  We study minimizers of the two-dimensional Ginzburg-Landau energy with
applied magnetic field, between the first and second critical fields. In this
regime, minimizing configurations exhibit densely packed hexagonal vortex
lattices, called Abrikosov lattices. We derive, in some asymptotic regime, a
limiting interaction energy between points in the plane, $W$, which we prove
has to be minimized by limits of energy-minimizing configurations, once
blown-up at a suitable scale. This is a next order effect compared to the
mean-field type results we previously established. The limiting "Coulombian
renormalized energy" $W$ is a logarithmic type of interaction, computed by a
"renormalization," and we believe it should be rather ubiquitous. We study
various of its properties, and show in particular, using results from number
theory, that among lattice configurations the hexagonal lattice is the unique
minimizer, thus providing a first rigorous hint at the Abrikosov lattice. Its
minimization in general remains open. The derivation of $W$ uses energy
methods: the framework of $\Gamma$-convergence, and an abstract scheme for
obtaining lower bounds for "2-scale energies" via the ergodic theorem.
