This is the first one of a series of articles in which we develop the theory
of Jacobi forms of lattice index, their close interplay with the arithmetic
theory of lattices and the theory of Weil representations. We hope to publish
this series eventually in an extended and combined way as a monograph.
  In this part we present the basic theory and first structure theorems. We
deduce explicit dimension formulas and give non-trivial explicit example