Hypergeometric Series and Gaussian Hypergeometric Functions - PowerPoint PPT Presentation

Hypergeometric Series and Gaussian Hypergeometric Functions Fang-Ting Tu , joint with Alyson Deines, Jenny Fuselier, Ling Long, Holly Swisher a Women in Numbers 3 project National Center for Theoretical Sciences, Taiwan Mini-workshop at LSU on Algebraic Varieties, Hypergeometric series, and Modular Forms Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 1 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins 2 F 1 -Hypergeometric Series � � a b • Let a , b , c ∈ R . The hypergeometric function 2 F 1 c ; z is defined by � � � ∞ a b ( a ) n ( b ) n ( c ) n n ! z n , 2 F 1 c ; z = n = 0 where ( a ) n = a ( a + 1 ) . . . ( a + n − 1 ) is the Pochhammer symbol. • Euler’s integral representation of the 2 F 1 with c > b > 0 � � � 1 a , b 1 x b − 1 ( 1 − x ) c − b − 1 ( 1 − λ x ) − a dx , 2 F 1 c ; λ = B ( b , c − b ) 0 where � 1 x a − 1 ( 1 − x ) b − 1 dx = Γ( a )Γ( b ) B ( a , b ) = Γ( a + b ) 0 is the Beta function. Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 2 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins Hypergeometric Functions over Finite Fields Let q = p s be a prime power. Let � F × q denote the group of multiplicative q . Extend χ ∈ � characters on F × F × q to F q by setting χ ( 0 ) = 0. Gaussian Hypergeometric Function. (Greene, 1984) Let λ ∈ F q , and A , B , C ∈ � F × q . � A � � := ε ( λ ) BC ( − 1 ) B • 2 F 1 C ; λ B ( x ) BC ( 1 − x ) A ( 1 − λ x ) , q q x ∈ F q where ε is the trivial character. • � A � � A χ � � B χ � � B q 2 F 1 C ; λ := χ ( λ ) , χ C χ q − 1 q χ ∈ � F × q � A � := B ( − 1 ) where J ( A , B ) is the normalized Jacobi sum of A , B q B . Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 3 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins Legendre Family For λ � = 0, 1, let E λ : y 2 = x ( x − 1 )( x − λ ) be the elliptic curve in Legendre normal form. • The periods of the Legendre family of elliptic curves are � ∞ dx Ω( E λ ) = � x ( x − 1 )( x − λ ) 1 • If 0 < λ < 1, then � 1 � 1 = Ω( E λ ) 2 2 2 F 1 1 ; λ . π If λ ∈ Q , and E λ has good reduction at prime p , we can express # E λ ( F p ) in terms of Gaussian hypergeometric functions. Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 4 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins Legendre Family over Finite Fields Legendre family of elliptic curves over F p : E λ : y 2 = x ( x − 1 )( x − λ ) � Trace of Frobenius: a p ( λ ) = p + 1 − # � E λ ( F p ) , λ � = 0 , 1 Koike 1992. If p is an odd prime, then � � η 2 η 2 p 2 F 1 ε ; λ = − η 2 ( − 1 ) a p ( λ ) , λ � = 0 , 1 , p where ε is the trivial character and η 2 is the quadratic character. Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 5 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins E λ : y 2 = x ( x − 1 )( x − λ ) , λ ∈ Q − { 0 , 1 } . • If 0 < λ < 1, then � 1 � 1 = Ω( E λ ) = Ω( E λ ) 2 2 2 F 1 1 ; λ 2 ) 2 . Γ( 1 π • If p is an odd prime with ord p ( λ ( λ − 1 )) = 0, then � � η 2 η 2 = − a p ( λ ) p η 2 ( − 1 ) = − a p ( λ ) 2 F 1 ε ; λ g ( η 2 ) 2 . p • If λ = 1 2 , p ≡ 1 mod 4, we have √ � 1 / 4 � � η 4 � 2 − η 2 ( 2 ) 2 π Ω( E λ ) = Re , a p ( λ ) = Re , 1 / 2 2 p η 2 where η 4 is a character of order 4. Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 6 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins For m ∈ Z + , define the truncated 2 F 1 -hypergeometric series by � � m � a b ( a ) k ( b ) k ( c k ) k ! λ k . 2 F 1 c ; λ := m k = 0 When a p ( λ ) is not divisible by p , Dwork shows that � 1 � � 1 � � 1 1 1 ; ˆ 1 ; ˆ 2 2 2 2 f p ( λ ) := lim s →∞ 2 F 1 λ 2 F 1 λ p s − 1 p s − 1 − 1 is the unit root of T 2 − a p ( λ ) T + p , where ˆ λ is the image of λ under the Teichmüller character. Example. When λ = − 1, p ≡ 1 ( mod 4 ) , � η 2 � η 2 • − a p ( − 1 ) = p · 2 F 1 ε ; − 1 = J ( η 4 , η 2 ) + J ( η 4 , η 2 ) . p Γ p ( 1 2 ) Γ p ( 1 4 ) • f p ( λ ) = , where Γ p ( · ) is the p -adic Gamma function. Γ p ( 3 4 ) Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 7 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins Motivations hypergeometric series ← → periods Gaussian hypergeometric series ← → Golais representations truncated hypergeometric series ← → unit roots Motivation Investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of hypergeometric algebraic varieties. • y N = x i ( 1 − x ) j ( 1 − λ x ) k • y n = ( x 1 x 2 · · · x n − 1 ) n − 1 ( 1 − x 1 ) · · · ( 1 − x n − 1 )( x 1 − λ x 2 x 3 · · · x n − 1 ) Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 8 / 32

