Hypergeometric Properties of Genus 3 Generalized Legendre Curves
aa r X i v : . [ m a t h . N T ] M a y HYPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZEDLEGENDRE CURVES
HEIDI GOODSON
Abstract.
Inspired by a result of Manin, we study the relationship between certain periodintegrals and the trace of Frobenius of genus 3 generalized Legendre curves. We show thatboth of these properties can be computed in terms of “matching” classical and finite fieldhypergeometric functions, a phenomenon that has also been observed in elliptic curves andmany higher dimensional varieties. Introduction
The motivation for this work comes from a particular family of elliptic curves. For λ = 0 , E λ : y = x ( x − x − λ ) . We compute a period integral associated to the Legendre elliptic curve given by integratingthe nowhere vanishing holomorphic 1-form ω = dxy over a 1-dimensional cycle containing λ .This period is a solution to a hypergeometric differential equation and can be expressed asthe classical hypergeometric series π = Z λ dxy = F (cid:18)
12 12 (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) . (1.1)See the exposition in [4] for more details on this.Koike [14, Section 4] showed that, for all odd primes p and λ ∈ Q \ { , } , the traceof Frobenius for curves in this family can be expressed in terms of Greene’s finite fieldhypergeometric function a E λ ( p ) = − φ ( − p · F (cid:18) φ φǫ (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) p , (1.2)where ǫ is the trivial character and φ is a quadratic character modulo p .Note the similarity between the period and trace of Frobenius expressions: the period isgiven by a classical hypergeometric series whose arguments are the fractions with denomi-nator 2 and the trace of Frobenius is given by a finite field hypergeometric function whosearguments are characters of order 2. This similarity is to be expected for curves by thefollowing result of Manin. Theorem 1.1. [16, Theorem 2]
The rows of the Hasse-Witt matrix satisfy every differentialrelation which is satisfied by the elements of a basis of the space of differentials of the firstkind (dual to a canonical basis of H ( M, O ) ) modulo the space of complete differentials. For elliptic curves, the Hasse-Witt matrix has a single entry: the trace of Frobenius. InCorollary 3.2 of [9], we show that the matching finite field and classical F hypergeometricexpressions in Equations 1.1 and 1.2 are congruent modulo p for odd primes. This resultwould imply merely a congruence between the finite field hypergeometric function expressionand the trace of Frobenius. The fact that Koike showed that we actually have an equality isvery intriguing and leads us to wonder for what other varieties this type of equality holds.Further examples of a correspondence between arithmetic properties of varieties and finitefield hypergeometric functions have been observed for algebraic curves and for Calabi-Yaumanifolds. For example, Fuselier [8] gave a finite field hypergeometric trace of Frobeniusformula for elliptic curves with j -invariant t , where t ∈ F p \ { , } . Lennon [15] extendedthis by giving a hypergeometric trace of Frobenius formula that does not depend on theWeierstrass model chosen for the elliptic curve. In [1], Ahlgren and Ono gave a formulafor the number of F p points on a modular Calabi-Yau threefold. We extended this work in[9, 10] by showing that the number of points on Dwork hypersurfaces over finite fields canbe expressed in terms of Greene’s finite field hypergeometric functions.In this paper we examine the connection between analytic and arithmetic properties ofalgebraic curves. We approach this story from two directions. First, in Sections 3 and 4, wedevelop tools that are needed to understand Manin’s statement in Theorem 1.1. Then, inSection 5, we apply Manin’s theory to the family of genus 3 generalized Legendre curves C λ : y = x ( x − x − λ ) . In this example, we see Manin’s theory in action since we obtain “matching” hypergeometricexpressions for the analytic and arithmetic data associated to these curves. We begin withbackground information in Section 2.2.
