Hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions

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by  Alyson Deines

HYPERGEOMETRIC SERIES, TRUNCATED HYPERGEOMETRIC SERIES,
AND GAUSSIAN HYPERGEOMETRIC FUNCTIONS
ALYSONDEINES,JENNY G. FUSELIER,LINGLONG, HOLLYSWISHER,FANG-TINGTU
5
1 Abstract. Inthispaper,weinvestigatetherelationshipsamonghypergeometricseries, truncated
0 hypergeometricseries, andGaussian hypergeometricfunctionsthroughsomefamilies of‘hypergeo-
2 metric’ algebraic varieties that are higher dimensional analogues of Legendre curves.

p
e
S
3 1. Introduction
2 1.1. Motivation. When a prime p satisfies p 1 (mod 6), the p-adic gamma value Γ 1 3 is a
≡ − p 3
] quadratic algebraic number with absolute value √p which can be written as a Jacobi sum. Thus,
T (cid:0) (cid:1)
Γ 1 6 is not a conjugate of Γ 1 3 in the sense of algebraic numbers [13]. However, considering
N p 3 − p 3
truncated hypergeometric series we have when p 1 (mod 6),
.
h (cid:0) (cid:1) (cid:0) (cid:1) ≡
at (1.1) F 13 31 13 ; 1 := p−1 −13 3 ( 1)k Γ 1 6 (mod p3),
m 3 2 1 1 k · − ≡ p 3
(cid:20) (cid:21)p 1 k=0(cid:18) (cid:19) (cid:18) (cid:19)
[ − X
which was shown by the third author and Ramakrishna in [31], while numerically we see that

2
v 2 2 2 p−1 2 3 1 3
4 (1.2) F 3 3 3 ; 1 := −3 ( 1)k Γ (mod p3),
3 2 p
6 1 1 k · − ≡ − 3
(cid:20) (cid:21)p 1 k=0(cid:18) (cid:19) (cid:18) (cid:19)
5 − X
3 and we will show this holds modulo p2 in this paper. By Dwork [16],
0
. 1 1 1 1 1 1 1 6
1 lim F 3 3 3 ; 1 F 3 3 3 ; 1 = Γ ,
3 2 3 2 p
0 s 1 1 1 1 3
→∞ (cid:20) (cid:21)ps 1 (cid:20) (cid:21)ps 1 1 (cid:18) (cid:19)
5 − . − −
1 while
: 2 2 2 2 2 2 1 3
v lim F 3 3 3 ; 1 F 3 3 3 ; 1 = Γ .
3 2 3 2 p
i s 1 1 1 1 − 3
X →∞ (cid:20) (cid:21)ps 1 (cid:20) (cid:21)ps 1 1 (cid:18) (cid:19)
− . − −
r When p 5 (mod 6), Dwork in [16] showed that there is a similar congruence that involves
a ≡1 1 1 2 2 2
both F 3 3 3 ; 1 and F 3 3 3 ; 1 . It is tempting to think of the param-
3 2 3 2
1 1 1 1
(cid:20) (cid:21)ps 1 (cid:20) (cid:21)ps 1 1
eters 1 and 2 as ‘conjug−ates of some sort’. Also, i−f −one considers the finite field analogue of
3 3
1 1 1
F 3 3 3 ; 1 due to Greene, what corresponds to 1 is a cubic character, which is determined
3 2 1 1 3
(cid:20) (cid:21)
up to a conjugate. Putting these together, it appears that Γ 1 3 is some sort of ‘conjugate’ of
− 3
Γ 1 6. One motivation of this paper is to investigate these seemingly contradicting phenomena
3 (cid:0) (cid:1)
via the relations between hypergeometric series, Gaussian hypergeometric functions and truncated
(cid:0) (cid:1)
hypergeometric series. These objects correspond to periods, Galois representations, and unit roots
(in the ordinary case) respectively.
2010 Mathematics Subject Classification. 33C20, 11T24, 11G25, 14J30.
Key words and phrases. Hypergeometric series, Gaussian hypergeometric functions, truncated hypergeometric
series, Gross-Koblitz formula, supercongruences, Galois representation.
1
2 ALYSONDEINES,JENNYG.FUSELIER,LINGLONG,HOLLYSWISHER,FANG-TINGTU
In recent work [15], the authors use the perspective of Wolfart [43] and Archinard [3] to consider
classical F -hypergeometric functions with rational parameters as periods of explicit generalized
2 1
Legendre curves
C[N;i,j,k] : yN = xi(1 x)j(1 λx)k.
