  We present a determination of the isospin-1/2 elastic $\pi K$ scattering
amplitudes in S and P partial waves using lattice Quantum Chromodynamics. The
amplitudes, constrained for a large number of real-valued energy points, are
obtained as a function of light-quark mass, corresponding to four pion masses
between 200 and 400 MeV. Below the first inelastic threshold, the P-wave
scattering amplitude is dominated by a single pole singularity that evolves
from being a stable bound-state at the highest quark mass into a narrow
resonance that broadens as the pion and kaon masses are reduced. As in
experiment, the S-wave amplitude does not exhibit an obviously resonant
behavior, but instead shows a slow rise from threshold, which is not
inconsistent with the presence of a $\kappa$/$K_0^\star(700)$-like resonance at
the considered quark masses. As has been found in analyses of experimental
scattering data, simple analytic continuations into the complex energy plane of
precisely-determined lattice QCD amplitudes on the real energy axis are not
sufficient to model-independently determine the existence and properties of
this state. The spectra and amplitudes we present will serve as an input for
increasingly elaborate amplitude analysis techniques that implement more of the
analytic structure expected at complex energies.