Preliminaries
In this section we recall definitions and properties of F classical and finite field hypergeo-metric functions. See, for example, [9, 11, 22] for extensions of this work to n +1 F n generalizedhypergeometric functions.We define the classical hypergeometric series by F (cid:18) a bc (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = ∞ X k =0 ( a ) k ( b ) k ( c ) k k ! x k , (2.1)where ( α ) = 1 and ( α ) k = α ( α + 1)( α + 2) . . . ( α + k − − abF + ( c − ( a + b + 1) x ) ddx F + x (1 − x ) d dx F = 0 (2.2)(see, for example, [22, Section 1.2]).Unless either a or b is a negative integer, classical hypergeometric series have an infinitenumber of terms. In some cases, for example when considering congruences or supercongru-ences, we may only need to consider a finite number of these terms. For a positive integer, YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 3 m , we define the hypergeometric series truncated at m to be F (cid:18) a bc (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) tr( m ) = m − X k =0 ( a ) k ( b ) k k !( c ) k x k . (2.3)In his 1987 paper [11], Greene introduced a finite field, character sum analogue of classicalhypergeometric series. Let F q be the finite field with q elements, where q is a power of an oddprime p . If χ is a multiplicative character of c F × q , extend it to all of F q by setting χ (0) = 0.For any two characters A, B of c F × q we define the normalized Jacobi sum by (cid:18) AB (cid:19) := B ( − q X x ∈ F q A ( x ) B (1 − x ) = B ( − q J ( A, B ) , (2.4)where J ( A, B ) = P x ∈ F q A ( x ) B (1 − x ) is the usual Jacobi sum.For any positive integer n and characters A, B, C in c F × q , Greene defined the finite fieldhypergeometric function F over F q by F (cid:18) A BC (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) q = qq − X χ (cid:18) Aχχ (cid:19)(cid:18)
BχCχ (cid:19) χ ( x ) . (2.5)An alternate definition, which is in fact Greene’s original definition, is given by F (cid:18) A BC (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) q = ǫ ( x ) BC ( − q X y B ( y ) BC (1 − y ) A (1 − xy ) . (2.6)Greene’s finite field hypergeometric functions were defined independently of those by Katz[12] and McCarthy [18], though relations between them have been demonstrated in [18].Greene shows that defining finite field hypergeometric functions in this way leads to manytransformation properties that mirror those of classical series. For example, classical F hypergeometric series satisfy the following identity [22, p. 48] F (cid:18) − m bc (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) = ( b ) m ( c ) m ( − x ) m F (cid:18) − m − c − m − b − m (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) The analogous statement for finite field hypergeometric functions is as follows
Theorem 2.1. [11, Theorem 4.2, ii]
For characters
A, B, C of F q and x ∈ F × q , F (cid:18) A BC (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) q = ABC ( − A ( x ) F (cid:18) A ACAB (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) q Note that these identities can be generalized to n +1 F n classical and finite field hypergeo-metric functions for n >
1. See Section 4 of [11] for other transformation and summationtheorems.In addition to having analogous transformation properties, “matching” classical and finitefield hypergeometric functions have also been shown to be congruent modulo p in manycases. The following theorem will be referenced in our discussion in Section 5.3. YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 4
Theorem 2.2. [9, Theorem 3.1]
Let m and d be integers with ≤ m < d . If p ≡ d ) and T is a generator for the character group c F × p then, for x = 0 , F (cid:18) md d − md (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) tr ( p ) ≡ − p F (cid:18) T mt T mt ǫ (cid:12)(cid:12)(cid:12)(cid:12) x (cid:19) p (mod p ) , where t = p − d . This builds on supercongruence results of Mortenson [19, 20] by considering hypergeomet-ric functions evaluated away from 1, though this result holds mod p instead of p . Furthercongruences and supercongruences between classical and finite field hypergeometric functionscan be found in [5, 9]. 3. Using the Lefschetz Number