λ − −
In [15], the main players are hypergeometric series and Gaussian hypergeometric functions. The
[N;i,j,k]
authorsuseGaussian F -hypergeometric functionstocountpointsof C over finitefieldsand
2 1 λ
hencecomputethecorrespondingGaloisrepresentations. Thisarithmeticinformationtogetherwith
[N;i,j,k]
the periods of C in terms of hypergeometric values yields information about the decompo-
λ
[N;i,j,k] [N;i,j,k]
sition of the Jacobian variety J constructed from the desingularization of C . When
λ λ
[N;i,j,k]
gcd(i,j,k) is coprime to N and N ∤ i+j +k, then J has a degree 2ϕ(N) ‘primitive’ factor
λ
Jnew, where ϕ is the Euler phi function. The authors prove the following theorem.
λ
Theorem 1 ([15]). Let N = 3,4,6 and 1 i,j,k < N with gcd(i,j,k) coprime to N and N ∤
≤
i+j+k. Then for each λ Q, the endomorphism algebra of Jnew contains a 4-dimensional algebra
∈ λ
over Q if and only if
N i N j k 2N i j k
B − , − B , − − − Q,
N N N N ∈
(cid:18) (cid:19) (cid:18) (cid:19)
.
Γ(a)Γ(b)
where B(a,b) = , and Γ() is the Gamma function.
Γ(a+b) ·
The second motivation for this paper is to explore the following higher dimensional analogues of
Legendre curves
C : yn = (x x x )n 1(1 x ) (1 x )(x λx x x ).
n,λ 1 2 n 1 − 1 n 1 1 2 3 n 1
··· − − ··· − − − ··· −
In particular, the curves C are known as Legendre curves. Up to a scalar multiple, the hyperge-
2,λ
ometric series
j j j
F n n ··· n ; λ
n n 1
− " 1 1 #
···
for any 1 j n 1, when convergent, can be realized as a period of C .
n,λ
≤ ≤ −
1.2. Results. Our first theorem shows that the number of rational points on C over finite fields
n,λ
F can be expressed in terms of Gaussian hypergeometric functions. For a definition of Gaussian
q
hypergeometric functions please see Section 2.3. 1 Let F denote the group of all multiplicative
×q
characters on F .
×q
c
Theorem 2. Let q = pe 1 (mod n) be a prime power. Let η be a primitive order n character
n
≡
and ε the trivial multiplicative character in F . Then
×q
cn 1
− ηi, ηi, , ηi,
#Cn,λ(Fq) = 1+qn−1+qn−1 nFn 1 n εn, ··· , εn, ;λ .
−
i=1 (cid:18) ··· (cid:19)q
X
Meanwhile, we are also interested in knowing how to use information from truncated hypergeo-
metric series to obtain information about the Galois representations and hence local zeta functions
of C . For instance, we have the following conjecture based on numerical evidence for the case
n,λ
λ = 1.
1ThesubscriptqforaGaussianhypergeometricfunctionrecordsthesizeofthecorrespondingfinitefieldandshould
not beconfused with thesubscript for truncated hypergeoemetric series which records thelocation of truncation.
TRUNCATED HYPERGEOMETRIC AND GAUSSIAN HYPERGEOMETRIC FUNCTIONS 3
Conjecture 3. Let n 3 be a positive integer, and p be prime such that p 1 (mod n). Then
≥ ≡
F n−n1 n−n1 ... n−n1 ; 1 := p−1 1−nn n( 1)kn Γ 1 n (mod p3).
n n 1 p
− 1 ... 1 k − ≡ − n
(cid:20) (cid:21)p 1 k=0(cid:18) (cid:19) (cid:18) (cid:19)
− X
Using the Gross-Koblitz formula [22], recalled in Section 2.4, we have for a prime p 1 (mod n),
≡
n
J(ηn,ηn)J(ηn,ηn2)···J(ηn,ηnn−2)= (−1)n−2+1+(nn−1)pΓp n1 ,
(cid:18) (cid:19)
where J(, ) denotes the standard Jacobi sum. We see that ( 1)n−2+1+(nn−1)p = 1 when n is odd
· · −
p 1
and ηn is an order n character of F×p such that ηn(x) ≡ x −n (mod p) for all x ∈ Fp. From the
perspective of Gro¨ssencharacters (Hecke characters) (see Weil [40]), this product of Jacobi sums
is associated with a linear representation χ of the Galois group Gal(Q/Q(e2πi/n)). We would like
to explore whether the Galois representation arising from C contains a factor that is related to
n,1
χ. When n = 3,4, the answer is positive. In proving these results the work of Greene [21] and
McCarthy [32] on finite field analogues of classical hypergeometric evaluation formulas plays an
essential role.
