Algebraic K-theory of the fraction field of topological K-theory

We compute the algebraic K-theory modulo p and v_1 of the S-algebra ell/p = k(1), using topological cyclic homology. We use this to compute the homotopy cofiber of a transfer map K(L/p) --> K(L_p), which we interpret as the algebraic K-theory of the "fraction field" of the p-complete Adams summand of topological K-theory. The results suggest that there is an arithmetic duality theorem for this fraction field, much like Tate--Poitou duality for p-adic fields.