A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this ... more A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data HD are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of-adic representations of the absolute Galois group of Q attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7F6(1) in Whipple's formula to the periods of these modular forms.
In this paper, we investigate the relationships among hypergeometric series, truncated hypergeome... more In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of 'hypergeometric' algebraic varieties that are higher dimensional analogues of Legendre curves.
We define a finite-field version of Appell-Lauricella hypergeometric functions built from period ... more We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al [14] in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell-Lauricella functions to establish a finite-field analogue of Koike and Shiga's cubic transformation [18] for the Appell hypergeometric function F1, proving a conjecture of Ling Long. We use our multivariable period functions to construct formulas for the number of Fp-points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite-field Appell-Lauricella functions.
8.2. A Pfaff-Saalschütz evaluation formula 8.3. A few analogues of algebraic hypergeometric formu... more 8.2. A Pfaff-Saalschütz evaluation formula 8.3. A few analogues of algebraic hypergeometric formulas 9. Quadratic or Higher Transformation Formulas 9.1. Some results related to elliptic curves 9.2. A Kummer quadratic transformation formula 9.3. The quadratic formula in connection with the Kummer relations 9.4. A finite field analogue of a theorem of Andrews and Stanton 9.5. Another application of Bailey cubic transformations 9.6. Another cubic 2 F 1 formula and a corollary 10. An application to Hypergeometric Abelian Varieties 11. Open Questions and Concluding Remarks 11.1. Numeric observations 12. Appendix 12.1. Bailey 3 F 2 cubic transforms 12.2. A proof of a formula by Gessel and Stanton References Index
In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions.... more In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions. The main novelty in our approach is the interpretation of identities among 2 F 1-hypergeometric functions as identities among automorphic forms on different Shimura curves.
Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanuja... more Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for 1/π. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.
Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanuja... more Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for 1/π. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.
In this note we will obtain defining equations of modular curves X 0 (2 2n). The key ingredient i... more In this note we will obtain defining equations of modular curves X 0 (2 2n). The key ingredient is a recursive formula for certain generators of the function fields on X 0 (2 2n).
Symmetry, Integrability and Geometry: Methods and Applications, 2018
For an odd prime p, let φ denote the quadratic character of the multiplicative group F × p , wher... more For an odd prime p, let φ denote the quadratic character of the multiplicative group F × p , where F p is the finite field of p elements. In this paper, we will obtain evaluations of the hypergeometric functions 2 F 1 φψ ψ φ ; x , x ∈ F p , x = 0, 1, over F p in terms of Hecke character attached to CM elliptic curves for characters ψ of F × p of order 3, 4, 6, 8, and 12.
We consider algebraic transformations of hypergeometric functions from a geometric point of view.... more We consider algebraic transformations of hypergeometric functions from a geometric point of view. Hypergeometric functions are shown to arise from the deRham realization of a hypergeometric motive. The $\ell$-adic realization of the motive gives rise to hypergeometric characters sums over finite fields. This helps to unify and explain some recent results about transformations of hypergeometric character sums.
In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analo... more In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analogue of the ideal lattices from number fields. In particular, we will construct the root lattices D_{4},
In this note we obtain defining equations of modular curves $X_0(2^{2n})$. The key ingredient is ... more In this note we obtain defining equations of modular curves $X_0(2^{2n})$. The key ingredient is a recursive formula for certain generators of the function fields on $X_0(2^{2n})$.
Let X D 0 (N), where (D, N) = 1, denote the Shimura curve associated to an Eichler order of level... more Let X D 0 (N), where (D, N) = 1, denote the Shimura curve associated to an Eichler order of level N, in an indefinite quaternion algebra over ޑ of discriminant D. Let W D,N be the group of all Atkin-Lehner involutions on X D 0 (N) and W D the subgroup consisting of Atkin-Lehner involutions w m with m | D. In this paper, we will determine Schwarzian differential equations associated to Shimura curves X D 0 (N)/ W D of genus zero in the cases where there exists a squarefree integer M > 1 such that X D 0 (M)/ W D is of genus zero.
In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analo... more In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analogue of the ideal lattices from number
In this report, we prove identities among the second, third and fourth twisted moments of twisted... more In this report, we prove identities among the second, third and fourth twisted moments of twisted Kloosterman sums, and certain hypergeometric functions over finite fields. In some cases, the moments can be expressed in terms of Hecke characters of imaginary quadratic fields.
Transactions of the American Mathematical Society, 2013
In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions.... more In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions. The main novelty in our approach is the interpretation of identities among 2 F 1-hypergeometric functions as identities among automorphic forms on different Shimura curves.
A well-known formula of Whipple relates certain hypergeometric values 7 F 6 (1) and 4 F 3 (1). In... more A well-known formula of Whipple relates certain hypergeometric values 7 F 6 (1) and 4 F 3 (1). In this paper, we revisit this relation from the viewpoint of the underlying hypergeometric data H D, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data H D are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of-adic representations of the absolute Galois group of Q attached to H D. For specialized choices of H D, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7 F 6 (1) in Whipple's formula to the periods of these modular forms. Keywords Hypergeometric functions • Whipple's 7 F 6 formula • Hypergeometric character sums • Galois representations and modular forms Mathematics Subject Classification 11F11 • 33C20 • 11F80 • 11F67 • 11T24 B Ling Long
A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this ... more A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data HD are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of-adic representations of the absolute Galois group of Q attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7F6(1) in Whipple's formula to the periods of these modular forms.
