Hypergeometric Functions and Hypergeometric Abelian Varieties

Hypergeometric Functions and Hypergeometric Abelian Varieties Fang-Ting Tu Louisiana State University September 29th, 2016 BIRS Workshop: Modular Forms in String Theory Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 1 / 33

Introduction Classical hypergeometric functions are well-understood. They are related to periods of algebraic varieties triangle groups, modular forms on arithmetic triangle groups Ramanujan type identities, combinatorial identities, physical applications... Hypergeometric functions over finite fields are developed theoretically by Evans, Greene, Katz, McCarthy, Ono,... They are related to L-functions of algebraic varieties character sum identities supercongruences (Apéry or Ramanujan type)... Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 2 / 33

Introduction For 0, 1, let E : y 2 = x(1 x)(1 x) be the elliptic curve in Legendre normal form. 1 0 1 1 1 0 0 1 1 1 0 1 A period of E is Ω(E ) = 1 0 dx 1 y = dx. 0 x(1 x)(1 x) and Ω(E ) π = 2 F 1 [ 1 2 ] 1 2 1 ; := ( 1 2 + n 1 ) 2 n. n n=0 Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 3 / 33

Introduction For Q and 0, 1 mod p, #Ẽ(F p ) = p + 1 a p (), where a p () = ( ) x(1 x)(1 x). p x F p The value a p () is the trace of Frobenius map; the p-th Fourier coefficient of certain modular form. a p () can be thought as a finite field analogue of the period Ω(E ) = 1 0 (x(1 x)(1 x)) 1/2 dx. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 4 / 33

Introduction 2F 1 -hypergeometric Function Let a, b, c Q. The hypergeometric function 2 F 1 [ a b c ; z ] by 2F 1 [ a b c ; z ] = n=0 (a) n (b) n (c) n n! zn, is defined where (a) n = a(a + 1)... (a + n 1) is the Pochhammer symbol. For fixed a, b, c and argument z, the function 2 F 1 [ a b c ; z ] can be viewed as a quotient of periods on certain algebraic varieties. satisfies a hypergeometric differential equation, whose monodromy group is a triangle group. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 5 / 33

2F 1 -Hypergeometric Functions Hypergeometric Differential Equation Hypergeometric Differential Equation For fixed a, b, c and argument z, the function 2 F 1 [ a b c ; z ] the hypergeometric differential equation HDE(a, b, c; z) : F + with 3 regular singularities at 0, 1, and. satisfies (a + b + 1)z c F ab + z(1 z) z(1 z) F = 0, Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 6 / 33

2F 1 -Hypergeometric Functions Hypergeometric Differential Equation Theorem (Schwarz) Let f, g be two independent solutions to HDE(a, b; c; ) at a point z H, and let p = 1 c, q = c a b, and r = a b. Then the Schwarz map D = f /g gives a bijection from H R onto a curvilinear triangle with vertices D(0), D(1), D( ), and corresponding angles pπ, qπ, rπ. H D(0) pπ D rπ D(1) qπ D( ) Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 7 / 33

2F 1 -Hypergeometric Functions Hypergeometric Differential Equation The universal cover of (p, q, r) is one of the followings: the unit sphere (p + q + r > 1); the Euclidean plane (p + q + r = 1); the hyperbolic plane (p + q + r < 1). When p, q, r are rational numbers in the lowest form with 0 = 1, let e i be the denominators of p, q, r arranged in the non-decreasing order, the monodromy group is isomorphic to the triangle group (e 1, e 2, e 3 ), where (e 1, e 2, e 3 ) := x, y x e 1 = y e 2 = (xy) e 3 = id. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 8 / 33

Example For 2 F 1 [ 1 2 1 2 2F 1 -Hypergeometric Functions ] Hypergeometric Differential Equation 1 ;, the triangle (p, q, r) = (0, 0, 0) is a hyperbolic triangle. The corresponding monodromy group is the arithmetic triangle group (,, ) Γ(2). Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 9 / 33

2 F1 -Hypergeometric Functions Hypergeometric Differential Equation Example " For 2 F1 1 84 13 84 1 2 # ;, the triangle (p, q, r ) = (1/2, 1/3, 1/7) is a hyperbolic triangle. The corresponding monodromy group is the arithmetic triangle group (2, 3, 7). Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 10 / 33

