hypergeometric functions over finite fields, i - icerm · motivation idictionaryhypergeometric over...

Motivation I Dictionary Hypergeometric over C Hypergeometric over Fq Another Perspective

Hypergeometric Functions over Finite Fields, I

Holly Swisher

Oregon State University

ICERM Research SeminarOctober 8, 2015

Joint work with Jenny Fuselier, Ling Long, Ravi Ramakrishna, Fang-Ting Tu

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Apery Supergongruence

Let A(n) denote the Apery numbers

A(n) =n∑j=0

(n+ j

j

)2(n

j

)2

.

Define the integers a(n) by∑n≥1

a(n)qn = q∏n≥1

(1− q2n)4(1− q4n)4.

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Apery Supergongruence

Beukers proved that

A

(p− 1

2

)≡ a(p) (mod p)

and conjectured that this should hold modulo p2.Ahlgren and Ono proved

A

(p− 1

2

)≡ a(p) (mod p2)

using a Gaussian hypergeometric evaluation.

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Gaussian Hypergeoemtric Functions

Greene defined Gaussian hypergoemetric functions asanalogues to classical

Gaussian hypergeometric functions are useful incounting points on curves

McCarthy, Evans, Katz, and more have also studiedfinite field analogues of hypergeometric functions

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Ramanujan Series for 1/π

In 1914, Ramanujan listed 17 infinite series representationsof 1/π, including for example

∞∑k=0

(4k + 1)(−1)k(12)3k

k!3=

2

π=

2

Γ(12

)2 .Several of these relate hypergeometric series to values ofthe gamma function.

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Hypergeometric Series

A hypergeometric series is a series∑cn such that the

quotients cn+1/cn are rational functions of n. This leads toan expression

∑n≥0

cn = c0 ·∞∑k=0

(α1)k(α2)k . . . (αs)k(β1)k . . . (βr)k

· λk

k!

= c0 ·s Fr[α1 . . . αsβ1 . . . βr

; λ

],

where for complex a and positive integer k,

(a)k := (a)(a+ 1) · · · (a+ k − 1).

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Ramanujan Revisited

In terms of our notation, our Ramanujan formula says

4F3

[54

12

12

12

14

1 1; −1

]=∞∑k=0

(4k + 1)(−1)k(12)3k

k!3

=2

π=

2

Γ(12

)2 .

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van Hamme’s p-adic analogues

In 1997, van Hamme developed p-adic analogues of severalRamanujan type series (supercongruences). In all, vanHamme conjectured 13 Ramanujan type supercongruences.Example: for odd primes p,

p−12∑

k=0

(4k + 1)(−1)k(12)3k

k!3≡ −p

Γp(12)2

(mod p3).

This is van Hamme’s (B.2) conjecture.

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Recent Work

Kilbourn (Modularity of Calabi-Yau 3-fold)

McCarthy and Osburn (Gaussian Hypergeometric)

Long (Hypergeometric evaluations and p-adic analysis)

Long and Ramakrishna (Hypergeometric evaluationsand p-adic analysis)

Osburn and Zudilin (WZ method)

S- (Hypergeometric evaluations and p-adic analysis)

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Theorem (S-)

The original method of Long to prove (B.2), can actually beused to prove over half of the van Hamme conjectures, andfurthermore can be used to prove extensions of some of theconjectures to additional sets of primes not previouslyconsidered, and to higher powers than previouslyconjectured by van Hamme.

Show van Hamme Table!

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Idea of Proof

The following is an identity of Whipple:

4F3

[a2

+ 1 a c da2

1 + a− c 1 + a− d ; −1

]=

Γ(1 + a− c)Γ(1 + a− d)

Γ(1 + a)Γ(1 + a− c− d).

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Our Goals

When obtaining finite field analogues of many classicalhypergeometric identities, the result often mirrors theclassical case somewhat. Our goals are to

Obtain finite field analogues which mirror the classicalcase as closely as possible. In particular, so that theproofs in many cases may be directly translated to thefinite field setting.

Highlight the geometric connection by developing theanalogues via periods of hypergeometric varieties.

Through a slight modification of the notation of bothGreene, and McCarthy we are able to achieve these goals.

