We use a relative Fourier-Mukai transform on elliptic K3 surfaces X to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of X reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on X which, in view of the triviality of the quantum cohomology of K3 surfaces, can be interpreted as mirror symmetry.From the mathematical viewpoint the novelty is that we exhibit another example of a Fourier-Mukai transform on K3 surfaces, whose properties are closely related to the geometry of the relative Jacobian of X.We denote byê : P 1 → X the canonical section; one hasê = ̟ • e. Moreover, we denote by π,π the projections of the fibred product X × P 1 X onto the factors.Remark 2.2. The Picard functor is also representable by an open dense subscheme J of X, the relative Jacobian J → P 1 of X → P 1 . If X s ⊂ X denotes the complement of the singular points of the fibres of π, then ̟ gives an isomorphism X s ∼ → J of schemes 3

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The main result of this paper is the explicit computation of the equations defining the moduli space of triples (C,p,), where C is an integral and complete algebraic curve, p a smooth rational point and a certain isomorphism. This is achieved by introducing algebraically infinite Grassmannians, tau and Baker-Ahkiezer functions and by proving an Addition Formula for tau functions.

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