Introduction

In this paper we make precise, in the case of rank 1 Fano 3-folds, the following programme:

The paper is based on the notes of the lecture series the author gave at the University of Cambridge in 2003. It expands the announcement, providing the background for and discussion of the modularity conjecture.

We start with basic material on mirror symmetry for Fano varieties. The quantum D–module and the regularized quantum D–module are introduced in Section 1. We state the mirror symmetry conjecture for Fano varieties. We give more conjectures implying, or implied by, the mirror symmetry conjecture. We review the algebraic Mellin transform of Loeser and Sabbah and define hypergeometric D–modules on tori.

In Section 2 we consider Fano 3-folds of Picard rank 1 and review Iskovskikh's classification into 17 algebraic deformation families. We apply the basic setup to Fano 3-folds to obtain the so called counting differential equations of type D3. We introduce DN equations as generalizations of these, discuss their properties and take a brief look at their singularities.

In Section 3, motivated by the Dolgachev-Nikulin-Pinkham picture of mirror symmetry for K3 surfaces, we introduce (N, d)-modular families; these are pencils of K3 surfaces whose Picard-Fuchs equations are the counting D3 equations of rank 1 Fano 3-folds.