A Cubic Transformation Formula for Appell-Lauricella Hypergeometric Functions over Finite Fields

0.23 MB·

by  Sharon Frechette

A CUBIC TRANSFORMATION FORMULA FOR
APPELL-LAURICELLA HYPERGEOMETRIC FUNCTIONS
OVER FINITE FIELDS
SHARONFRECHETTE, HOLLYSWISHER,ANDFANG-TINGTU
7
1 Abstract. Wedefineafinite-fieldversionofAppell-Lauricellahypergeometricfunctionsbuiltfrom
0 period functions in several variables, paralleling the development by Fuselier, et. al [14] in the
2
single variable case. We develop geometric connections between these functions and the family

n of generalized Picard curves. In our main result, we use finite-field Appell-Lauricella functions
a to establish a finite-field analogue of Koike and Shiga’s cubic transformation [18] for the Appell
J hypergeometric function F1, proving a conjecture of Ling Long. We use our multivariable period

9 functions to construct formulas for the number of Fp-points on the generalized Picard curves. We
1 also give some transformation and reduction formulas for the period functions, and consequently
for the finite-fieldAppell-Lauricella functions.
]
T
N
.
h
t 1. Introduction and Statement of Results
a
m
Classical hypergeometric functions are among the most versatile of all special functions. These
[
functions and their finite-field analogues have numerous applications in number theory and geom-

2 etry. For instance, finite-field hypergeometric functions play a role in proving congruences and
v supercongruences, they count points modulo p over algebraic varieties and affine hypersurfaces,
6
and in certain instances they provide formulas for the Fourier coefficients of modular forms.
2
(n)
5 In this paper, we define functions F as a finite-field version of the Appell-Lauricella hyperge-
D
4 (n)
ometric functions F . We develop the theory of these functions, with a focus on their geometric
0 D
. connections to the generalized Picard curves. This parallels the construction (by the second and
1
0 third authors, et. al.) in [14], which was an effort to unify and improve on the interplay between
7 classical and finite-field hypergeometric functions in the single-variable setting.
1
Our results are motivated by a conjecture of Ling Long, related to results by Koike and Shiga.
:
v In [18], Koike and Shiga applied Appell’s F -hypergeometric function in two variables to establish
1
i
X a new three-term arithmetic geometric mean result (AGM), related to Picard modular forms. As
a consequence of this cubic AGM, Koike and Shiga proved the following cubic transformation for
r
a Appell’s F -function. Let x,y C, and let ω be a primitive cubic root of unity. Then
1
∈
1 1 1
(1) F ; , ; 1 1 x3, 1 y3
1
3 3 3 − −
(cid:20) (cid:12) (cid:21)
(cid:12) 3 1 1 1 1+ωx+ω2y 3 1+ω2x+ωy 3
(cid:12)
= F ; , ; 1 , .
1
1+x+y 3 3 3 1+x+y 1+x+y
" #
(cid:12) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:12)
As an application of Appell-Lauricella functions over fi(cid:12)nite fields, we prove the following finite-field
analogue of Koike and Shiga’s transformation, as conjectured by Ling Long.
2010 Mathematics Subject Classification. 11F24, 33C05, 33C70, 11T20.
Key words and phrases. Hypergometric functions, finitefield.
1
Theorem 1.1. Let p 1 (mod 3) be prime, let ω be a primitive cubic root of unity, and let η be
3
≡
a primitive cubic character in F×. If λ,µ F satisfy 1+λ+µ = 0, then
p p
∈ 6
η ; η η η ; η η 1+ωλ+ω2µ 3 1+ω2λ+ωµ 3
F(2) 3 3 3 ; 1 λ3,1 cµ3 = F(2) 3 3 3 ; , .
D ε − − D ε 1+λ+µ 1+λ+µ
" #
(cid:20) (cid:21) (cid:18) (cid:19) (cid:18) (cid:19)
When λ = µ, we have the following corollary.
