$|V_{us}|$ from $K$ decays in theory

Leptonic and semileptonic kaon decays represent till now the golden modes for the extraction of the Cabibbo angle from experiments, provided the relevant hadronic quantities, namely the ratio of the kaon and pion leptonic decay constants, $f_K / f_\pi$, and the semileptonic vector form factor at zero four-momentum transfer, $f_+(0)$, are computed accurately from QCD. In the last years, using large-scale lattice QCD simulations, the determination of both $f_K / f_\pi$ and $f_+(0)$ has reached an impressive level of precision ($\approx 0.3 \%$), which is expected to be further improved in the near future. However, at a permille level of precision both electromagnetic and strong isospin-breaking effects cannot be neglected anymore. In this contribution a new lattice strategy aiming at determining QCD+QED effects in the inclusive leptonic decay rates of charged pseudoscalar mesons is briefly illustrated, and the preliminary results obtained in the case of $\pi_{\ell 2}$ and $K_{\ell 2}$ decays are presented. As for the semileptonic $K_{\ell 3}$ decays, it is pointed out that the determination of the relevant form factors in the full kinematical range covered by the experiments is necessary for a more stringent test of the Standard Model. In this respect the perspectives based on the only existing results from the European Twisted Mass Collaboration (ETMC) are quite encouraging.