Landlands has proposed a vast generalisation of the classical reciprocity theorems of number theory. In joint work with Dinakar Ramakrishnan we propose a step beyond these conjectures as explained below.
Langlands’ programme says that the L-function associated to a (pure irreducible) motive of a variety defined over the rational numbers occurs as the L-function of a certain automorphic representation. In the case, when the motive has rank 2, the automorphic representation is the one associated with a modular form. The question we ask is whether we can geometrically represent the modular form by a variety whose cohomology is particularly simple. In other words, we would like the cohomology of the variety to be made of Hodge cycles except for the motive of the modular form.
We have constructed a number of examples to examine this possibility. Moreover, some classical examples also fit into this mold. In a joint paper one example that has been presented is the case of the quotient of the third power of an elliptic curve by a suitable action of the alternating group on four letters.