Corollary 1.2. For p 1 (mod 3) prime, and ω as above, if λ F satisfies 1+2λ = 0, then
p
≡ ∈ 6
η η2 η η2 1 λ 3
F 3 3 ; 1 λ3 = F 3 3 ; − .
2 1 2 1
ε − ε 1+2λ
" #
(cid:20) (cid:21) (cid:18) (cid:19)
The result of Corollary 1.2 was first established in [14], using a different method of proof. It is
a finite-field version of the cubic transformation
1 2 3 1 2 1 x 3
(2) F 3 3 ; 1 x3 = F 3 3 ; − ,
2 1 2 1
1 − 1+2x 1 1+2x
" #
(cid:20) (cid:21) (cid:18) (cid:19)
proved by Borwein and Borwein [9], [10] for x R with 0 < x < 1, as a cubic analogue of Gauss’
∈
quadratic AGM.
Taking the approach used in [14], our finite-field Appell-Lauricella hypergeometric functions are
(n)
defined in terms of finite-field period functions P . These period functions are naturally related
D
to periods of the generalized Picard curves
(3) C[N;i,j,k] : yN = xi(1 x)j(1 λ x)k1 (1 λ x)kn,
λ − − 1 ··· − n
defined for distinct complex numbers λ ,...,λ = 0,1 and positive integers N,i,j,k ,...,k with
1 n 1 n
6
(n)
gcd(N,i,j,k ,...,k ) = 1 and N ∤ i + j + k + + k . As a consequence, the P functions
1 n 1 ··· n D
are ideally suited for counting F -points on Picard curves; in Theorem 5.3, we express these point
p
(n)
counts in terms of finite sums of P values.
D
Transformation and reduction formulas for classical hypergeometric functions have been success-
fully translated to the finite-field setting, first by Greene and also by authors such as McCarthy,
and Fuselier et. al. (See [15], [23], [14] for details.) Transformation formulas for classical Appell-
Lauricella hypergeometric functions can also be translated into the finite-field setting, using the
(n) (n)
samemethods. Wecarryoutthisprocess,provingseveralidentities fortheP -andF -functions.
D D
We note that another version of finite-field Appell-Lauricella functions is independently defined by
He [16] and Li, et. al. [21], which closely follows Greene’s definition. For their version, they estab-
lish several degree 1 transformation and reduction formulas, including some that are analogous to
the identities in this paper.
The paper is organized as follows: Section 2 includes definitions and properties needed for work-
ing with classical and finite-field Appell-Lauricella hypergeometric functions, such as Gamma and
Jacobifunctions,characters over finitefields,andclassicalhypergeometricfunctionsofonevariable.
(n)
In Section 3, we recall the definitions of classical Appell-Lauricella functions F and some of their
D
(n)
transformation properties, and we define period functions P to capture the relationship between
D
Appell-Lauricella functions and Picard curves. Section 4 introduces the finite-field periodfunctions
(n) (n)
P and finite-field Appell-Lauricella functions F , as well as several of their transformation and
D D
reduction formulas. In Section 5, we prove Theorem 5.3 which gives thenumberof F -points on the
p
[N;i,j,k]
generalized Picard curves. We also compute the genus of the generalized Picard curves C in
λ
this section. In Section 6, we give several degree 1 transformation and reduction formulas for the
(n) (n)
P and F functions. Finally, in Section 7, we give the proofs of Theorem 1.1 and Corollary 1.2.
D D
2
2. Preliminaries
In this section, we recall the necessary background for defining and working with classical and
finite-field hypergeometric functions. For further details, we refer the reader to [4], [28], for the
classical setting, and [15], [14] for the finite-field setting.
For a C and n Z , define the Pochhammer symbol (a) by
≥0 n
∈ ∈
1 if n = 0,
(a) :=
n
(a(a+1) (a+n 1) if n 1,
··· − ≥
and for x C with Re(x) > 0, define the Gamma function Γ(x) by
∈
∞
Γ(x) = tx−1e−tdt.