Ahlgren and Ono [1] show that for any odd prime p,
η2, η2, η2, η2
(1.3) p3 F 4 4 4 4;1 = a(p) p,
· 4 3 ε, ε, ε − −
(cid:18) (cid:19)p
where a(p) is the pth coefficient of the weight 4 Hecke cuspidal eigenform
η(2z)4η(4z)4,
with η(z) being the Dedekind eta function. Here, we show the following.
Theorem 4. Let η , η or η denote characters of order 2, 3, or 4, respectively, in F .
2 3 4 ×q
(1) Let q 1 (mod 3) be a prime power. Then
≡ c
η , η , η
q2 F 3 3 3;1 =J(η ,η )2 J(η2,η2).
· 3 2 ε, ε 3 3 − 3 3
(cid:18) (cid:19)q
(2) Let q 1 (mod 4) be a prime power. Then
≡
η , η , η , η
q3 F 4 4 4 4;1 = J(η ,η )3+qJ(η ,η ) J(η ,η )2.
· 4 3 ε, ε, ε 4 2 4 2 − 4 2
(cid:18) (cid:19)q
Here we observe J(η ,η )2 = η ( 1)J(η ,η )J(η ,η 2). To prove Theorem 4 we use the work
4 2 4 4 4 4 4
−
of Greene [21] and McCarthy [32], except in case (2) when q 5 (mod 8), in which we use
≡
Gro¨ssencharacters and representation theory. The reason we do this is because a key ingredient of
our proof is Theorem 1.6 of McCarthy [32], for which we assume η is a square, i.e., q 1 (mod 8).
4
≡
Combining this with the theory of Galois representations, we can reach our conclusion when q 5
≡
(mod 8). We wish to point out that the above results can be interpreted in terms of Galois repre-
sentations. 2 Result(1) describesthe trace of the Frobeniuselement at q in Gal(Q/Q(√ 3)) under
−
a 2-dimensional Galois representation arising from the second ´etale cohomology of C in terms of
3,1
Jacobi sums (and hence Gro¨ssencharacters); while (2) describes a 3-dimensional Galois representa-
tion of Gal(Q/Q(√ 1)) arising from the third ´etale cohomology of C in terms of Jacobi sums.
4,1
−
2Inadifferentlanguage, ourresultscorrespondtotheexplicitdescriptionsofsomemixedweighthypergeometric
motivesarisingfromexponentialsumswhichareinitiatedbyKatz[24],andareexplicitlyformulatedandimplemented
byagroupofmathematiciansincludingBeukers,Cohen,Rodriguez-Villegasandothers(fromprivatecommunication
with H. Cohen and F. Rodriguez-Villegas). Here we can use the explicit algebraic varieties to compute the Galois
representationsdirectly. Adifferentalgebraicmodelforthealgebraicvarietiesisgiveninthefollowingrecentpreprint
[9].
4 ALYSONDEINES,JENNYG.FUSELIER,LINGLONG,HOLLYSWISHER,FANG-TINGTU
Both cases are exceptional. Consequently we can describe the local zeta function of C and C
3,1 4,1
completely. For instance when p 1 (mod 3) is prime, by the Hasse-Davenport relation for Jacobi
≡
sums (see [23]), the local zeta function of C over F is
3,1 p
1
Z (T,p)=
C3,1 (1 T)(1+(α +α )T +pT2)(1 p2T)(1 (α2 +α2)T +p2T2)
− p p − − p p
where α = J(η ,η ). Note that the factor (1+(α +α )T +pT2) appearing in the denominator
p 3 3 p p
has roots of absolute value 1/√p; meanwhile following Weil’s conjecture (see [23]) such a term
should appear in the numerator instead. We believe the discrepancy is due to the fact that we are
not computing using the smooth model of C as no resolution of singularities is involved so far.
n,λ
Similarly, we have for any prime p 1 (mod 4)
≡
Z (T,p) = (1+(β3+β3)T +p3T2)(1+(β +β)pT +p3T2)
C4,1 p p p
(1 (β2+β2)T +p2T2)(1 a(p)T +p3T2)(1 pT)
− p p − − ,
· (1 T)(1 p3T)
− −
where a(p) as in (1.3) and β = J(η ,η ). The factor corresponding to
p 4 2
y2 = (x x x )3(1 x )(1 x )(1 x )(x x x )
1 2 3 1 2 3 1 2 3
− − − −
is
(1 a(p)T +p3T2)(1 pT)
Z (T,p) = − − ,
C4o,l1d (1 T)(1 p3T)
− −
and new primitive portion is
ZC4n,e1w(T,p) = (1+(βp3+β3p)T +p3T2)(1+(βp +β)pT +p3T2)(1−(βp2+β2p)T +p2T2).