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Binomial Theorem

By the binomial theorem,

(1− z)a =∞∑k=0

(a

k

)(−z)k =

∞∑k=0

(−a)kk!

zk.

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Gamma Function and Beta Function

Recall, For Re(z) > 0,

Γ(z) :=

∫ ∞0

tz−1e−t dt,

For Re(x) > 0,Re(y) > 0, define

B(x, y) :=

∫ 1

0

tx−1(1− t)y−1 dt,

which is related to Γ(z) via

B(x, y) =Γ(x)Γ(y)

Γ(x+ y).

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Properties of Gamma

Euler’s reflection formula. For complex z,

Γ(z)Γ(1− z) =π

sin(πz).

Furthermore, Γ satisfies the duplication formula

Γ(z)Γ

(z +

1

2

)= 21−2z√πΓ(2z),

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Properties of Gamma

Gauss’ more general multiplication formula is

Γ(ma)(2π)(m−1)/2

= mma−1/2Γ(a)Γ

(a+

1

m

)· · ·Γ

(a+

m− 1

m

).

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Finite Fields

Fq finite field, where q = pe is a power of an odd prime

χ multiplicative character on F×qF×q cyclic group of all multiplicative characters on F×qε trivial character, defined by ε(a) = 1 for a 6= 0

φ quadratic character on F×qNote: We extend characters on F×q to all of Fq by settingχ(0) = 0, including ε(0) = 0.

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Useful definition

Further, it is useful to define δ on either F×q or Fq by

δ(χ) :=

{1 if χ = ε,

0 if χ 6= ε;δ(x) :=

{1 if x = 0,

0 if x 6= 0,

This definition will allow us (as in Greene) to describeformulas which hold for all characters, without having toseparate cases involving trivial characters.

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Gauss and Jacobi Sums

Gauss and Jacobi Sums are analogues of the Gamma andBeta functions.Let ζp be a primitive pth root of unity, and A,B ∈ F×q .Then,

g(A) :=∑x∈F×

q

A(x)ζtr(x)p ,

J(A,B) :=∑x∈Fq

A(x)B(1− x).

In general the Gauss sums depend on the choice of ζp, theJacobi sums do not.

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Analogues of Euler’s Reflection Formula

The relation between the two is

J(A,B) =g(A)g(B)

g(AB)+ (q − 1)B(−1)δ(AB).

Analogue of Euler’s Reflection Formula:

g(A)g(A) = qA(−1)− (q − 1)δ(A),

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Analogue of Gauss’ Multiplication Formula

Let m ∈ N and q = pe be a prime power with q ≡ 1

(mod m). For any multiplicative character ψ ∈ F×q ,∏χ∈F×

q

χm=ε

g(χψ) = −g(ψm)ψ(m−m)∏χ∈F×

q

χm=ε

g(χ).

Letting m = 2 gives the duplication formula analogue

g(A)g(φA) = g(A2)g(φ)A(4).

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Analogues of rising factorials and binomial coefficients

Let

(A)χ :=g(Aχ)

g(A),(

A

χ

):= −χ(−1)J(A,χ).

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Analogues, continued

For A,χ1, χ2 ∈ F×q ,

(A)χ1χ2 = (A)χ1(Aχ1)χ2 ,

which is the analogue of the identity

(a)n+m = (a)n(a+ n)m,

for integers n,m ≥ 0.

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Analogue of Binomial Coefficient Identity

For any characters A,B on F×q ,

J(A,B) = A(−1)J(A,BA),

i.e. (A

B

)=

(A

AB

)is the analogue of

(nm

)=(

nn−m

).

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Proof

Analogue of Euler’s reflection formula

Relation to Gauss sum

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Dictionary between C and Fq settings

n ∈ Z ↔ trivial character ε1N ↔ a primitive character ηN of order N

a = iN , b = j

N ↔ A,B ∈ F×q , A = ηiN , B = ηjNxa ↔ A(x)

xa+b ↔ A(x)B(x) = AB(x)

a+ b ↔ A ·B−a ↔ A

Γ(a) ↔ g(A)

(a)n = Γ(a+ n)/Γ(a) ↔ (A)χ = g(Aχ)/g(A)∫ 1

0dx ↔

∑x∈F

Γ(a)Γ(1− a) = πsin aπ , a /∈ Z ↔ g(A)g(A) = A(−1)q, A 6= ε

(ma)mn = mmn∏mi=1

(a+ i

m

)n↔ (Am)χm = χ(mm)

∏mi=1(Aηim)χ

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Hypergeometric Functions

Recall,

n+1Fn

[a1 a2 · · · an+1

b1 · · · bn; z

]:=

∞∑k=0

(a1)k(a2)k · · · (an+1)k(b1)k · · · (bn)kk!

zk.