2F 1 -Hypergeometric Functions Generalized Legendre Curves Euler s integral representation of the 2 F 1 with c > b > 0 2P 1 [ a b c ; ] = 1 0 x b 1 (1 x) c b 1 (1 x) a dx = 2 F 1 [ a b c ; ] B(b, c b), where B(a, b) = 1 0 x a 1 (1 x) b 1 dx = Γ(a)Γ(b) Γ(a + b) is the Beta function. Following Wolfart, we can realize 2 P 1 [ a b c ; ] as a period of C [N;i,j,k] : y N = x i (1 x) j (1 x) k, where N = lcd(a, b, c), i = N(1 b), j = N(1 + b c), k = Na. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 11 / 33

Examples 2F 1 -Hypergeometric Functions Generalized Legendre Curves The function B ( [ 12 84, 29 ) 1 84 84 2F 1 ] 13 84 1 2 ; is a period of the curve C [84;71,55,1] : y 84 = x 71 (1 x) 55 (1 x). For the curve C [6;4,3,1] : y 6 = x 4 (1 x) 3 (1 x), B ( [ ] 1 3, 1 ) 1 1 6 3 2 2F 1 ; is a period. 5 6 the corresponding triangle group is Γ (3, 6, 6) Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 12 / 33

2F 1 -Hypergeometric Functions Generalized Legendre Curves Motivation Study the arithmetic of generalized Legendre curves y N = x i (1 x) j (1 zx) k, which are parameterized by Shimura curves; general hypergeometric varieties y N = x i 1 1 x in n (1 x 1 ) j1 (1 x n ) jn (1 zx 1 x 2 x 3 x n ) k. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 13 / 33

Period Functions over Finite Fields Let p be a prime, and q = p s. Let F q denote the group of multiplicative characters on F q. Extend χ F q to F q by setting χ(0) = 0. Definition Let F q, and A, B, C F q. Define 2P 1 ( A ) B C = B(x)BC(1 x)a(1 x). x F q This is a finite field analogue of 2P 1 [ a b c ; ] = 1 0 x b 1 (1 x) c b 1 (1 x) a dx Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 14 / 33

Let X [N;i,j,k] be the smooth model of C [N;i,j,k] : y N = x i (1 x) j (1 x) k. Let q be an odd prime power, and let i, j, k be natural numbers with 1 i, j, k < N. Further, let η N F q be a character of order N. Then for F q \ {0, 1}, # X [N;i,j,k] N 1 = 1 + q + m=1 ( # X [N;i,j,k] N 1 η km = 1 + q + N 2P 1 x F q η m N (x i (1 x) j (1 x) k ). m=1 ηn im η (i+j)m N ). Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 15 / 33

Let X [N;i,j,k] be the smooth model of C [N;i,j,k] : y N = x i (1 x) j (1 x) k. For each d N, there is a natural covering from C [N;i,j,k] For a given curve C [N;i,j,k], we let J new be the subvariety of Jac(X [N;i,j,k] ) which is not induced from C [d;i,j,k] for all d N, d < N. to C [d;i,j,k]. dim J new = ϕ(n). Let K be the Galois closure of Q(). For any fixed prime l, one can construct a compatible family of degree-2ϕ(n) representations ρ new l () : G K := Gal(K /K ) GL 2ϕ(N) (Z l ) via the Tate module of J new. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 16 / 33

For any good prime p O K with residue field F q, Trρ new l ()(Frob p ) = m (Z /NZ) 2P 1 ( η km N ηn im η (i+j)m N ) Let ζ be a primitive Nth root of unity. The map A ζ : (x, y) (x, ζ 1 y) induces an action on the ρ l. Consequently, ρ new l () GK (ζ) = gcd(m,n)=1 Here σ m,l () is 2-dimensional when (n, N) = 1. σ m,l (). Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 17 / 33

Theorem (Fuselier, Long, Ramakrishna, Swisher, T.) If gcd(m, N) = 1, then ( η km Trσ m,l (Frob q ) and 2 P N 1 agree up to different embeddings of Q(ζ N ) in C Theorem (Deines, Fuselier, Long, Swisher, T.) ηn im η m(i+j) N ) Let N = 3, 4, 6, and N i + j + k. Then for each Q, the endomorphism algebra of J new contains a quaternion algebra over Q if and only if ( i B N, j ) / ( N k B N N, i + j + k N ) Q. N Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 18 / 33

( η km For a given, let S m := 2 P N 1 ηn im η (i+j)m N ). Atkin-Li-Liu-Long: If End(J new ) contains a quaternion algebra, then the function is a character. F(η N ) := S 1 /S N 1 = J(ηN i, ηj N )/J(η k N, ηi+j+k N ) Yamamoto: The quotient ( i B N, j N is algebraic. ) /B ( k N, i + j + k ) N Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 19 / 33