The Lefschetz number associated to a map from a manifold to itself essentially keeps trackof the number of fixed points of the map. Let f : M → M be a differentiable map on thecompact differentiable manifold M such that the graph of f meets the diagonal transversely.Then the Lefschetz number L ( f ) can be computed in two ways: L ( f ) = X p ∈ M σ p ( f ) = ∞ X n =0 ( − n tr[ f ∗ : H n ( M, C ) → H n ( M, C )] , (3.1)where σ p ( f ) = (cid:26) f ( p ) = p ± f ) meets diagonal with positive/negative orientation . When the map f is the Frobenius map on a curve, then L ( f ) measures the number ofpoints on the curve over a finite field F q . In this field we have ( x, y ) = ( x q , y q ), so that anypoint on the curve will be a fixed point of the map.We will rewrite both expressions for the Lefschetz number in order to show the relation-ship between the period associated to a curve and its point count. We follow the work ofClemens [4, Chapter 2].We start by rewriting P p ∈ M σ p ( f ). Let J p ( f ) be the Jacobian of f at the point p . Thetransversality of f at p implies that (identity − f ) has maximal rank at p . This is the rankof I − J p ( f ) at the point p , which is a matrix that gives us information about the orientationof the map f . Thus, we can write σ p ( f ) = sign det ( I − J p ( f )). Clemens shows that thisdeterminant can also be expressed asdet ( I − J p ( f )) = n X r =0 ( − r tr( ∧ r J p ( f )) (3.2)so that we can write X p ∈ M σ p ( f ) = X p,r ( − r tr( ∧ r J p ( f )) | det ( I − J p ( f )) | . Denote the restrictions of J p ( f ) to type (1,0) (holomorphic) and type (0,1) (anti-holomorphic)parts of J p ( f ) by J ′ p ( f ) and J ′′ p ( f ), respectively. Clemens notes that if the manifold M is YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 5 a K¨ahler manifold then we can replace the de Rham complex by the Dolbeault complex on M . Thus Equation 3.1 becomes X p,r ( − r tr( ∧ r J ′′ p ( f )) | det ( I − J p ( f )) | = ∞ X n =0 ( − n tr[ f ∗ | H n ( M, O ) ] , (3.3)We also have that X r ( − r tr( ∧ r J ′′ p ( f )) = det ( I − J ′′ p ( f )) , as we did in Equation 3.2, anddet ( I − J p ( f )) = det ( I − J ′ p ( f )) det ( I − J ′′ p ( f )) . Hence, X p,r ( − r tr( ∧ r J ′′ p ( f )) | det ( I − J p ( f )) | = X p det ( I − J ′′ p ( f )) | det ( I − J ′ p ( f )) det ( I − J ′′ p ( f )) | = X p | det ( I − J ′ p ( f )) | = X p fixed | det ( I − J ′ p ( f )) | . Thus, P σ p ( f ) can be expressed in terms of the holomorphic part of J p ( f ).We specialize to the case where f is the Frobenius map and the manifold is an algebraiccurve C . Note that J p ( f ) = 0 since d ( x p ) /dx = px p − = 0 in F p . Hence, | det ( I − J ′ p ( f )) | = 1and P p ∈ C σ p ( f ) = the number of fixed points of f . Since f is a map on C and x p = x if andonly if x ∈ F p , the number of fixed points of f will be exactly the number of F p -points on C plus the point at infinity. Thus, X p ∈ C σ p ( f ) = 1 + the number of F p -points on C. We now rewrite the expression P ∞ n =0 ( − n tr[ f ∗ | H n ( C, O ) ] for the case we are considering.Recall that, in general, H n ( M, O ) = 0 whenever n > dim( M ). Equation 3.3 then becomes1 + the number of F p -points on C = 1 − tr[ f ∗ | H ( C, O ) ] , i.e. the number of F p -points on C = − tr[ f ∗ | H ( C, O ) ] . (3.4)In Section 4 we will see that the right-hand-side of this equation is related to the periodsof an algebraic curve. 4. The Hasse-Witt Matrix
In this section, we piece together the work of [4] and [16]. Let g be the genus of thealgebraic curve C . The Hasse-Witt matrix of C is the g × g matrix of the Frobenius mapwith respect to a basis of regular differentials of the first kind. Thus, the trace of this matrixwill give us tr[ f ∗ | H ( C, O ) ] (trace is independent of basis). In this section we aim to describethe Hasse-Witt matrix in greater detail. YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 6
The genus g of the curve is equal to both the dimension of the space H ( C, O ) of 1-cyclesand the dimension of the space of regular 1-forms on C . We will choose dual bases for thesetwo spaces (dual with respect to a residue pairing). Let P , . . . , P g be a set of distinct pointson C such that the divisor D = P P i is nonspecial. It is noted by Manin [16, Section 1.5]that we may identify H ( C, O ) with the space of functions that have poles at worst at thepoints P , . . . , P g . Thus, we can choose a basis h , . . . , h g for H ( C, O ), where each h i is afunction with a simple pole at P i and no other poles (except at infinity). Thus, h i has Taylorseries expansion, h i = 1 x − x i + X l ≥ c i,l ( x − x i ) l , where P i = ( x i , y i ) is a point on C as above [4, Section 2.12]. Similarly, to each point P i = ( x i , y i ) we can associate a differential ω i , which is to say we can write ω i locally at thepoint P i : ω i = dx + X r ≥ a i,r ( x − x i ) r dx. The bases { ω i } i and { h i } i are dual with