  We explore the role played by the spectral curves associated with Higgs pairs
in the context of the Nahm transform of doubly-periodic instantons defined in
"Construction of doubly-periodic instantons" (math.DG/9909069) and "Nahm
transform for doubly-periodic instantons" (math.DG/9910120). More precisely, we
show how to construct a triple consisting of an algebraic curve plus a line
bundle with connection over it from a doubly-periodic instanton, and that these
coincide with the Hitchin's spectral data associated with the Nahm transformed
Higgs bundle.