These functions satisfy an inductive integral relation ofEuler when Re(bn) > Re(an+1) > 0:

n+1Fn

[a1 a2 · · · an+1

b1 · · · bn; z

]= B(an+1, bn − an+1)

−1

·∫ 1

0

tan+1−1(1−t)bn−an+1−1·nFn−1[a1 a2 · · · an

b1 · · · bn−1; zt

]dt.

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Relationship to periods

By the binomial theorem, we may define for general a ∈ C,

1P0[a; z] := (1− z)−a =∞∑k=0

(a)kk!

zk = 1F0[a; z].

In particular, when a, b, c ∈ Q the integral

2P1

[a b

c; z

]:=

∫ 1

0

tb−1(1− t)c−b−1 1P0 [a; zt] dt

=

∫ 1

0

tb−1(1− t)c−b−1(1− zt)−adt

can be realized as a period of a corresponding generalizedLegendre curve.

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Generalized Legendre Curves

By work of Wolfart, for λ 6= 0, 1, the curve containing thisperiod can be given by the smooth model of

C[N ;i,j,k]λ : yN = xi(1− x)j(1− λx)k,

where N is the least common denominator of a, b, and c,and the integers i, j, k are defined by i = N · (1− b),j = N · (1 + b− c), and k = N · a.

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By the inductive integral relation of Euler for 2F1 we seethat when Re(c) > Re(b) > 0,

2P1

[a b

c; z

]=

∫ 1

0

tb−1(1− t)c−b−1 · 1P0[a; zt]dt

=

∫ 1

0

tb−1(1− t)c−b−1 · 1F0[a; zt]dt

= B(b, c− b) ·2 F1

[a b

c; z

].

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Inductively one can define the (higher) periods n+1Pnsimilarly by

n+1Pn

[a1 a2 · · · an+1

b1 · · · bn; z

]:=∫ 1

0

tan+1−1(1−t)bn−an+1−1nPn−1

[a1 a2 · · · an

b1 · · · bn−1; zt

]dt.

Again using the beta function, one can show that whenRe(bi) > Re(ai+1) > 0 for each i ≥ 1,

n+1Fn

[a1 a2 · · · an+1

b1 · · · bn; z

]=

n∏i=1

B(ai+1, bi − ai+1)−1 · n+1Pn

[a1 a2 · · · an+1

b1 · · · bn; z

].

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Guiding Principle

By definition, any given n+1Fn function satisfies two niceproperties:

1) The leading coefficient is 1;

2) The roles of upper entries ai (resp. lower entries bj) aresymmetric.

The n+1Pn period functions do not satisfy these propertiesin general. The n+1Fn hypergeometric functions can thusbe viewed as periods that are normalized so that bothproperties 1) and 2) are satisfied.

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Finite Field Setting

For any function f : Fq −→ C, by the orthogonality ofcharacters f has a unique representation

f(x) = δ(x)f(0) +∑χ∈F×

q

fχχ(x),

where

fχ =1

q − 1

∑y∈Fq

f(y)χ(y).

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Analogue of the binomial formula for (1− z)a

If A ∈ F×q and x ∈ Fq, then we have

A(1− x) = δ(x) +1

q − 1

∑χ∈F×

q

J(A,χ)χ(x)

=

1

q − 1

∑χ∈F×

q

J(A,χ)χ(x), if x 6= 0,

1, if x = 0.

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New definition of hypergeometric functions over Fq

This definition is very parallel to the classical setting. Westart similarly with

1P0[A;x; q] := A(1− x),

and then inductively define

n+1Pn[A1 A2 . . . An+1

B1 . . . Bn;λ; q

]:=∑

y∈Fq

An+1(y)An+1Bn(1−y)·nPn−1[A1 A2 . . . An

B1 . . . Bn−1;λy; q

].