Z0
Γ(a+n)
Note that (a) = for all nonnegative integers n.
n
Γ(a)
For r 0, and parameters a ,a ,...,a , b ,b ,...,b C and x < 1, the classical hypergeo-
1 2 r+1 1 2 r
≥ ∈ | |
metric function F in these parameters is defined by
r+1 r
∞
a a a (a ) (a ) (a )
(4) F 1 2 ··· r+1 ; z := 1 n 2 n··· r+1 n xn.
r+1 r
b b (b ) (b ) (b ) n!
(cid:20) 1 ··· r (cid:21) n=0 1 n 2 n··· r n
X
The study of these versatile functions goes back to the likes of Euler and Gauss, and among
other things, the F -hypergeometric functions occur as solutions to hypergeometric differential
r+1 r
equations and periods of algebraic varieties, among other things.
Finite-field hypergeometric functions in one variable were defined by Greene [15] as analogues
to the classical version. Turning our attention to this setting, let F be a finite field where q = pe,
q
with p an odd prime and e N. Let F× be the group of multiplicative characters on F×. Extend
∈ q q
any character χ on F× to F by defining χ(0) = 0. Let ε denote the trivial character. Following
q q
c
Greene [15], for x F and χ F×, we define
q q
∈ ∈
1c if χ = ε, 1 if x = 0,
(5) δ(χ) := and δ(x) :=
(0 if χ = ε, (0 if x = 0.
6 6
We will frequently have need of the following orthogonality relations for characters:
Lemma 2.1. For all A,B,χ F× and all x F , we have
∈ q ∈ q
(1) A(x)B(x) = (q 1)δ(AB).
−
xX∈Fq
(2) χ(x) = (q 1)δ(1 x).
− −
×
χX∈Fq
For A,B, F×, define the Jacobi sum
q
∈
(6) c J(A,B) := A(x)B(1 x).
−
xX∈Fq
Greene proved the following result, which is a finite field analogue of the binomial theorem:
3
Theorem 2.2 (Theorem 2.3 of [15]). For any character A F× and x F , we have
q q
∈ ∈
1
(7) A(1 x) = δ(x)+ J(A,cχ)χ(x).
− q 1
− χX∈Fc×q
This shows that the Jacobi sum may be viewed as an analogue of the binomial coefficient, so for
A, χ F× we define
q
∈
A
(8) c := χ( 1)J(A,χ)
χ − −
(cid:18) (cid:19)
This definition, given in [14], differs from Greene’s version by a factor of q. With this notation,
−
the result of Theorem 2.2 may be written as
1 A
(9) A(1 x)= δ(x) χ( x).
− − q 1 χ −
− χX∈Fc×q (cid:18) (cid:19)
Greene gives another version of Theorem 2.2 which we require in proving certain tranformation
formulas in Section 6.
Theorem 2.3 (Equation (2.11) of [15]). For all A,B,χ F and all x F , we have
q q
∈ ∈
q Aχ
B(x)AB(1 x) = c χ(x).
− q 1 Bχ
− χX∈Fcq(cid:18) (cid:19)
ConvertingthistheoremtoJacobisums,usingouralternatedefinitionofthebinomialcoefficients,
we obtain the following property
B( 1)
(10) B(x)AB(1 x) = − J(Aχ,Bχ)χ( x).
− q 1 −
− χX∈Fcq
Several identities involving Jacobi sums will also be of frequent use. Note that the binomial coef-
ficient versions of these identities hold for our version as well as Greene’s, since the factors of q
−
will simply cancel.
Lemma 2.4 (See (2.6)–(2.8) in [15]). For any characters A,B,C F×,
q
∈
A A
(1) J(A,B) = A( 1)J(A,BA), or = . c
− B AB
(cid:18) (cid:19) (cid:18) (cid:19)
A BA
(2) J(A,B) = B( 1)J(BA,B), or = B( 1) .
