, we see two periods that are “missed"by the Griffiths–Dwork method, and here they clearly correspond to the two additional trivial factorscoming from the q term in the point count. Step 1: Computing and clustering the characters.
Again, we compute the solutions to the systemof congruences given by Theorem 3.3.3. We obtain(a) If q , the solution set is(i) the set S = { k (1 , , , , −
4) : k ∈ Z /q × Z } .(b) If q ≡ , then clusters of solutions are(i) the set S = { k (1 , , , , −
4) : k ∈ Z /q × Z } and(ii) four sets of the form S = { k (1 , , , , −
4) + q × (1 , , , ,
0) : k ∈ Z /q × Z } . Step 2: Counting points on the open subset with nonzero coordinates.
Lemma B.4.3.
Suppose ψ ∈ F × q . For q odd, we have: (a) If q , then (B.4.4) U L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − ) . (b) If q ≡ , then (B.4.5) U L ,ψ ( F q ) = q − q + 3 + H q ( , , ; 0 , , | ψ − ) + 4 qH q ( , , , ; 0 , , , | ψ ) . Proof.
When q , we know that there is only one cluster of characters, S . ByLemma 3.4.4, we know that(B.4.6) U L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) . When q ≡ we have two types of clusters of characters. By Lemmas 3.4.4 and B.4.8, U L ,ψ ( F q ) = X s ∈ S ω ( a ) − s c s + 4 X s ∈ S ω ( a ) − s c s = q − q + 3 + H q ( , , ; 0 , , | ψ − ) (B.4.7) + 4 qH q ( , , , ; 0 , , , | ψ ) . (cid:3) e now just need a hypergeometric way to write the point count associated to the cluster S . Lemma B.4.8. If q ≡ and q is odd then (B.4.9) X s ∈ S ω ( a ) − s c s = qH q ( , , , ; 0 , , , | ψ ) Proof.
By using the hybrid hypergeometric definition, this equality is found quickly:(B.4.10) X s ∈ S ω ( a ) − s c s = 1 qq × q − X k =0 ω ( − ψ ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( − k )= 1 qq × q − X k =0 ω (4 ψ ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( − k )= qq × q − X k =0 ω (4 ψ ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( k + q × ) q g ( − k )= qq × q − X k =0 ω (4 ψ ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( k + q × ) g ( q × ) g ( q × ) g ( q × ) g ( q × ) g ( − k )= qH q ( , , , ; 0 , , , | ψ ) The last line uses the hybrid definition (Definition 3.2.7) of the hypergeometric function H q ( , , , ; 0 , , , | ψ ) . Even though the hypergeometric function is defined over Q , we can get to the relation much morequickly using the hybrid definition. (cid:3) Step 3: Count points when at least one coordinate is zero.
Lemma B.4.11. If q is odd and not 7, then X L ,ψ ( F q ) − U L ,ψ ( F q ) = 6 q − . Proof.
First, we count the number of rational points when exactly variable equals zero. Withoutloss of generality, assume x = 0 . Then we want solutions of x x + x x = 0 which we can solve for x . Since x is completely determined by x and x , we can normalize x = 1 and see there are exactly q − solutions when only x is zero. By symmetry, this shows that thereare q − solutions when exactly one variable equals zero. If two consecutive variables are zero (say x = x = 0 ) then we then want solutions of the form x x = 0 which implies that a third variableequals zero. Thus there are 4 solutions with 3 variables equaling zero and no solutions when exactlytwo variables equal zero and those variables are consecutive. Lastly, if two non-consecutive variablesare zero then any other solution works. For any pair of non-consecutive variables (of which thereare two), we then have q − solutions. Therefore X L ,ψ ( F q ) − U L ,ψ ( F q ) = 4 q − q −
1) = 6 q − . (cid:3) tep 4: Combine Steps 2 and 3 to find conclusion. We now prove Proposition 3.6.3.
Proof of Proposition 3.6.3.
Combining Lemmas B.4.3 and B.4.11, we have(a) If q , then(B.4.12) X L ,ψ ( F q ) = ( q − q + 3 + H q ( , , ; 0 , , | ψ − )) + (6 q − q + 3 q + 1 + H q ( , , ; 0 , , | ψ − ) . (b) If q ≡ , then(B.4.13) X L ,ψ ( F q ) = ( q − q + 3 + H q ( , , ; 0 , , | ψ − ) + 4 qH q ( , , , ; 0 , , , | ψ )) + (6 q − q + 3 q + 1 + H q ( , , ; 0 , , | ψ − ) + 4 qH q ( , , , ; 0 , , , | ψ ) . This completes the proof. (cid:3)
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