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Observation

There is asymmetry here among the Ai (resp. Bj) bydefinition. However, we start with the analogue of theperiod, rather than the hypergeometric function becausethe periods are very related to point-counting.

When there is no ambiguity in our choice of field Fq, wewill leave out the q in this notation.

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When n = 1 we have,

2P1

[A1 A2

B1;λ

]=∑y∈Fq

A2(y)A2B1(1− y)A1(1− λy).

To be explicit,

2P1

[A1 A2

B1;λ

]=

J(A2, A2B1), if λ = 01

q − 1

∑χ∈F×

q

J(A1, χ)J(A2χ,B1A2)χ(λ), if λ 6= 0.

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A general formula for the n+1Pn period function is

n+1Pn[A1 A2 . . . An+1

B1 . . . Bn;λ

]=

(−1)n+1

q − 1

·

(n∏i=1

Ai+1Bi(−1)

) ∑χ∈F×

q

(A1χ

χ

)(A2χ

B1χ

)· · ·(An+1χ

Bnχ

)χ(λ)

+ δ(λ)n∏i=1

J(Ai+1, Ai+1Bi).

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We normalize the finite field period functions to haveconstant term 1 as in the classical case.

2F1

[A1 A2

B1;λ

]:=

1

J(A2, B1A2)· 2P1

[A1 A2

B1;λ

].

The 2F1 function satisfies

1) 2F1

[A1 A2

B1; 0

]= 1;

2) 2F1

[A1 A2

B1;λ

]= 2F1

[A2 A1

B1;λ

],

if A1, A2 6= ε, and A1, A2 6= B1.

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More generally, we define

n+1Fn[A1 A2 · · · An+1

B1 · · · Bn

; λ

]:=

1∏ni=1 J(Ai+1, BiAi+1)

n+1Pn[A1 A2 . . . An+1

B1 . . . Bn;λ

].

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Definition of Primitive

A period function n+1Pn[A1 A2 . . . An+1

B1 . . . Bn;λ

]or the

corresponding n+1Fn[A1 A2 . . . An+1

B1 . . . Bn;λ

]is said to be

primitive if Ai 6= ε and Ai 6= Bj for all i, j. Like the n = 1case, the value of the n+1Fn functions at λ = 0 is 1, and thecharacters Ai (resp. Bj) are commutative for primitive

n+1Fn.

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Comparison to Greene and McCarthy

Our period functions are closely related to Greene’sGaussian hypergeometric functions and both can be usedto count points. The relationship is

n+1Pn[A1, A2, . . . , An+1

B1, . . . , Bn;λ

]= qn

(n∏i=1

Ai+1Bi(−1)

)n+1Fn

(A1, A2, . . . , An+1

B1, . . . , Bn;λ

)G+ δ(λ)

n∏i=1

J(Ai+1, Ai+1Bi).

In the primitive case, the normalized n+1Fn-hypergeometricfunction is the same as McCarthy’s hypergeometricfunction over finite fields when λ 6= 0.

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Results

Theorem (Fuselier, Long, Ramakrishna, S-, Tu)

Suppose we have a classical formula relating two 2F1

functions that satisfies the property that it can be provedusing only the binomial theorem, the reflection andmultiplication formulas, or their corollaries, under somespecific technical conditions. Then, there exists an explicitN ∈ N, and a finite set Σ ⊂ Q, such that for each primepower q ≡ 1 (mod N), the finite field analogue of theformula over Fq holds for almost all x ∈ Fq except a fewdetermined by Σ.

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Example

Let A be any character on F×q such that A6 is not trivial,then for any z ∈ Fq such that z − 1 or z − 1/3 is not zero inFq, then we can show

2F1

[A A

φ;

27z(1− z)2

4

]= 2F1

[A3 A3

φ;

3z

4

],

which is an analogue of the following formula in theclassical case of Gessel and Stanton

2F1

[a −a

12

;27z(1− z)2

4

]= 2F1

[3a −3a

12

;3z

4

].

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Back to the Beginning

There is another natural perspective one can take whenstudying supercongruences arising from Ramanujan’s seriesrepresentations for 1/π, and this is through the geometry ofabelian varieties. In particular, the original Ramanujanformulas for 1/π are all related to elliptic curves withcomplex multiplication (CM). More next week!

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Thank you!

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