respect to the pairing ( ω i , h j ) = Res( h j ω i , P i ), theresidue at P i , since Res( h j ω i , P i ) = (cid:26) i = j i = j. Let K be the matrix of scalar products [( ω i , h j )]. We can write the Hasse-Witt matrix H as H = KH = [( ω i , f ∗ h j )] , where the map f ∗ sends each h i ( x ) to h i ( x p ) = 1( x − x i ) p + X l ≥ b i,l ( x − x i ) pl . Thus, tr[ f ∗ | H ( C, O ) ] = P gi =1 ( ω i , f ∗ h i ). In fact we can say even more about this matrix. Notethat if i = j then ( ω i , f ∗ h j ) = Res( f ∗ h j ω i , P i ) = 0since h j , and therefore f ∗ h j , is holomorphic at the point P i . Thus, the Hasse-Witt matrix isa diagonal matrix with this choice of basis.These diagonal entries can be expressed in terms of coefficients in the expansions of thedifferentials. We have that Res( f ∗ h j ω i , P i ) is the coefficient of 1 / ( x − x i ) in the expansion f ∗ h i ω i = x − x i ) p + X l ≥ b i,l ( x − x i ) pl ! X r ≥ a i,r ( x − x i ) r ! dx. Thus, ( ω i , f ∗ h i ) = a i,p − , so that tr[ f ∗ | H ( C, O ) ] = P gi =1 a i,p − . YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 7 Generalized Legendre Curves
We now apply this theory to a particular family of curves. We look at a specific case ofgeneralized Legendre curves given by C λ : y = x ( x − x − λ ) . When viewed as a projective curve, it is given by the homogeneous equation Y = ZX ( X − Z )( X − λZ )by sending ( x, y ) → ( X/Z, Y /Z ). When written in this form we see that the curve isnonsingular in P . Thus, by a well-known genus formula for nonsingular curves, the genusof C λ is g = (4 − − = 3.5.1. Period Computation.
In this section we give formulas for certain period integralsassociated to genus 3 generalized Legendre curves. The periods we are interested in areobtained by choosing dual bases of the space of holomorphic differentials and the space ofcycles H ( C λ , O ) and integrating the differentials over each cycle. Note that Barman andKalita developed a hypergeometric formula for one of these period integrals in [3] usingtrigonometric substitution.Using the method described in [2, Section 2] yields the following basis for the space ofdifferentials (cid:26) ω = xdxy , ω = dxy , ω = dxy (cid:27) . Theorem 5.1.
The periods of the genus 3 generalized Legendre curve are π = F (cid:18) / / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) , π = F (cid:18) / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) , π = F (cid:18) / / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) . Proof.
As noted in Section 4, we can write each ω i locally at a distinct point P i on the curve.We compute the periods π , π , π of C λ by integrating each differential ω i over a cycle in H ( C λ , O ) that contains the point P i and not the other P j . Such a cycle exists since thechosen points are distinct.We follow the work of Clemens [4, Section 2.10] to find differential equations satisfied bythe periods and then give combinatorial expressions for them. We show the computation for π and omit the work for the remaining periods. Starting with the differential ω = dxy = ( x ( x − x − λ )) − / dx, we take derivatives with respect to λ to get ∂∂λ (( x ( x − x − λ )) − / ) = − x − / ( x − − / ( x − λ ) − / ∂ ∂λ (( x ( x − x − λ )) − / ) = x − / ( x − − / ( x − λ ) − / . YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 8
We wish to find a linear combination of ω and its derivatives that gives an exact differ-ential. To do this, we rewrite the following differential d (cid:18) x / ( x − / ( x − λ ) / ( x − λ ) (cid:19) = d (cid:0) x / ( x − / ( x − λ ) − / (cid:1) = (cid:20) x − / ( x − / ( x − λ ) − / + 14 x / ( x − − / ( x − λ ) − / − x / ( x − / ( x − λ ) − / (cid:21) = 13 ( x − dω dλ + 13 x dω dλ − x ( x − d ω dλ = − ω − λ + 1) dω dλ − λ ( λ − d ω dλ . By integrating both sides and then multiplying by 3/4, we see that π satisfies F π = 0,where F = − / − λ ) ddλ + λ (1 − λ ) d dλ . (5.1)Note that this is a hypergeometric differential equation. We solve for a, b, c in Equation2.2 and find that a = 3 / , b = 5 / c = 3 /
2. This gives us the followingexpression for the period π = F (cid:18) / / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) . Similarly, we find that π satisfies F π = 0, where F = −
316 + (1 / − λ ) ddλ + λ (1 − λ ) d dλ (5.2)and can be expressed as π = F (cid:18) / / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) , and that π satisfies F π = 0, where F = −
14 + (1 − λ ) ddλ + λ (1 − λ ) d dλ (5.3)and can be expressed as π = F (cid:18) / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) . (cid:3) Point Count.
