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Let G = SL_2(K) with K a local function field of characteristic 2. We reviewArtin-Schreier theory for the field K, and show that this leads to aparametrization of certain L-packets in the smooth dual of G. We relate this toa recent geometric conjecture. The L-packets in the principal series areparametrized by quadratic extensions, and the supercuspidal L-packets ofcardinality 4 are parametrized by biquadratic extensions. Each supercuspidalpacket of cardinality 4 is accompanied by a singleton packet for SL_1(D). Wecompute the depths of the irreducible constituents of all these L-packets forSL_2(K) and its inner form SL_1(D).

L-packets and depth for SL_2(K) with K a local function field of characteristic 2