− B − B
(cid:18) (cid:19) (cid:18) (cid:19)
A B
(3) = AB( 1) .
B − A
(cid:18) (cid:19) (cid:18) (cid:19)
(4) J(A,B)J(C,A) = B( 1)J(C,B)J(BC,AB) δ(A)(q 1)+δ(BC)(q 1).
− − − −
3. Classical Appell-Lauricella functions
As generalizations of the classical F -hypergeometric series, Appell [5], [6], [7] introduced four
2 1
two-variable hypergeometric series, each with a different type of coupled Pochhammer symbol
4
coefficients. These four series were later generalized to several variables by Lauricella [20]. We
require the Appell functions of the first type, defined for x < 1, y < 1 by
| | | |
∞
(a) (b ) (b )
(11) F (a; b , b ; c x, y)= m+n 1 m 2 n xmyn.
1 1 2
| (c) m!n!
m+n
m,n=0
X
Lauricella’s series of type D give a natural generalization of F to n variables and are closely
1
related to generalized Picard curves. Following the literature, we refer to these generalizations
as Appell-Lauricella functions, as defined below. For a comprehensive survey of Appell-Lauricella
functions, we refer the reader to the article by Schlosser [27], and to the monograph by Slater [28].
Definition 3.1. For n 2, a,c,b ,b ,...b C, and x < 1,..., x < 1, the Appell-Lauricella
1 2 n 1 n
≥ ∈ | | | |
(n)
function F is given by
D
∞
a; b ... b (a) (b ) (b )
F(n) 1 n ; x ,...,x := i1+···in 1 i1··· n in xi1 xin.
D c 1 n (c) i ! i ! 1 ··· n
(cid:20) (cid:21) i1,.X..,in=0 i1+···in 1 ··· n
(2) (n)
We note that F = F . For consistency we use the F notation throughout this paper.
1 D D
(n)
Parallel to the classical one-variable setting, the F -functions have the following one-variable
D
integral representation, due to Picard when n = 2, and Lauricella in the general case.
Theorem 3.2 (Picard [25], Lauricella [20]). For Re(c) > Re(a) > 0 and x < 1, x < 1,...,
1 2
| | | |
x < 1, we have
n
| |
a; b ... b Γ(c) 1
F(n) 1 n ; x ,...,x = ta−1(1 t)c−a−1(1 x t)−b1 (1 x t)−bndt.
D c 1 n Γ(a)Γ(c a) − − 1 ··· − n
(cid:20) (cid:21) − Z0
(n)
From this we see that when F = F is evaluated at distinct parameters λ ,...,λ = 0 or 1, it
1 D 1 n 6
is naturally related to a period of the generalized Picard curve
C[N;i,j,k] : yN = xi(1 x)j(1 λ x)k1 (1 λ x)kn,
λ − − 1 ··· − n
(n)
as defined in (3). We define the following n-variable period functions P , in order to demonstrate
D
this relationship.
Definition 3.3. For n 2 and arbitrary constants a,c,b ,b ,...b C, the period functions
1 2 n
≥ ∈
(n)
P corresponding to the Appell-Lauricella functions are given by
D
a; b ... b 1
P(n) 1 n ; x ,...,x := ta−1(1 t)c−a−1(1 x t)−b1 (1 x t)−bndt.
D c 1 n − − 1 ··· − n
(cid:20) (cid:21) Z0
The period functions are a suitably-chosen normalization of the classical hypergeometric func-
tions. Specifically, Definition 3.3 immediately implies the following
a; b ... b Γ(a)Γ(c a) a; b ... b
(n) 1 n (n) 1 n
P ; x ,...,x = − F ; x ,...,x .
