* Mann et al. (2008) Mann, M.E., Zhang, Z., Hughes, M.K., Bradley, R.S., Miller, S.K., Rutherford, S., Ni, F., 2008. Proxy-based reconstructions of hemispheric and global surface temperature variations over the past two millennia. Proceedings of the National Academy of Sciences 105(36) 13252-13257.
  * Masson-Delmotte et al. (2018) Masson-Delmotte, V., Zhai, P., Pörtner, H.-O., Roberts, D., Skea, J., Shukla, P., Pirani, A., Moufouma-Okia, W., Péan, C., Pidcock, R., Connors, S., Matthews, J., Chen, Y., X.Zhou, M.G., Lonnoy, E., Maycock, T., Tignor, M., , (eds.), T.W., 2018. IPCC, 2018: Global Warming of 1.5°C. An IPCC Special Report on the impacts of global warming of 1.5 °C above pre-industrial levels and related global greenhouse gas emission pathways, in the context of strengthening the global response to the threat of climate change, sustainable development, and efforts to eradicate poverty. Cambridge University Press.
  * Masson-Delmotte et al. (2021) Masson-Delmotte, V., Zhai, P., Pirani, A., Connors, S., Péan, C., Berger, S., Caud, N., Chen, Y., Goldfarb, L., Gomis, M., Huang, M., Leitzell, K., Lonnoy, E., Matthews, J., Maycock, T., Waterfield, T., Yelekci, O., Yu, R., (eds.), B.Z., 2021. IPCC, 2021: Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press.
  * Matthes et al. (2017) Matthes, K., Funke, B., Andersson, M.E., Barnard, L., Beer, J., Charbonneau, P., Clilverd, M.A., deWit, T.D., Haberreiter, M., Hendry, A., Jackman, C.H., Kretzschmar, M., Kruschke, T., Kunze, M., Langematz, U., Marsh, D.R., Maycock, A.C., Misios, S., Rodger, C.J., Scaife, A.A., Seppälä, A., Shangguan, M., Sinnhuber, M., Tourpali, K., Usoskin, I., vande Kamp, M., Verronen, P.T., Versick, S., 2017. Solar forcing for CMIP6 (v3.2). Geoscientific Model Development 10, 2247–2302.
  * Mauritsen et al. (2019) Mauritsen, T., Bader, J., Becker, T., Behrens, J., Bittner, M., Brokopf, R., et al. (2019). Developments in the MPI-M Earth System Model version 1.2 (MPI-ESM1.2) and its response to increasing CO2. Journal of Advances in Modeling Earth Systems, 11, 998–1038.
  * Mauritsen and Roeckner (2020) Mauritsen, T., Roeckner, E. (2020). Tuning the MPI-ESM1.2 global climate model to improve the match with instrumental record warming by lowering its climate sensitivity. Journal of Advances in Modeling Earth Systems, 12, e2019MS002037.
  * McCarthy and Caesar (2023) McCarthy, G.D., Caesar, L., 2023. Can we trust projections of AMOC weakening based on climate models that cannot reproduce the past? Philosophical Transaction of the Royal Society A 381: 20220193.
  * McKitrick and Christy (2020) McKitrick, R., Christy, J., 2020. Pervasive warming bias in CMIP6 tropospheric layers. Earth and Space Science 7(9).
  * McNutt (2015) McNutt, M., 2015. The beyond-two-degree inferno. Science 349, 7.
  * Mears and Wentz (2016) Mears, C.A., Wentz, F.J., 2016\. Sensitivity of satellite-derived tropospheric temperature trends to the diurnal cycle adjustment. Journal of Climate 29(10), 3629–3646.
  * Meinshausen et al. (2020) Meinshausen, M., Nicholls, Z. R. J., Lewis, J., Gidden, M. J., Vogel, E., Freund, M., Beyerle, U., Gessner, C., Nauels, A., Bauer, N., Canadell, J. G., Daniel, J. S., John, A., Krummel, P. B., Luderer, G., Meinshausen, N., Montzka, S. A., Rayner, P. J., Reimann, S., Smith, S. J., van den Berg, M., Velders, G. J. M., Vollmer, M. K., and Wang, R. H. J., 2020\. The shared socio-economic pathway (SSP) greenhouse gas concentrations and their extensions to 2500. Geoscientific Model Development 13, 3571–3605.
  * Meinshausen et al. (2022) Meinshausen, M., Lewis, J., McGlade, C., Gütschow, J., Nicholls, Z., Burdon, R., Cozzi, L., Hackmann, B., 2022. Realization of Paris Agreement pledges may limit warming just below 2 °C. Nature 604, 304–309.
  * Meinshausen et al. (2023) Meinshausen, M., Schleussner, C.-F., Beyer, K., Bodeker, G., Boucher, O., Canadell, J. G., Daniel, J. S., Diongue-Niang, A., Driouech, F., Fischer, E., Forster, P., Grose, M., Hansen, G., Hausfather, Z., Ilyina, T., Kikstra, J. S., Kimutai, J., King, A., Lee, J.-Y., Lennard, C., Lissner, T., Nauels, A., Peters, G. P., Pirani, A., Plattner, G.-K., Pörtner, H., Rogelj, J., Rojas, M., Roy, J., Samset, B. H., Sanderson, B. M., Séférian, R., Seneviratne, S., Smith, C. J., Szopa, S., Thomas, A., Urge-Vorsatz, D., Velders, G. J. M., Yokohata, T., Ziehn, T., and Nicholls, Z.: A perspective on the next generation of Earth system model scenarios: towards representative emission pathways (REPs), Geosci. Model Dev. Discuss. [preprint], https://doi.org/10.5194/gmd-2023-176, in review, 2023.