In this paper, we investigate the relationships among hypergeometric series, truncated hypergeome... more In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of 'hypergeometric' algebraic varieties that are higher dimensional analogues of Legendre curves.
We define a finite-field version of Appell-Lauricella hypergeometric functions built from period ... more We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al [14] in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell-Lauricella functions to establish a finite-field analogue of Koike and Shiga's cubic transformation [18] for the Appell hypergeometric function F1, proving a conjecture of Ling Long. We use our multivariable period functions to construct formulas for the number of Fp-points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite-field Appell-Lauricella functions.
8.2. A Pfaff-Saalschütz evaluation formula 8.3. A few analogues of algebraic hypergeometric formu... more 8.2. A Pfaff-Saalschütz evaluation formula 8.3. A few analogues of algebraic hypergeometric formulas 9. Quadratic or Higher Transformation Formulas 9.1. Some results related to elliptic curves 9.2. A Kummer quadratic transformation formula 9.3. The quadratic formula in connection with the Kummer relations 9.4. A finite field analogue of a theorem of Andrews and Stanton 9.5. Another application of Bailey cubic transformations 9.6. Another cubic 2 F 1 formula and a corollary 10. An application to Hypergeometric Abelian Varieties 11. Open Questions and Concluding Remarks 11.1. Numeric observations 12. Appendix 12.1. Bailey 3 F 2 cubic transforms 12.2. A proof of a formula by Gessel and Stanton References Index
In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions.... more In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions. The main novelty in our approach is the interpretation of identities among 2 F 1-hypergeometric functions as identities among automorphic forms on different Shimura curves.
Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanuja... more Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for 1/π. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.
Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanuja... more Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for 1/π. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples.
In this note we will obtain defining equations of modular curves X 0 (2 2n). The key ingredient i... more In this note we will obtain defining equations of modular curves X 0 (2 2n). The key ingredient is a recursive formula for certain generators of the function fields on X 0 (2 2n).
Symmetry, Integrability and Geometry: Methods and Applications, 2018
For an odd prime p, let φ denote the quadratic character of the multiplicative group F × p , wher... more For an odd prime p, let φ denote the quadratic character of the multiplicative group F × p , where F p is the finite field of p elements. In this paper, we will obtain evaluations of the hypergeometric functions 2 F 1 φψ ψ φ ; x , x ∈ F p , x = 0, 1, over F p in terms of Hecke character attached to CM elliptic curves for characters ψ of F × p of order 3, 4, 6, 8, and 12.
We consider algebraic transformations of hypergeometric functions from a geometric point of view.... more We consider algebraic transformations of hypergeometric functions from a geometric point of view. Hypergeometric functions are shown to arise from the deRham realization of a hypergeometric motive. The $\ell$-adic realization of the motive gives rise to hypergeometric characters sums over finite fields. This helps to unify and explain some recent results about transformations of hypergeometric character sums.
In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analo... more In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analogue of the ideal lattices from number fields. In particular, we will construct the root lattices D_{4},
In this note we obtain defining equations of modular curves $X_0(2^{2n})$. The key ingredient is ... more In this note we obtain defining equations of modular curves $X_0(2^{2n})$. The key ingredient is a recursive formula for certain generators of the function fields on $X_0(2^{2n})$.
Let X D 0 (N), where (D, N) = 1, denote the Shimura curve associated to an Eichler order of level... more Let X D 0 (N), where (D, N) = 1, denote the Shimura curve associated to an Eichler order of level N, in an indefinite quaternion algebra over ޑ of discriminant D. Let W D,N be the group of all Atkin-Lehner involutions on X D 0 (N) and W D the subgroup consisting of Atkin-Lehner involutions w m with m | D. In this paper, we will determine Schwarzian differential equations associated to Shimura curves X D 0 (N)/ W D of genus zero in the cases where there exists a squarefree integer M > 1 such that X D 0 (M)/ W D is of genus zero.
In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analo... more In this paper, we will discuss ideal lattices from a denite quaternion algebra, which is an analogue of the ideal lattices from number
In this report, we prove identities among the second, third and fourth twisted moments of twisted... more In this report, we prove identities among the second, third and fourth twisted moments of twisted Kloosterman sums, and certain hypergeometric functions over finite fields. In some cases, the moments can be expressed in terms of Hecke characters of imaginary quadratic fields.
Transactions of the American Mathematical Society, 2013
In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions.... more In this paper, we will obtain new algebraic transformations of the 2 F 1hypergeometric functions. The main novelty in our approach is the interpretation of identities among 2 F 1-hypergeometric functions as identities among automorphic forms on different Shimura curves.
A well-known formula of Whipple relates certain hypergeometric values 7 F 6 (1) and 4 F 3 (1). In... more A well-known formula of Whipple relates certain hypergeometric values 7 F 6 (1) and 4 F 3 (1). In this paper, we revisit this relation from the viewpoint of the underlying hypergeometric data H D, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data H D are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of-adic representations of the absolute Galois group of Q attached to H D. For specialized choices of H D, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7 F 6 (1) in Whipple's formula to the periods of these modular forms. Keywords Hypergeometric functions • Whipple's 7 F 6 formula • Hypergeometric character sums • Galois representations and modular forms Mathematics Subject Classification 11F11 • 33C20 • 11F80 • 11F67 • 11T24 B Ling Long
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