2F 1 -hypergeometric Functions over Finite Fields Define 2F 1 [ A B C ; ] = which is a finite field analogue of 2F 1 [ a b c ; ] = Γ(a) g(a) B(a, b) J(A, B) 1 J(B, CB) 2 P 1 1 B(b, c b) 2 P 1 ( ) A B C, [ a b ] c ; Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 20 / 33

Transformation Formulas and Applications a = i N A = η i N Γ(a) g(a) B(a, b) = Γ(a)Γ(b) Γ(a+b) J(A, B) = g(a)g(b) g(ab) if A A Γ(a)Γ(1 a) = π sin aπ, a / Z g(a)g(a) = A( 1)q, A ε Gauss multiplication formula Γ(ma)(2π) (m 1)/2 = m ma 1 2 Γ(a)Γ ( a + 1 m) Γ ( a + m 1 m χ F q χ m =ε g(χψ) = g(ψ m )ψ(m m ) χ F g(χ) q χ m =ε Hasse-Davenport relation ) Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 21 / 33

Transformation Formulas and Applications Theorem (Fuselier-Long-Ramakrishna-Swisher-T.) Let q be an odd prime power, z 1 F q, φ be the quadratic character, and A F q of order larger than 2. Then 2F 1 [ A φa φ ; z ] = { 0, if z is not a square, A 2 (1 + z) + A 2 (1 z), if z is a square. This is the analogue to the classical result [ ] a a + 1 2 2F 1 ; z = 1 ( (1 + z) 2a + (1 z) 2a). 1 2 2 Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 22 / 33

Transformation Formulas and Applications Theorem (Fuselier-Long-Ramakrishna-Swisher-T.) Let q be an odd prime power, z 1 F q, φ be the quadratic character, and A F q of order larger than 2. Then ( ) [ ] 1 A φa A φa J(φA, A) 2 P 1 φ z = 2 F 1 φ ; z = { 0, if z is not a square, A 2 (1 + z) + A 2 (1 z), if z is a square. This is the analogue to the classical result [ ] a a + 1 2 2F 1 ; z = 1 ( (1 + z) 2a + (1 z) 2a). 1 2 2 Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 22 / 33

Transformation Formulas and Applications Theorem (T.-Yang) Let p be a prime congruent to 1 modulo 4, p be a prime ideal of Z[i] lying above p. Let ψ p be the quartic multiplicative character on F p satisfying ψ p (x) x (p 1)/4 mod p, for every x Z[i]. Then, for a 0, 1, if one of a and 1 a is not a square in F p, we have 2P 1 ( φψp ) φ a = 0, ψ p and 2P 1 ( φψp ) φ a = 2ψ p ( 1)φ (1 + b) χ(p), ψ p if a = b 2 for some b F p, where χ is the Hecke character associated to the elliptic curve E : y 2 = x 3 x satisfying χ(p) p for all primes p of Z [i]. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 23 / 33

Transformation Formulas and Applications The curve C 4 : y 4 = x(x 1)(x a) has genus 3 and it is a 2-fold cover of the following 3 elliptic curves C 2 : y 2 = x(x 1)(x a), E + : y 2 = x 3 + (1 + b) 2 x, E : y 2 = x 3 + (1 b) 2 x. We have similar results for ψ p of order N = 3, 6, 8, and 12. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 24 / 33

Quadratic Formula Transformation Formulas and Applications Theorem (Fuselier, Long, Ramakrishna, Swisher, and T.) Let B, D F q, and set C = D 2. When D φ, B D and x ±1, we have [ ] [ ] DφB D C(1 x) 2 F 1 CB ; 4x B (1 x) 2 = 2 F 1 CCB ; x. This is the analogue to the classical result (1 z) c 2F 1 [ 1+c 2 b c 2 c b + 1 ; 4z (1 z) 2 ] [ ] b c = 2 F 1 c b + 1 ; z. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 25 / 33

Transformation Formulas and Applications Analogue to the classical result [ ] a a 1 ( ) 1 2a 2 1 + 1 z 2F 1 2a ; z =. 2 Let q be an odd prime power, z F q, φ be the quadratic character, and A F q of order larger than 2. Then 2F 1 [ A ] φa A 2 ; z = { 0, ( if φ(1 z) = 1, A 2 1+ ) 1 z 2 + A 2 ( 1 1 z 2 ), if φ(1 z) = 1. Remark The Galois perspective tells us that "using the dictionary directly" does not work. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 26 / 33

