Dermot McCarthy

On a supercongruence conjecture of Rodriguez-Villegas

In examining the relationship between the number of points over

EXTENDING GAUSSIAN HYPERGEOMETRIC SERIES TO THE p-ADIC SETTING

\mathbb{F}_p

Transformations of well-poised hypergeometric functions over finite fields

on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the

3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES

p

A finite field hypergeometric function associated to eigenvalues of a Siegel eigenform

-th Fourier coefficient of a modular form.

Binomial Coefficient – Harmonic Sum Identities Associated to Supercongruences

We define a function which extends Gaussian hypergeometric series to the

Hypergeometric type identities in the -adic setting and modular forms

p

SUMMATION IDENTITIES AND SPECIAL VALUES OF HYPERGEOMETRIC SERIES IN THE p-ADIC SETTING

-adic setting. This new function allows results involving Gaussian hypergeometric series to be extended to a wider class of primes. We demonstrate this by providing various congruences between the function and truncated classical hypergeometric series. These congruences provide a framework for proving the supercongruence conjectures of Rodriguez-Villegas.

Supercongruence conjectures of Rodriguez-Villegas

Abstract We define a hypergeometric function over finite fields which is an analogue of the classical generalized hypergeometric series. We prove that this function satisfies many transformation and summation formulas. Some of these results are analogous to those given by Dixon, Kummer and Whipple for the well-poised classical series. We also discuss this functionʼs relationship to other finite field analogues of the classical series, most notably those defined by Greene and Katz.

Multiplicative relations for Fourier coefficients of degree 2 Siegel eigenforms

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.