In this section we will compute the number of points on the curve C λ in two ways. We first specify the work in Section 4 to the curve C λ . Then, we will computethe number of points using character sums.Recall Manin’s result (see Theorem 1.1 of this paper) that tells us that the rows of theHasse-Witt matrix satisfy every differential equation satisfied by the periods of a curve. InSection 4 we specifically chose bases for the spaces of differentials and cycles that were dualto each other, which results in a diagonal Hasse-Witt matrix. Thus, the sum of the rows ofthe Hasse-Witt matrix is exactly the trace of the matrix, which in this case is the trace of YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 9
Frobenius. Hence, the trace of Frobenius must satisfy the same differential equations as theperiods.Moreover, since the space of differentials is 3-dimensional and we have developed differen-tial equations for three C -linearly independent periods, it must be the case that the trace ofFrobenius is a C -linear combination of the periods:tr[ f ∗ | H ( M, O ) ] ≡ X i =1 c i,p ( λ ) π i (mod p ) , where each c i,p ( λ ) ∈ C may depend on the order p of the field and on the parameter λ of thecurve. Using Equation 3.4 we can conclude thatnumber of F p -points on C λ ≡ − X i =1 c i,p ( λ ) π i (mod p ) , which we showed in Section 5.1 is C -linear combination of classical hypergeometric series.In fact each of the classical hypergeometric series are congruent to truncated series when wereduce mod p . Thus, we have proved the following theorem. Theorem 5.2. C λ ≡ − c ,p ( λ ) F (cid:18) / / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) tr ( p ) − c ,p ( λ ) F (cid:18) / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) tr ( p ) − c ,p ( λ ) F (cid:18) / / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) tr ( p ) (mod p ) , where C λ is the number of F p -points plus the point at infinity and c i,p ( λ ) ∈ C . This gives us the number of points on the curve modulo the order of the field we areworking over. We have not solved for the coefficients c i,p ( λ ), though it is perhaps possibleto using methods similar to Clemens’ exposition on Legendre elliptic curves in [4, Section2.11]. Rather than go through this computation, we instead compute the exact number ofpoints on C λ using character sums. Theorem 5.3.
Let q be a prime power such that q ≡ . Let T ∈ c F × q be a generatorof the character group and let ψ = T q − . Then C λ = q + 1 + qǫ ( λ ) X m =1 ψ m ( − · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q . Remark.
This result may follow from [5, Theorem 11], though our equation for the generalizedLegendre curve is written in a slightly different form. In [5], the genus 3 generalized Legendrecurve is written as y = x (1 − x )(1 − λx ) . The resulting point count formulas are identical, so we should be able to find a transformationbetween the two curves.
YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 10
Proof.
