2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59369We study U(r) instantons on elliptic surfaces with a section and show that they are in one-one correspondence with spectral data consisting of a curve in the dual elliptic surface and a line bundle on that curve. We use relative Fourier-Mukai transforms to analyse their properties and, in the case of the K3 and abelian surfaces, we show that the moduli space of instantons has a natural Lagrangian fibration with respect to the canonical complex symplectic structure.17 pages. Final version to appear in CrelleAlgebraic GeometryMathematical Physicstext