Part (1) of Theorem 4 explains why Γ (1)3 appears to be a conjugate of Γ (1)6. There are
− p 3 p 3
two ways to specify a cubic character in F when p 1 (mod 3), i.e. η (x) x(p 1)/3 (mod p) for
×p 3 −
≡ ≡
all x F or η (x) x2(p 1)/3 (mod p). Either way gives an embedding of
p 3 −
∈ ≡
c
η , η , η
p2 F 3 3 3;1
· 3 2 ε, ε
(cid:18) (cid:19)p
to Z . Then the image of the Gaussian hypergeometric function is congruent to Γ (1)3 or Γ (1)6
p − p 3 p 3
respectively via the Gross-Koblitz formula [22, 37]. Using this formula, we also prove the following
result which relates Gaussian hypergeometric functions to truncated hypergeometric series.
Lemma 5. Let r,n,j be positive integers with 1 j < n. Let p 1 (mod n) be prime and η F
n ×p
≤ ≡ ∈
such that η (x) xj(p 1)/n (mod p) for each x F . Then,
n − p
≡ ∈
c
η , η , , η
pr−1·rFr−1 n εn, ··· , εn;x ≡
(cid:18) ··· (cid:19)p
n j n j n j 1
( 1)r+1 F −n −n ··· −n ;
r r 1
− · − " 1 ··· 1 x#(p−1)(nn−j)
+( 1)r+1+(p−n1)jr x(p−1)n−nj xp−n1j (mod p);
− −
(cid:16) (cid:17)
TRUNCATED HYPERGEOMETRIC AND GAUSSIAN HYPERGEOMETRIC FUNCTIONS 5
η , η , , η
pr−1·rFr−1 n εn, ··· , εn;x ≡
(cid:18) ··· (cid:19)p
1 1 1 1
( 1)r+1 pr F ··· ; (mod p).
− · r+1 r 2n j 2n j x
" − − #
n ··· n p 1
−
Thus, (1.1) and (1.2) hold modulo p. It is shown in [31] that (1.1) holds modulo p3. These kinds
of stronger congruences are known as supercongruences as they are stronger than what the theory
of formal groups can predict. We will establish a few here. In particular, we prove the claim that
Conjecture 3 holds modulo p2.
Theorem 6. Conjecture 3 holds modulo p2. Namely, for n 3, and p 1 (mod n) prime,
≥ ≡
F n−n1 n−n1 ... n−n1 ; 1 := p−1 1−nn n( 1)kn Γ 1 n (mod p2).
n n 1 p
− 1 ... 1 k − ≡ − n
(cid:20) (cid:21)p 1 k=0(cid:18) (cid:19) (cid:18) (cid:19)
− X
Remark. Theorem 6 also holds for n = 2, due to Mortenson [35].
We note that in [33, Defn. 1.4], McCarthy defines a new function G [ ] in terms of sums and
n n
···
ratios of p-adic Gama functions. Recently, the second author and McCarthy produced families of
congruences between these G functions and truncated hypergeometric series [19]. New identities
n n
for these functions have also recently been obtained by McCarthy, et. al. [7] and it is possible they
could be used to prove Conjecture 3 in full.
For the truncated hypergeometric series related to C we have another result.
4,1
Theorem 7. For each prime p 1 (mod 4),
≡
4F3 14 41 41 14 ; 1 = p−1 −14 4 ( 1)p−41Γp 1 Γp 1 6 (mod p4).
1 1 1 k ≡ − 2 4
(cid:20) (cid:21)p 1 k=0(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
− X
Corresponding to Ahlgren and Ono’s result (1.3), Kilbourn [25] shows the supercongruence
1 1 1 1 p 1 1 4
−
F 2 2 2 2 ; 1 := −2 a(p) (mod p3),
4 3
1 1 1 k ≡
(cid:20) (cid:21)p 1 k=0(cid:18) (cid:19)
− X
where a(p) is defined as in (1.3).
Supercongruences are not only intellectually appealing, they are of very practical use for our
computations. For instance, Theorem 7 corresponds to the properties of the third ´etale cohomol-
ogy group of C as mentioned earlier. By the Hasse-Weil bounds for them, which are constant
4,1
multiplies of p and p3/2 respectively, the supercongruence results allowed us to compute the traces
of Frobenius without any ambiguity, from which we were able to nail down the local zeta functions
of C andC and discover Theorem4numerically beforeprovingit. Therearea variety of differ-
3,1 4,1
ent techniques for proving such results and each has its own strength. See [11] for another Women
in Numbers (WIN) project on supercongruences, which was motivated by the work of Zudilin [44]
and his conjectures. We prove Theorems 6 and 7 by deforming truncated hypergeometric series
using hypergeometric evaluation identities (several of them are due to Whipple [41, 2]) together
with p-adic analysis via harmonic sums and p-adic Gamma functions. This technique is originated
in [10] and is later formulated explicitly in [31].
