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For positive integer p=k+2, we construct a logarithmic extension of the^sl(2)_k conformal field theory of integrable representations by taking thekernel of two fermionic screening operators in a three-boson realization of^sl(2)_k. The currents W^-(z) and W^+(z) of a W-algebra acting in the kernelare determined by a highest-weight state of dimension 4p-2 and charge 2p-1, anda (theta=1)-twisted highest-weight state of the same dimension 4p-2 and charge-2p+1. We construct 2p W-algebra representations, evaluate their characters,and show that together with the p-1 integrable representation characters theygenerate a modular group representation whose structure is described as adeformation of the (9p-3)-dimensional representation $R_{p-1} \oplus C^2\tensor R_{p-1} \oplus R_{p-1} \oplus C^2 \tensor R_{p-1} \oplus C^3 \tensorR_{p-1}$, where R_{p-1} is the SL(2,Z)-representation on integrablerepresentation characters and R_{p-1} is a (p+1)-dimensionalSL(2,Z)-representation known from the logarithmic (p,1) model. The dimension9p-3 is conjecturally the dimension of the space of torus amplitudes, and theC^n with n=2 and 3 suggest the Jordan cell sizes in indecomposable W-algebramodules. Under Hamiltonian reduction, the W-algebra currents map into thecurrents of the triplet W-algebra of the logarithmic (p,1) model.Comment: 54 pages, amsart++, xy, graphicx, times&mathptmx. V2: references  added, some changes in the presentatio

Toward logarithmic extensions of $$\widehat{s\ell }(2)_k $$ conformal field models