Transformation Formulas and Applications Let X [6;4,3,1] and X [12;9,5,1] be the smooth models of respectively. y 6 = x 4 (1 x) 3 (1 ), and y 12 = x 9 (1 x) 5 (1 ), Theorem (Fuselier, Long, Ramakrishna, Swisher, and T.) Let Q such that 0, ±1. Let J,1 new (resp. Jnew 4 be the primitive (1 ) 2,2) part of the Jacobian variety of X [6;4,3,1] (resp. X [12;9,5,1] ). Then J new 4 Jnew (1 ) 2,2,1 Jnew,1 over some number field depending on. Their corresponding arithmetic traingle groups are (2, 6, 6) 2 (3, 6, 6) Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 27 / 33

Generalized Period Functions As an analogue of the classical hypergeometric series, we inductively define [ ] A0 A 1... A n n+1p n ; := B 1... B n y F q A n (y)a n B n (1 y) np n 1 [ ] A0 A 1... A n 1 ; y B 1... B n 1 [ ] A0 A 1 A n n+1f n ; B 1 B n [ ] 1 A0 A := n i=1 J(A i, B i A i ) n+1 P 1... A n n ;, B 1... B n where A i, B j F q, and F q. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 28 / 33

Generalized Period Functions Example: Consider the higher dimensional analogue of the legendre curve: C n, : y n = (x 1 x 2 x n 1 ) n 1 (1 x 1 ) (1 x n 1 )(1 x 1 x 2 x 3 x n 1 ) C 2, are known as Legendre curves. [ ] j j j Up to a scalar multiple, n F n n n n 1 1 1 ; for any 1 j n 1, when convergent, can be realized as a period of C n+1,. 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 n 1 ( #C n, (F q ) = 1 + q n 1 η i + n ηn np i η i ) n n 1 ε ε ;. i=1 Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 29 / 33

Generalized Period Functions Local L-functions of C 3,1 and C 4,1 Theorem (Deines, Long, Fuselier, Swisher, T.) Let η 3, and η 4 denote characters of order 3, or 4, respectively, in F q. Let q 1 (mod 3) be a prime power. Then ( ) η3 η 3 η 3 3P 2 ε ε ; 1 = J(η 3, η 3 ) 2 J(η3 2, η2 3 ). (Greene s transformation formula) Let q 1 (mod 4) be a prime power. Then ( ) η4 η 4 η 4 η 4 4P 3 ε ε ε ; 1 = J(η 4, φ) 3 + qj(η 4, φ) J(η 4, φ) 2. (McCarthy s finite field version of Whipple s formula) Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 30 / 33

Generalized Period Functions C 4,1 : y 4 = (x 1 x 2 x 3 ) 3 (1 x 1 )(1 x 2 )(1 x 3 )(x 1 x 2 x 3 ) Ahlgren-Ono For any odd prime p, ( η 2 4P 4 η4 2 η4 2 η 2 ) 4 3 ε ε ε ; 1 = a(p) p, where a(p) is the pth coefficient of the weight-4 Hecke eigenform η(2z) 4 η(4z) 4, with η(z) being the Dedekind eta function. The factor of the zeta function Z C4,1 (T, p) 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 Z C old 4,1 (T, p) = (1 a(p)t + p3 T 2 )(1 pt ) (1 T )(1 p 3. T ) Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 31 / 33

Generalized Period Functions ( ) η4 η 4 P 4 η 4 η 4 3 ε ε ε ; 1 = J(η 4, φ) 3 + qj(η 4, φ) J(η 4, φ) 2 Hasse-Davenport relation Let F be a finite field and F s an extension field over F of degree s. If χ ε F and χ s = χ N Fs/F a character of F s. Then ( g(χ)) s = g(χ s ). The factor of Z C4,1 (T,p) corresponding to new part is (1 + (β 3 p + β 3 p)t + p 3 T 2 ) (1 + (β p + β p )pt + p 3 T 2 ) (1 (β 2 p + β 2 p)t + p 2 T 2 ), where β p = J(η 4, φ). Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 32 / 33

Generalized Period Functions Batyrev-Van Straten: Calabi-Yau manifolds whose Picard Fuchs equations are hypergeometric functions of the form [ ] { d1 1 d 1 d 2 1 d 2 1 4F 3 ; z, d 1, d 2 1 1 1 2, 1 3, 1 4, 1 }. 6 Conjectures: z = 1 [Cohen] 4F 3 [ η3 η 3 η 4 η 4 ε ε ε ; 1 ] = J(η 3, η 3 ) 3 J(η 3, η 3 ) 3 + η 12 ( 1)p [Long] Numerically, Long finds the weight 4 cuspidal Hecke forms corresponding to d 1 = 1 2 and d 2 = 1 3, 1 4, 1 6. Fang Ting Tu (LSU) Hypergeometric Functions Sep 29, 2016 33 / 33