6 ALYSONDEINES,JENNYG.FUSELIER,LINGLONG,HOLLYSWISHER,FANG-TINGTU
1.3. Outline of this paper. Section 2 contains some background material. In Section 3, we
consider thefamiliar setting of Legendrecurves. Thissection serves as ashowcase of ourtechniques
withoutgettingintotoomuchtechnicality. WeproveTheorem2andLemma5inSection4. Section
5 is devoted to proving the results on Gaussian hypergeometric functions in Theorem 4. In Section
6, we prove Theorem 6, based on an idea of Zudilin (private communication), and then prove
Theorem 7. Sections 4, 5, and 6 are technical in nature. In Section 7 we end with some remarks
including a few conjectures based on our numerical data computed using Sage.
2. Preliminaries
2.1. Generalized hypergeometric series and truncation. For a positive integer r, and α ,
i
β C with β ..., 3, 2, 1 , the (generalized) hypergeometric series F is defined by
i i r r 1
∈ 6∈{ − − − } −
α1 α2 ... αr ∞ (α1)k(α2)k...(αr)k λk
F ; λ :=
r r 1
− (cid:20) β1 ... βr−1 (cid:21) Xk=0 (β1)k...(βr−1)k · k!
where (a) := 1 and (a) := a(a+1) (a+k 1) are rising factorials. This series converges for
0 k
··· −
λ < 1.
| |
When we truncate the above sum at k = m, we use the subscript notation
α α ... α m (α ) (α ) ...(α ) λk
1 2 r 1 k 2 k r k
F ; λ := .
r r 1
− (cid:20) β1 ... βr−1 (cid:21)m Xk=0 (β1)k...(βr−1)k · k!
We note that the books by Slater [38], Bailey [5], and Andrews, Askey and Roy [2] are excellent
sources for information on classical hypergeometric series.
The following gives an alternate truncation for hypergeometric series modulo powers of primes.
Lemma 8. Let n 2 be a positive integer, j an integer 1 j < n, and p 1 (mod n) prime.
≥ ≤ ≡
Then for x Z ,
p
∈
j j j j j j
F n n ··· n ; x F n n ··· n ; x (mod pr).
r r 1 r r 1
− " 1 ··· 1 #j(p 1) ≡ − " 1 ··· 1 #p 1
n − −
j(p 1)
Proof. The lemma follows from the fact that when − +1 k (p 1), the rising factorial
n ≤ ≤ −
j pZ , since it contains the factor p j , while (1) is not divisible by p. (cid:3)
n k ∈ p n k
(cid:16) (cid:17) (cid:16) (cid:17)
2.2. Euler’sintegral formulaand higher generalization. WhenRe(β ) > Re(α ) > 0,Euler’s
1 2
integral representation for F [2] states that
2 1
α α Γ(β ) 1
F 1 2 ; λ = 1 xα2 1(1 x)β1 α2 1(1 λx) α1dx.
2 1 − − − −
β Γ(α )Γ(β α ) − −
(cid:20) 1 (cid:21) 2 1− 2 Z0
More generally, one has that when Re(β )> Re(α ) > 0 (see [2, (2.2.2)])
r r+1
α α ... α Γ(β )
1 2 r+1 r
(2.1) F ; λ =
r+1 r
β ... β Γ(α )Γ(β α )
(cid:20) 1 r (cid:21) r+1 r − r+1
1 α α ... α
xαr+1 1(1 x)βr αr+1 1 F 1 2 r ; λx dx.
− − − r r 1
·Z0 − − (cid:20) β1 ... βr−1 (cid:21)
From the above two integral formulas, one can derive that for each 1 j n 1 the series
≤ ≤ −
j j j
F n n ··· n ; λ , with a suitable beta quotient factor, is a period of C .
n n 1 n,λ
− " 1 1 #
···
TRUNCATED HYPERGEOMETRIC AND GAUSSIAN HYPERGEOMETRIC FUNCTIONS 7
2.3. Gaussian hypergeometric functions. Let p be prime and let q = pe. We extend any
character χ F to all of F by setting χ(0) = 0, including the trivial character ε, so that
×q q
∈
ε(0) = 0. For A,B F , let J(A,B) := A(x)B(1 x) denote the Jacobi sum and define
c∈ ×q x Fq −
∈
P
c
A B( 1) B( 1)
:= − J(A,B) = − A(x)B(1 x).
