Abstract
We will describe in a special case the conjectural relationship among automorphic forms, l-adic representations, and motives. To make the discussion concrete, we shall restrict ourselves to weight 2 modular forms for GL2. In this case the modular forms can be thought of either as certain harmonic forms on products of the upper half complex planes and hyperbolic three spaces or as cohomology classes for certain quotients of these products. As such, they are relatively concrete and often computable topological objects. Similarly, we shall restrict attention to irreducible two-dimensional l-adic representations that are de Rham with Hodge-Tate numbers 0 and -1, and to certain abelian varieties.
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