I will report on the progress of our program, led by Chentao Yang, of high-z studies of multi-line H₂O and other molecules in strongly lensed high-z Herschel galaxies from the H-ATLAS survey. High-z H₂O detections with IRAM/PdBI will be recalled in 15 objects, as well as its comparison with Herschel detections in local ULIRGs. We have shown that H₂O is easily detectable at all redshifts. The H₂O line luminosity increases as L_IR^~1.2, high-excitation lines (E_up/k ~ 300K) remain strong in the majority of the sources, but not in all and at higher excitation. H₂O lines provide thus an important diagnosis in extreme starbust (and possibly AGN) conditions of warm, dense cores/clumps of these galaxies, which are often more extreme than local equivalents. Modeling H₂O rotational excitation allows us to constrain physical parameters there such as warm dust opacity and temperature and H₂O abundance.

    Extensions and work in progress will be presented in various directions: H₂O⁺ lines are easy to detect and provide important constraints on the H₂O chemistry. We have just got the first IRAM/NOEMA map at high angular resolution of two H₂O lines plus CO(10-9) and H₂O⁺ in one source; it displays striking differences in the spatial distribution of the FIR continuum and various lines. Multi-line CO detections at IRAM/30m provides a complementary view of their excitation in many of our H₂O galaxies. Observations of lines connected to the ground state, mostly in absorption, of various molecules (H₂O, H₂O⁺, CH⁺, OH) present exciting prospects for probing the external layers of the galaxy and of its core and their dynamics. High-J detections of HCN, HCO⁺ and HNC provide a complementary diagnostic of their low-J lines for the dense, star-forming gas.