D c 1 n Γ(c) D c 1 n
(cid:20) (cid:21) (cid:20) (cid:21)
The Appell-Lauricella functions satisfy many transformation and reduction properties that are
analogous to those satisfied by the classical hypergeometric functions. For instance, when n = 2,
we have the following properties which we generalize to the finite-field setting in Section 6.
5
Proposition 3.4 (Analogue of Pfaff-Kummer transformation, (8.3.2) in [28]). For a,b ,b ,c C
1 2
∈
and all x,y for which the series are defined,
a; b b c a; b b x y
F(2) 1 2 ; x,y = (1 x)−b1(1 y)−bb F(2) − 1 2 ; , .
D c − − D c x 1 y 1
(cid:20) (cid:21) (cid:20) − − (cid:21)
Proposition 3.5 (Analogue of Euler’s transformation, (8.3.6) in [28])).
a; b b c a; c b b b x y
F(2) 1 2 ; x, y = (1 x)c−a−b1(1 y)−b2F(2) − − 1− 2 2 ; x, − .
D c − − D c 1 y
(cid:20) (cid:21) (cid:20) − (cid:21)
Proposition 3.6 (Reduction Formulas, (8.3.1.1) in [28]).
a; b b c a c b b
F(2) 1 2 ; x, x = (1 x)c−a−b1−b2 F − − 1− 2 ; x ,
D c − 2 1 c
(cid:20) (cid:21) (cid:20) (cid:21)
a b +b
1 2
= F ; x .
2 1
c
(cid:20) (cid:21)
Note that the second equality in Proposition 3.6 follows immediately from the integral represen-
tation in Theorem 3.2, and Euler’s integral transformation formula for F .
2 1
4. Finite-field Appell-Lauricella functions
(n)
In this section, we definefinite-field analogues of the Appell-Lauricella period functions P and
D
(n)
hypergeometric functions F given in Section 3. This parallels the one-variable construction in
D
[14]. There, theauthors definedperiodfunctions P , and thesewere suitably normalized to give
n+1 n
hypergeometric functions F .
n+1 n
Definition 4.1. For n 2 and A,C,B ,...B F×, we define the Appell-Lauricella period
1 n q
≥ ∈
(n)
functions P over F by
D q
c
A; B B
(n) 1 n
P ··· ; λ ,...,λ := A(y)CA(1 y)B (1 λ y)B (1 λ y) B (1 λ y).
D C 1 n − 1 − 1 2 − 2 ··· n − n
(cid:20) (cid:21) yX∈Fq
Note that this definition is symmetric in the characters B ,...,B . Also note that when n= 1,
1 n
this definition recovers the P -period function in [14], but with the first two parameters reversed.
2 1
That is,
A; B B A
(1)
P ; λ = P ; λ .
D C 2 1 C
(cid:20) (cid:21) (cid:20) (cid:21)
(Although the P -function is not symmetric in A and B, the F -function is symmetric in these
2 1 2 1
(n)
parameters.) Moreover, when n 2, if λ = 0 or 1 for some i, the P -function reduces to a
≥ i D
(n−1)
P -function in the remaining variables. Without loss of generality (by symmetry in the B ), if
D i
λ = 0 then B (1 λ y)= 1, and we have
n n n
−
A; B B A; B B
(n) 1 n (n−1) 1 n−1
P ··· ; λ ,...,λ ,0 = P ··· ; λ ,...,λ .
D C 1 n−1 D C 1 n−1
(cid:20) (cid:21) (cid:20) (cid:21)
Thus when each λ = 0, the period function simply reduces to the Jacobi sum J(A,CA).
i
Similarly, when λ = 1, then B (1 λ y) = B (1 y) and when c = 0 or 1, we therefore have
n n n n
− −
A; B B A; B B
(n) 1 n (n−1) 1 n−1
P ··· ; λ ,...,λ ,1 = P ··· ; λ ,...,λ
D C 1 n−1 D CB 1 n−1
(cid:20) (cid:21) (cid:20) n (cid:21)
Hereafter, we assume that λ = 0 or 1, whenever n 2.
i
6 ≥
6
As is done in the case of the classical Appell-Lauricella hypergeometric functions, we normalize
(n)
the period function P by dividing out its value when all λ = 0, and we define the Appell-
D i
(n)
Lauricella hypergeometric functions F as follows.