B q q −
(cid:18) (cid:19) xX∈Fq
In [21], Greene defines a finite field analogue of hypergeometric series called Gaussian hyperge-
ometric functions, defined below.
Definition 9 ([21] Defn. 3.10). If n is a positive integer, x F , and A ,A ,...,A ,
q 0 1 n
∈
B ,B ,...,B F , then
1 2 n ×q
∈
cA , A , ..., A q A χ A χ A χ
F 0 1 n;x := 0 1 ... n χ(x).
n+1 n(cid:18) B1, ..., Bn (cid:19)q q−1 c (cid:18) χ (cid:19)(cid:18)B1χ(cid:19) (cid:18)Bnχ(cid:19)
χXF×q
∈
Greene showcases a variety of identities satisfied by his Gaussian hypergeometric functions,
many of which provide direct analogues for transformations of classical hypergeometric series. For
example, he provides a finite field analogue of (2.1), shown below.
Theorem 10 (Greene [21]). For characters A ,A ,...,A ,B ,...,B in F , and x F ,
0 1 n 1 n ×q q
∈
c
A , A , ..., A
F 0 1 n;x =
n+1 n B , ..., B
(cid:18) 1 n (cid:19)q
A B ( 1) A , A , ..., A
n nq − · y nFn−1(cid:18) 0 B11, ..., Bnn−−11;xy(cid:19)q ·An(y)AnBn(1−y).
X
Toextend Greene’s program, McCarthy provides amodification of Greene’s functions below. We
let g(χ) = χ(x)ζTr(x) denote the Gauss sum of χ, and Tr the usual trace map form F to F .
p q p
xX∈Fq
Definition 11. [32] For characters A ,A ,...,A ,B ,...,B in F ,
0 1 n 1 n ×q
n cn
F A0, A1, ..., An;x ∗ := 1 g(Aiχ) g(Bjχ)g(χ)χ( 1)n+1χ(x).
n+1 n(cid:18) B1, ..., Bn (cid:19)q q−1 c i=0 g(Ai) j=1 g(Bj) −
χXF×q Y Y
∈
McCarthy makes explicit how the two hypergeometric functions are related, via the following.
Proposition 12 (McCarthy [32]). If A = ε and A = B for each 1 i n, then
0 i i
6 6 ≤ ≤
n 1
F A0, A1, ..., An;x ∗ = Ai − F A0, A1, ..., An;x .
n+1 n(cid:18) B1, ..., Bn (cid:19)q "i=1(cid:18)Bi(cid:19) #n+1 n(cid:18) B1, ..., Bn (cid:19)q
Y
McCarthy uses this hypergeometric function to provide analogues to classical formulas of Dixon,
Kummer, and Whipple for well-poised classical hypergeometric series [32]. For example, consider
Whipple’s classical transformation below:
8 ALYSONDEINES,JENNYG.FUSELIER,LINGLONG,HOLLYSWISHER,FANG-TINGTU
Theorem 13 (Whipple [41]). If one of 1+ 1a b, c, d, e is a negative integer, then
2 −
a b c d e
F ;1
5 4 1+a b 1+a c 1+a d 1+a e
(cid:20) − − − − (cid:21)
Γ(1+a c)Γ(1+a d)Γ(1+a e)Γ(1+a c d e)
= − − − − − −
Γ(1+a)Γ(1+a d e)Γ(1+a c d)Γ(1+a c e)
− − − − − −
1+ 1a b c d e
· 4F3 2 − 1+ 1a c+d+e a 1+a b ;1 .
(cid:20) 2 − − (cid:21)
McCarthy provides a finite field analogue to this result using his hypergeometric series.
Theorem 14 (McCarthy, Thm. 1.6 of [32]). Let A,B,C,D,E F such that, when A is a square,
×q
∈
A= ε, B = ε, B2 = A, CD = A, CE = A, DE =A, and CDE =A. Then, if A is not a square,
6 6 6 6 6 6 6
c
A, B, C, D, E ∗
F ;1 = 0,
5 4 AB, AC, AD AE
(cid:18) (cid:19)q
and if A is a square,
A, B, C, D, E ∗
F ;1 =
5 4 AB, AC, AD AE
(cid:18) (cid:19)q
g(A)g(ADE)g(ACD)g(ACE) RB, C, D, E ∗
F ;1
g(AC)g(AD)g(AE)g(ACDE) 4 3 R ACDE, AB
RX2=A (cid:18) (cid:19)q
g(ADE)g(ACD)g(ACE)q A, B ∗
+ F ; 1 .