To prove the result, we first express the number of points as a sum of characters overthe finite field F q . C λ ( F q ) = X x ∈ F q (cid:8) y ∈ F q (cid:12)(cid:12) y = x ( x − x − λ ) (cid:9) + 1= X x ∈ F q −{ , ,λ } X m =0 ψ m ( x ( x − x − λ )) ! + 1 + 3= X x ∈ F q −{ , ,λ } ǫ ( x ( x − x − λ )) + X x ∈ F q −{ , ,λ } X m =1 ψ m ( x ( x − x − λ )) ! + 4= q − X x ∈ F q −{ , ,λ } X m =1 ψ m ( x ( x − x − λ )) ! + 4For each m we have X x ∈ F q −{ , ,λ } ψ m ( x ( x − x − λ )) = X x ∈ F q −{ , ,λ } ψ m ( x ) ψ m ( x − ψ m ( x − λ ) . We work to rewrite the summand and get= X x ∈ F q −{ , ,λ } ψ m ( x ) ψ m (1 − x ) ψ m ( λ − x ) ψ m ( − ψ m ( − X x ∈ F q −{ , ,λ } ψ m ( x ) ψ m (1 − x ) ψ m (1 − λ x ) ψ m ( λ )= ψ m ( λ ) X x ∈ F q −{ , ,λ } ψ m ( x ) ψ m (1 − x ) ψ m (1 − λ x )= ψ m ( λ ) X x ∈ F q −{ , ,λ } ψ m ( x ) ψ − m ψ m (1 − x ) ψ m (1 − λ x ) , which we recognize as being the hypergeometric function expression= ψ m ( λ ) · qǫ (cid:0) λ (cid:1) ψ m ( − · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q = ψ m ( − λ ) · qǫ (cid:0) λ (cid:1) · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q . YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 11
We use Theorem 2.1 to write F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q = ψ − m + m +2 m ( − ψ m (cid:18) λ (cid:19) · F (cid:18) ψ − m ψ − m +2 m ψ − m − m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q = ψ m (cid:18) λ (cid:19) · F (cid:18) ψ − m ψ m ψ − m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q = ψ m (cid:18) λ (cid:19) · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q . Thus, for each m we have ψ m ( − λ ) · qǫ (cid:0) λ (cid:1) · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q = ψ m ( − λ ) · qǫ (cid:0) λ (cid:1) · ψ m (cid:18) λ (cid:19) · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q = q · ψ m ( − ǫ ( λ ) · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q . Putting this back into the formula for C λ gives C λ ( F q ) = q − X m =1 q · ψ m ( − ǫ ( λ ) · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q + 4= q + 1 + qǫ ( λ ) X m =1 ψ m ( − · F (cid:18) ψ − m ψ m ψ m (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q . (cid:3) Period - Point Count Connection.
We notice two phenomena here that also occurwhen one computes periods and point counts for Legendre elliptic curves. The first is that,remarkably, the number of points on the curve can be expressed in terms of finite field hy-pergeometric functions with input given by λ . In fact we get equality, not just a congruence,between the number of points and a finite field hypergeometric expression. This phenomenonalso occurs for families of curves not expressible in Legendre form over Q (see, for example,[7, 8, 15]). In fact, this phenomenon seems to extend to some higher dimensional Calabi-Yaumanifolds as is shown in [1, 9, 10, 17, 21] leading us to wonder if this will be the case for alarge class of algebraic varieties.The second phenomenon is that in computing the point count in two different ways, we geta congruence between classical and finite field hypergeometric expressions. We can say a bitmore on this: it seems as though we can identify congruences between particular summandsthat “match”. For example, we saw in Theorem 5.1 that one period of the curve C λ can beexpressed as π = F (cid:18) / / (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) . YPERGEOMETRIC PROPERTIES OF GENUS 3 GENERALIZED LEGENDRE CURVES 12
We saw in Theorem 5.3 that one of the summands in the point count for C λ is qT q − ( − F (cid:18) T q − T q − ǫ (cid:12)(cid:12)(cid:12)(cid:12) λ (cid:19) q . Note that since q ≡ T q − ( −
1) = 1. Theorem 2.2 then tells us that thisexpression is congruent modulo p to the negative of the classical hypergeometric series π .The classical and finite field hypergeometric expressions – including the two that are notcovered by Theorem 2.2 – for the generalized Legendre curves “match” in the same way thatperiod and trace of Frobenius expressions match for elliptic curves: we replace the fraction ab with a character of order b raised to the a th power. This phenomenon also seems to extendto some other curves (see [19]) and to higher dimensional Calabi-Yau manifolds (see, forexample, [9, 10, 13, 17, 20]. By testing values in Sage [6], we know that it is not the casethat congruences exist between arbitrary (matching) truncated hypergeometric series andfinite field hypergeometric functions. This leads us to wonder when we can expect to have acongruence between these two types of series. References [1] Scott Ahlgren and Ken Ono. A Gaussian hypergeometric series evaluation and Ap´ery number congru-ences.
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