D
Definition 4.2. For n 2 and A,C,B ,B ,...B F×, we define the Appell-Lauricella hyper-
1 2 n q
≥ ∈
geometric functions over F by
q
c
A; B B 1 A; B B
(n) 1 n (n) 1 n
F ··· ; λ ,...,λ := P ··· ; λ ,...,λ .
D C 1 n J(A,CA) · D C 1 n
(cid:20) (cid:21) (cid:20) (cid:21)
(n)
We also give expressions for the P -period functions in terms of Jacobi sums.
D
Proposition 4.3 (JacobisumRepresentationsforPeriodFunctions). The following identitieshold.
(1) (Equation 48 in [14]) For A, B, C F×, and l F ,
q q
∈ ∈
A; B A( 1)
P(1) ; λ = − Jc(Bχ,χ)J(Aχ,Cχ)χ(λ)+δ(λ)J(A,CA)
D C q 1
(cid:20) (cid:21) − χX∈Fc×q
AC( 1) Bχ Aχ
= − χ(λ)+δ(λ)J(A,CA)
q 1 χ Cχ
− χX∈Fc×q (cid:18) (cid:19)(cid:18) (cid:19)
(2) For A,B , ,B ,C F× and λ , ,λ F×,
1 ··· n ∈ q 1 ··· n ∈ q
A; B B
P(n) 1 ··· cn ; λ ,...,λ
D C 1 n
(cid:20) (cid:21)
n
A( 1)
= − J(Aχ χ χ ,Cχ χ χ ) J(B χ ,χ )χ (λ )
(q 1)n 1 2··· n 1 2··· n i i i i i
!
− χ1,..X.,χn∈Fc×q Yi=1
n
AC( 1) Aχ χ χ B χ
= ( 1)n+1 − 1 2··· n i i χ (λ )
− (q 1)n Cχ χ χ χ i i
− χ1,..X.,χn∈Fc×q (cid:18) 1 2··· n(cid:19) Yi=1(cid:18) i (cid:19) !
Proof. Applying (7) to the character B, we see that when λ y = 0 we have
i
6
1 1
B (1 λ y)= J(B ,χ )χ (λ y) = J(B χ ,χ )χ ( λ y).
i i i i i i i i i i i
− q 1 q 1 −
− χXi∈Fc×q − χXi∈Fc×q
Using this fact along with Definition 4.1, we have
A; B B
(n) 1 n
P ··· ; λ ,...,λ
D C 1 n
(cid:20) (cid:21)
n
1
= J(Aχ χ χ ,CA) J(B χ ,χ )χ ( λ )
(q 1)n 1 2··· n i i i i − i
!
− χ1,..X.,χn∈Fc×q Yi=1
n
A( 1)
= − J(Aχ χ χ ,Cχ χ χ ) J(B χ ,χ )χ (λ )
(q 1)n 1 2··· n 1 2··· n i i i i i
!
− χ1,..X.,χn∈Fc×q Yi=1
7
The desired identity follows by applying (8). (cid:3)
(2)
The case n = 2 is used often in Section 6, in proving transformation properties for the P -
D
functions. We state it separately here for convenience.
Corollary 4.4. For A,B ,B , C F× and λ , λ F , we have
1 2 q 1 2 q
∈ ∈
A; B B AC( 1) B χ B ψ Aχψ
(2) 1 2 c 1 2
P ; λ ,λ = − − χ(λ )ψ(λ )
D C 1 2 (q 1)2 χ ψ Cχψ 1 2
(cid:20) (cid:21) − χ,Xψ∈Fc×q (cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19)
AC( 1) B χ Aχ
1
+δ(λ ) − χ(λ )
2 1
q 1 χ Cχ
− χX∈Fc×q (cid:18) (cid:19)(cid:18) (cid:19)
AC( 1) B χ Aχ
2
+δ(λ ) − ψ(λ )
1 2
q 1 χ Cχ
− χX∈Fc×q (cid:18) (cid:19)(cid:18) (cid:19)
+δ(λ )δ(λ )J(A,CA).
