Abstract. For primes p ≡ 1 (mod 12), we present an explicit relation between the traces of Frobenius on a family of elliptic curves with j-invariant 1728 t and values of a particular 2 F 1 -hypergeometric function over F p . We also give a formula for traces of Hecke operators on spaces of cusp forms of weight k and level 1 in terms of the same traces of Frobenius. This leads to formulas for Ramanujan's τ -function in terms of hypergeometric functions.

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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.