Introduction 2 F 1 -Hypergeometric Series/Functoins Generalized Legendre Curves Let N ≥ 2, and i , j , k be natural numbers with 1 ≤ i , j , k < N . For the smooth model X λ of the curve C λ : y N = x i ( 1 − x ) j ( 1 − λ x ) k , λ ∈ Q − { 0 , 1 } • a period can be chosen as � k � � � N − i 1 − i N , 1 − j N N P ( λ ) = B 2 F 1 ; λ , 2 N − i − j N N • Let η ∈ � F × q be a character of order N . Then � η − km � N − 1 � η im η mj ( − 1 ) 2 F 1 # X λ ( F q )” = ” 1 + q + q η m ( i + j ) ; λ . q m = 1 Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 9 / 32

Main Results Generalized Hypergeometric Series/Functions Generalized Hypergeometric Series/Functions • For a positive integer n , and α i , β i ∈ C with β i �∈ Z − , the hypergeometric series n + 1 F n is defined by � � n � � ∞ α 0 α 1 . . . α n ( α 0 ) k ( α i ) k · λ k n + 1 F n ; λ := ( 1 ) k ( β i ) k β 1 . . . β n k = 0 i = 1 where ( a ) 0 := 1, ( 1 ) k = k ! , and ( a ) k := a ( a + 1 ) · · · ( a + k − 1 ) . • If n is a positive integer, and A i , B i ∈ � F × q , then � A 0 � � A 0 χ � � A i χ � � � n A 1 . . . A n q n + 1 F n B n ; λ := χ ( λ ) . B 1 . . . q − 1 χ B i χ q χ ∈ � i = 1 F × q Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 10 / 32

Main Results Generalized Hypergeometric Series/Functions Euler’s Integral Formulae When Re ( β r ) > Re ( α r ) > 0 , � � α 0 α 1 . . . α n n + 1 F n ; λ = β 1 . . . β n � � � 1 α 0 α 1 . . . α n − 1 Γ( β n ) x α n − 1 ( 1 − x ) β n − α n − 1 n F n − 1 ; λ x dx Γ( α n )Γ( β n − α n ) β 1 . . . β n − 1 0 For characters A 0 , A 1 , . . . , A n , B 1 , . . . , B n in � F × q , � A 0 , � A 1 , . . . , A n n + 1 F n B n ; λ = B 1 , . . . , q � A 0 , � � A n B n ( − 1 ) A 1 , . . . , A n − 1 · A n ( x ) A n B n ( 1 − x ) · n F n − 1 B n − 1 ; λ x . B 1 , . . . , q q x Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 11 / 32

Main Results Generalized Hypergeometric Series/Functions Higher Dimensional Analogues of Legendre Curves y n = ( x 1 x 2 · · · x n − 1 ) n − 1 ( 1 − x 1 ) · · · ( 1 − x n − 1 )( x 1 − λ x 2 x 3 · · · x n − 1 ) C n ,λ : • C 2 ,λ are known as Legendre curves. � j � j j · · · n n n • Up to a scalar multiple, n F n − 1 ; λ for any 1 · · · 1 1 ≤ j ≤ n − 1, when convergent, can be realized as a period of C n ,λ . Theorem (Deines, Long, Fuselier, Swisher, T.) Let q = p e ≡ 1 ( mod n ) be a prime power. Let η n be a primitive order n character and ε the trivial multiplicative character in � F × q . Then � η i � n − 1 � η i η i n , n , · · · , n , # C n ,λ ( F q ) = 1 + q n − 1 + q n − 1 n F n − 1 ε, ; λ . ε, · · · , q i = 1 Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 12 / 32

Main Results Generalized Hypergeometric Series/Functions Local L -functions of C 3 , 1 and C 4 , 1 Theorem (Deines, Long, Fuselier, Swisher, T.) Let η 2 , η 3 , and η 4 denote characters of order 2, 3, or 4, respectively, in � F × q . • Let q ≡ 1 ( mod 3 ) be a prime power. Then � η 3 , � η 3 , η 3 q 2 · 3 F 2 = J ( η 3 , η 3 ) 2 − J ( η 2 3 , η 2 ε ; 1 3 ) . ε, q • Let q ≡ 1 ( mod 4 ) be a prime power. Then � η 4 , � η 4 , η 4 , η 4 q 3 · 4 F 3 = J ( η 4 , η 2 ) 3 + qJ ( η 4 , η 2 ) − J ( η 4 , η 2 ) 2 . ε ; 1 ε, ε, q Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 13 / 32

Main Results Generalized Hypergeometric Series/Functions Ahlgren-Ono. For any odd prime p , � η 2 � η 2 η 2 η 2 4 , 4 , 4 , p 3 · 4 F 3 4 ε ; 1 = − a ( p ) − p , ε, ε, p where a ( p ) is the p th coefficient of the weight-4 Hecke eigenform η ( 2 z ) 4 η ( 4 z ) 4 , with η ( z ) being the Dedekind eta function. The factor of Z C 4 , 1 corresponding to y 2 = ( x 1 x 2 x 3 ) 3 ( 1 − x 1 )( 1 − x 2 )( 1 − x 3 )( x 1 − x 2 x 3 ) is 4 , 1 ( T , p ) = ( 1 − a ( p ) T + p 3 T 2 )( 1 − pT ) Z C old . ( 1 − T )( 1 − p 3 T ) Fang Ting Tu (NCTS) Hypergeometric Functions April 14th, 2015 14 / 32