1 2
5. Geometric Interpretation
(n)
5.1. Generalized Picard curves. The integral representation of the period function P can be
D
naturally viewed in terms of a period of the smooth model of the generalized Picard curve given in
(3) and restated here for convenience
C[N;i,j,k] : yN = xi(1 x)j(1 λ x)k1 (1 λ x)kn.
λ − − 1 ··· − n
The curve C[N;i,j,k] has singularities when x = 0,1, , and 1 , for 1 j n. Let X[N;i,j,k] denote
λ ∞ λi ≤ ≤ λ
its desingularization, or smooth model.
[N;i,j,k]
To compute the genus of X , we utilize the following theorem of Archinard [3].
λ
Theorem 5.1 ([Theorem 4.1 of [3]). Let X be the desingularization of the irreducible projective
N
algebraic plane curve C defined over C by the affine equation
N
r
(12) yN = (x λ )Ai,
i
−
i=0
Y
with λ ,...,λ C such that λ = λ for all i= j 0,...,r . Let N,A ,...,A N satisfy
0 r i j 0 r
∈ 6 6 ∈ { } ∈
r
N = A and gcd(N,A ,...,A ) = 1.
K 0 r
6
k=0
X
Then the Euler characteristic of X (C) is given by
N
r r
χ(X (C)) = rN +gcd(N,N A )+ gcd(N,A ),
N k j
− −
k=0 j=0
X X
and the genus of X by
N
r r
1
(13) g[X ]= (X (C)) = 1+ (rN gcd(N,N A ) gcd(N,A )).
N N k j
2 − − −
k=0 j=0
X X
8
[N;i,j,k]
Our curves X are isomorphic to the curves X given in Theorem 5.1, and so by applying
λ N
[N;i,j,k]
this theorem, we find that the genus of X is given by
λ
n n
[N;i,j,k] 1
(14) g(X ) = 1+ (n+1)N gcd(N,i+j+ k ) gcd(N,i) gcd(N,j) gcd(N,k ) .
λ 2 − j − − − j
(cid:18) j=1 j=1 (cid:19)
X X
In particular, the classical Picard curve, C : y3 = x2(1 x)(1 λ x)(1 λ x), has genus 3.
λ1,λ2 − − 1 − 2
[N;i,j,k]
5.2. Counting points on C over finite fields. Finite-field hypergeometric functions have
λ
been used by many authors in recent years to count points on affine hypersurfaces and algebraic
varieties, with a number of applications. (See [1], [2], [13], [14], [24], [29], [12], among others.)
Here, we use a technique similar to the one used by Fuselier, et. al. in [14] to count points on
generalized Legendre curves, which in turn is based on the point-counting method used by Vega in
[29]. We first require the following well-known result on character sums.
Lemma 5.2 (Proposition 8.1.5 of [17]). Let p be a prime and a F 0 . If n (p 1) then
p
∈ \{ } | −
# x F xn = a = χ(a),
p
{ ∈ | }
χn=ε
X
where the sum runs over all characters χ F× of order dividing n.
p
∈
[N;i,j,k] [N;i,j,k]
Let #C (F ) denote the number of Fc-points on the generalized Picard curve C . We
λ p p λ
have the following formula in terms of Appell-Lauricella period functions.
Theorem 5.3. Let p 1 (mod N)be a prime, and let η F× be a primitive Nth-order character.