g(C)g(D)g(E)g(AC)g(AD)g(AE) 2 1 AB −
(cid:18) (cid:19)q
Gaussian hypergeometric functions have been used to count points on different types of varieties
over F and they are related to coefficients of various modular forms [26, 36, 17, 18, 27, 39, 6, 1].
q
We use Greene’s hypergeometric functions to count points on C in Section 4.1. We make use of
n,λ
McCarthy’s hypergeometric function, as well as the previous theorem, Theorem 14, in the proof of
Theorem 4 in Section 5. Values of McCarthy’s normalized version of the hypergeometric function
over finitefields have also been shown to berelated to eigenvalues of Siegel modularforms of higher
degree [34].
2.4. p-adic Gamma functions and the Gross-Koblitz formula. We first recall the p-adic
Γ-function. The p-adic Γ-function is defined for n N by
∈
Γ (n):= ( 1)n j,
p
−
0<j<n
Y
p∤j
and extends to x Z by defining Γ (0) := 1, and for x = 0,
p p
∈ 6
Γ (x) := lim Γ (n),
p p
n x
→
where n runs through any sequence of positive integers p-adically approaching x.
We recall some basic properties for Γ () which will be useful later (see Theorem 14 of [31]).
p
·
Proposition 15. Let x Z . We have the following facts about Γ .
p p
∈
a) Γ (0) = 1
p
b) Γ (x+1)/Γ (x) = x unless x pZ in which case the quotient takes value 1.
p p p
− ∈ −
c) Γ (x)Γ (1 x) = ( 1)a0(x) where a (x) is the least positive residue of x modulo p,
p p 0
− −
TRUNCATED HYPERGEOMETRIC AND GAUSSIAN HYPERGEOMETRIC FUNCTIONS 9
d) Given p > 11, there exist G (x),G (x) Z such that for any m Z ,
1 2 p p
∈ ∈
(mp)2 (mp)3
Γ (x+mp) Γ (x) 1+G (x)mp+G (x) +G (x) (mod p4).
p p 1 2 3
≡ 2 6
(cid:20) (cid:21)
e) G (x) = G (1 x) and G (x)+G (1 x) = 2G (x)2,
1 1 2 2 1
− −
f) If x y (mod pn), then Γ (x) Γ (y) (mod pn).
p p
≡ ≡
We note that c) above implies in particular that for any integer n > 1 and prime p 1 (mod n),
≡
1 1 1+(n 1)p
Γp Γp 1 =( 1) n− .
n − n −
(cid:18) (cid:19) (cid:18) (cid:19)
Thus when n is odd, Γ (1)Γ (1 1)= 1.
p n p − n −
We now recall the Gross-Koblitz formula [22, 37] in the case of F . Let
p
ϕ: F Z
×p → ×p
be the Teichmu¨ller character such that ϕ(x) x (mod p). The Gross-Koblitz formula states that
≡
the Gauss sum g(ϕ j) defined using the Dwork exponential as the additive character satisfies
−
j
(2.2) g ϕ j = πjΓ ,
− − p p p 1
(cid:18) − (cid:19)
(cid:0) (cid:1)
where 0 j p 2, and π C is a root of xp 1+p = 0.
p p −
≤ ≤ − ∈
3. In the setting of Legendre curves
We first briefly explain the relationships between Gaussian hypergeometric functions, hypergeo-
metric series, and truncated hypergeometric series using the Legendre curve
C :y2 = x (1 x )(x λ),
2,λ 1 1 1
− −
which is isomorphic to the more familiar form y2 = x(x 1)(x λ) over Q(√ 1). It is well-known
− − −
that one period of C is given by
2,λ
1 1
π F 2 2 ; λ .
2 1
· 1
(cid:20) (cid:21)
Assume λ Q and η F is of order 2. It follows from the Taniyama-Shimura-Wiles theorem
2 ×p
∈ ∈
[42] that for good primes p,
c
η , η
a (λ) = p+1 #C (F )= η (x (1 x )(x λ)) = p F 2 2;λ
p − 2,λ p − 2 1 − 1 1− − ·2 1 ε
xX1∈Fp (cid:18) (cid:19)p
isthepthcoefficientofaweight2cuspidalHeckeeigenformthatcanbecomputedfromacompatible
familyof2-dimensionalℓ-adicGalois representations ofG := Gal(Q/Q)constructedfromtheTate
Q
module of C via L-series. This gives a correspondence between the F Gaussian hypergeometric
2,λ 2 1
functions and the Galois representations arising from C .
