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We propose a manifestly U-duality invariant modular form for the D^6 R^4interaction in the effective action of type IIB string theory compactified onT^2. It receives perturbative contributions upto genus three, as well asnon-perturbative contributions from D-instantons and (p,q) string instantonswrapping T^2. Our construction is based on constraints coming from stringperturbation theory, U-duality, the decompactification limit to ten dimensions,and the equality of the perturbative part of the amplitude in type IIA and typeIIB string theories. Using duality, parts of the perturbative amplitude arealso shown to match exactly the results obtained from eleven dimensionalsupergravity compactified on T^3 at one loop. We also obtain parts of the genusone and genus k amplitudes for the D^{2k} R^4 interaction for arbitrary k > 3.We enhance a part of this amplitude to a U-duality invariant modular form.

The D^6 R^4 term in type IIB string theory on T^2 and U-duality