N p
≡ ∈
Then
c
#C[N;i,j,k](F ) = 1+p + N−1 P(n) ηNmi; ηN−mki ··· ηN−mkn ; λ ,...,λ .
λ p D mi+mj 1 n
" η #
m=1 N
X
Proof. Let f(x,λ) = xi(1 x)j(1 λ x)k1 (1 λ x)kn. With 1 representing the point at infinity,
1 n
− − ··· −
we have
#C[N;i,j,k](F )= 1+ # y F yN = f(x,λ) ,
λ p { ∈ p | }
xX∈Fp
= 1+# x F f(x)= 0 + # y F yN = f(x,λ) ,
p p
{ ∈ | } { ∈ | }
xX∈Fp
f(x)6=0
N−1
= 1+# x F f(x)= 0 + ηm(f(x,λ)), by Lemma 5.2,
{ ∈ p | } N
xX∈FpmX=0
N−1
= 1+ # x F f(x)= 0 + ε(f(x)) + ηm(f(x,λ)).
{ ∈ p | } N
(cid:20) xX∈Fp (cid:21) xX∈FpmX=1
The terms in the bracket combine to give p as follows: Since ε(0) = 0, then for each x F , either
p
∈
f(x) = 0 and this contributes 1 to the count given by the first term in the bracket, or f(x) = 0
6
9
and this contributes 1 to the count given by the second term. Therefore,
N−1
#C[N;i,j,k](F ) = 1+p+ ηm(f(x,λ)),
λ p N
mX=1xX∈Fp
N−1
= 1+p+ ηm(xi(1 x)j(1 λ x)k1 (1 λ x)kn),
N − − 1 ··· − n
mX=1xX∈Fp
N−1
= 1+p+ ηmi(x)ηmj(1 x)ηmk1(1 λ x) ηmkn(1 λ x),
N N − N − 1 ··· N − n
mX=1xX∈Fp
N−1 ηmi; η−mki η−mkn
= 1+p + P(n) N N ··· N ; λ ,...,λ
D mi+mj 1 n
" η #
m=1 N
X
(cid:3)
[N;i,j,k]
Counting F -points on the smooth model X requires resolving the singularities at 0,1, ,
p λ ∞
and 1 ,..., 1 , and determining the contribution arising from each one. The contributions arising
λ1 λn
from these singularities affect the polynomial part 1+p in the count above, but more importantly,
they do not change the hypergeometric functions that appear. For brevity’s sake, we omit the
details here.
6. Some Degree 1 Reduction and Transformation Formulas
(n) (n)
In this section, we consider the behavior of the P and F functions under transformations in
D D
the variables λ , as well as the simplifications that result when some of the characters are trivial or
i
are repeated. The following results are finite-field analogues of the classical results found in Slater
[28]. This development mirrors the translation of identities for classical hypergeometric functions
to the finite field setting (see [15] and Section 8 of [14]).
6.1. Transformation Formulas. In order to establish these expressions, we first recall some nec-
essaryfactsabouttheonevariableperiodfunctions P definedin[14]. Thefollowingproposition
n+1 n
from [14] is due to Greene (Theorem 4.4 of [15]) using slightly different notation.
Proposition 6.1. [14] For any characters A, B, C F×, and λ F , we have
q q
∈ ∈
A B CcB CA
(15) P ; λ = ABC( 1)C(λ) P ; λ +δ(λ)J(B,CB),
2 1 2 1
C − " C #
(cid:20) (cid:21)
A B A CA 1
(16) P ; λ = ABC( 1)A(λ) P ; +δ(λ)J(B,CB),
2 1 2 1
C − " BA λ#
(cid:20) (cid:21)
A B A B
(17) P ; λ = B( 1) P ; 1 λ .
2 1 2 1
C − ABC −
(cid:20) (cid:21) (cid:20) (cid:21)
(n)
The behavior of P under the transformation λ 1 λ for each i is given in the following
D i 7→ − i
result, analogous to equation (17) in Proposition 6.1.
10