In 1914, Ramanujan listed 17 infinite series representations of 1/π, including for example Several of Ramanujan's formulas relate hypergeometric series to values of the gamma function.In the 1980s, it was discovered that Ramanujan's formulas provided efficient means for calculating digits of π. In 1987, Borwein and Borwein [3] proved all 17 of Ramanujan's identities, while Chudnovsky and Chudnovsky [5] derived additional series for 1/π. Digits of π were calculated in both papers resulting in a new world record at the time by the Chudnovskys of 2, 260, 331, 336 digits. All of these Ramanujan-type formulas for 1/π are related to elliptic curves with complex multiplication (CM).In 1997, van Hamme [18] developed p-adic analogs, for primes p, of several Ramanujan-type series. Analogs of this type are called Ramanujan-type supercongruences, and relate truncated sums of hypergeometric series to values of the p-adic gamma function. In a recent paper [4], the author along with S. Chisholm, A. Deines, L. Long, and G. Nebe proves a general p-adic analog of Ramanujan-type supercongruences modulo p 2 AbstractIn 1997, van Hamme developed p-adic analogs, for primes p, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the p-adic gamma function, and are called Ramanujan-type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods.Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions.

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Acknowledgments vi Chapter 1. Introduction 1.1. Overview 1.2. Organization and the main results Chapter 2. Preliminaries for the Complex and Finite Field Settings 2.1. Gamma and beta functions 2.2. Gauss and Jacobi sums 2.3. Lagrange inversion 2.4. A dictionary between the complex and finite field settings Chapter 3. Classical Hypergeometric Functions 3.1. Classical development 3.2. Some properties of hypergeometric functions with n = 1 Chapter 4. Finite Field Analogues 4.1. Periods in the finite field setting 4.2. Hypergeometric varieties 4.3. Hypergeometric functions over finite fields 4.4. Comparison with other finite field hypergeometric functions Chapter 5. Some Related Topics on Galois Representations 5.1. Absolute Galois groups and Galois representations 5.2. Grössencharacters in the sense of Hecke 5.3. Notation for the N th power residue symbol 5.4. Jacobi sums and Grössencharacters Chapter 6. Galois Representation Interpretation 6.1. Galois interpretation for 1 P 0 6.2. Generalized Legendre curves and their Jacobians 6.3. Galois interpretation for 2 P 1 6.4. Some special cases of 2 P 1 -functions 6.5. Galois interpretation for n+1 F n 6.6. Zeta functions and hypergeometric functions over finite fields 6.7. Summary Chapter 7. A finite field Clausen formula and an application 7.1. A finite field version of the Clausen formula by Evans and Greene 7.2. Analogues of Ramanujan type formulas for 1/π Chapter 8. Translation of Some Classical Results iii iv CONTENTS 8.1. Kummer's 24 Relations 8.2. A Pfaff-Saalschütz evaluation formula 8.3. A few analogues of algebraic hypergeometric formulas Chapter 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 Chapter 10. An application to Hypergeometric Abelian Varieties Chapter 11. Open Questions and Concluding Remarks 11.1. Numeric observations Chapter 12. Appendix 12.1. Bailey 3 F 2 cubic transforms 12.2. A proof of a formula by Gessel and Stanton Bibliography Index

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.

Please cite this article in press as: A. Deines et al., Generalized Legendre curves and quaternionic multiplication, J. Number Theory (2015), http://dx. AbstractThis paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of Q contains a quaternion algebra.