2,λ
When a (λ) is not divisible by p, i.e. p is ordinary for C , then a result of Dwork [16] says that
p 2,λ
1 1 1 1
(3.1) lim F 2 2 ; λˆ F 2 2 ; λˆ
2 1 2 1
s 1 1
→∞ (cid:20) (cid:21)ps 1 (cid:20) (cid:21)ps 1 1
− . − −
is the unit root of T2 a (λ)T + p, where λˆ = ϕ(λ) is the image of λ under the Teichmu¨ller
p
−
character.
10 ALYSONDEINES,JENNYG.FUSELIER,LINGLONG,HOLLYSWISHER,FANG-TINGTU
Since λ = 1 is a singular case, we study the case when λ = 1, for which the corresponding
−
Legendre curve admits complex multiplication. Let p 1 (mod 4) be prime, then by [21, (4.11)],
≡
η , η
p F 2 2; 1 = J(η ,η )+J(η ,η ),
· 2 1 ε − 4 2 4 2
(cid:18) (cid:19)p
whereη is a primitive order 4 character of F .3 In the perspectiveof Dwork, thevalue (3.1) agrees
4 ×p
with the unit root of T2 +(J(η ,η )+J(η ,η ))T +p, which is Γp(21)Γp(41) by the Gross-Koblitz
4 2 4 2 Γp(34)
formula. Using Lemma 5, we have the following.
Proposition 16. For each prime p 1 (mod 4)
≡
1 1 Γ 1 Γ 1 Γ 1
F 2 2 ; 1 p 2 p 4 = p 4 (mod p).
2 1 1 − ≡ Γ 3 −Γ 1 Γ 3
(cid:20) (cid:21)p−21 (cid:0) p(cid:1) 4 (cid:0) (cid:1) p 2 (cid:0) p(cid:1) 4
Proof. By Lemma 8 and Lemma 5, we obtain(cid:0)th(cid:1)at (cid:0) (cid:1) (cid:0) (cid:1)
1 1 1 1
η , η
F 2 2 ; 1 F 2 2 ; 1 p F 2 2; 1 (mod p).
2 1 1 − ≡ 2 1 1 − ≡ − · 2 1 ε −
(cid:20) (cid:21)p−21 (cid:20) (cid:21)p−1 (cid:18) (cid:19)p
It remains to show
(3.2) p F η2, η2; 1 Γp 21 Γp 14 (mod p).
· 2 1 ε − ≡ − Γ 3
(cid:18) (cid:19)p (cid:0) p(cid:1) 4 (cid:0) (cid:1)
By the relations
(cid:0) (cid:1)
p F η2, η2; 1 = J(η ,η )+J(η ,η )= g(η2) g(η4)2+g(η4)2 ,
· 2 1 ε − 4 2 4 2 g(η )g(η )
(cid:18) (cid:19)p (cid:0) 4 4 (cid:1)
using the Gross-Koblitz formula, we see that
p F η2, η2; 1 = −πpp−21Γp 21 (πp3p−21Γp 43 2+πpp−21Γp 41 2)
· 2 1(cid:18) ε − (cid:19)p (cid:0) (cid:1)πpp−1Γp 41(cid:0)Γ(cid:1)p 34 (cid:0) (cid:1)
= Γp 12 (−pΓp 34 2(cid:0)+Γ(cid:1)p 41(cid:0) 2(cid:1)),
− Γ 1 Γ 3
(cid:0) (cid:1) p 4(cid:0) (cid:1)p 4 (cid:0) (cid:1)
which yields the result. (cid:3)
(cid:0) (cid:1) (cid:0) (cid:1)
Using a different technique via hypergeometric evaluation identities, one can prove the following
stronger result. We will also outline this strategy here (for details, see [31]). First we deform the
truncated hypergeometric series p-adically so that it becomes a whole family of terminating series
that can be written as a quotient of Pochhammer symbols (a) via appropriate hypergeometric
k
evaluation formulas. We further rewrite the quotient of Pochhammer symbols as a quotient of
p-adic Gamma values using the functional equation of p-adic Gamma functions. Then we use
harmonic sumsto analyze the terminating series on one side and usethe Taylor expansion of p-adic
Gamma functions on the other side. Now picking suitable members in the deformed family, we
compare both sides to get a linear system which allows us to conclude the desired congruence.
Proposition 17. For any prime p 1 (mod 4),
≡
1 1 Γ (1)
F 2 2 ; 1 p 4 (mod p2).
2 1 1 − ≡ −Γ (1)Γ (3)
(cid:20) (cid:21)p−21 p 2 p 4
3When p≡3 (mod 4), F η2, η2;−1 =0.
2 1(cid:18) ε (cid:19)
p