  * Mignot et al. (2021) Mignot, J., Hourdin, F., Deshayes, J., Boucher, O., Gastineau, G., Musat, I., et al. (2021). The tuning strategy of IPSL-CM6A-LR. Journal of Advances in Modeling Earth Systems, 13, e2020MS002340.
  * Mitchell et al. (2020) Mitchell, D., Lo, Y.E., Seviour, W., Haimberger, L., Polvani, L., 2020. The vertical profile of recent tropical temperature trends: Persistent model biases in the context of internal variability. Environmental Research Letters 15(10), 1040b4.
  * Moberg et al. (2005) Moberg, A., Sonechkin, D., Holmgren, K., Datsenko, N.D., Karlén, W., 2005. Highly variable Northern Hemisphere temperatures reconstructed from low- and high-resolution proxy data. Nature 433, 613–617.
  * Möller (1963) Möller, F., 1963. On the influence of changes in the CO2 concentration in air on the radiation balance of the earth’s surface and on the climate. Journal of Geophysical Research 68, 3877–3886.
  * Morice et al. (2012) Morice, C.P., Kennedy, J.J., Rayner, N.A., Jones, P.D., 2012. Quantifying uncertainties in global and regional temperature change using an ensemble of observational estimates: The HadCRUT4 dataset. Journal of Geophysical Research, 117, D08101.
  * Morice et al. (2021) Morice, C.P., Kennedy, J.J., Rayner, N.A., Winn, J.P., Hogan, E., Killick, R.E., Dunn, R. J.H., Osborn, T.J., Jones, P.D., Simpson, I.R., 2021. An updated assessment of near-surface temperature change from 1850: The HadCRUT5 data set. Journal of Geophysical Research: Atmospheres 126(3), e2019JD032361.
  * Monckton et al. (2015) Monckton, C., Soon, W.-H., Legates, D. R., Briggs, W. M., 2015. Why models run hot: Results from an irreducibly simple model. Science Bulletin, 60, 122–135.
  * Neff et al. (2001) Neff, U., Burns, S.J., Mangini, A., Mudelsee, M., Fleitmann, D., Matter, A., 2001. Strong coherence between solar variability and the monsoon in Oman between 9 and 6 kyr ago. Nature 411, 290–293.
  * Nijsse et al. (2020) Nijsse, F. J. M.M., Cox, P.M., Williamson, M.S., 2020. Emergent constraints on transient climate response (TCR) and equilibrium climate sensitivity (ECS) from historical warming in CMIP5 and CMIP6 models. Earth System Dynamics 11(3), 737–750.
  * O’Neill et al. (2022) O’Neill, P., Connolly, R., Connolly, M., Soon, W., Chimani, B., Crok, M., de Vos, R., Harde, H., Kajaba, P., Nojarov, P., Przybylak, R., Rasol, D., Skrynyk, O., Skrynyk, O., Štěpánek, P., Wypych, A., Zahradníček , P., 2022. Evaluation of the Homogenization Adjustments Applied to European Temperature Records in the Global Historical Climatology Network Dataset. Atmosphere 13, 285.
  * PAGES2k Consortium (2017) PAGES2k Consortium, 2017. A global multiproxy database for temperature reconstructions of the Common Era. Scientific Data 4:170088.
  * Penza et al. (2022) Penza, V., Berrilli, F., Bertello, L., Cantoresi, M., Criscuoli, S., Giobbi, P., 2022. Total solar irradiance during the last five centuries. The Astrophysical Journal 937(2), 84.
  * Piao et al. (2019) Piao, S., Wang, X., Park, T., Chen, C., Lian, X., He, Y., Bjerke, J.W., Chen, A., Ciais, P., Tømmervik, H., Nemani, R.R., Myneni, R.B., 2019. Characteristics, drivers and feedbacks of global greening. Nature Reviews Earth & Environment 1(1), 14–27.
  * Pielke (2021) Pielke, R., 2021. How to Understand the New IPCC Report: Part 1, Scenarios. The Honest Broker. https://rogerpielkejr.substack.com/p/how-to-understand-the-new-ipcc-report
  * Pielke and Ritchie (2021a) Pielke, R., Ritchie, J., 2021a. Distorting the view of our climate future: The misuse and abuse of climate pathways and scenarios. Energy Research and Social Science 72, 101890.
  * Pielke and Ritchie (2021b) Pielke, R., Ritchie, J., 2021b. How climate scenarios lost touch with reality. Issues in Science and Technology 37(4), 75–83.
  * Pielke et al. (2022) Pielke, R., Burgess, M.G., Ritchie, J., 2022. Plausible 2005–2050 emissions scenarios project between 2 °C and 3 °C of warming by 2100. Environmental Research Letters 17, 024027.
  * Pörtner et al. (2022) Pörtner, H.-O., Roberts, D., Tignor, M., Poloczanska, E., Mintenbeck, K., Alegría, A., Craig, M., Langsdorf, S., Löschke, S., Möller, V., Okem, A., (eds.), B.R., 2022\. IPCC, 2022: Climate Change 2022: Impacts, Adaptation and Vulnerability. Contribution of Working Group II to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press.
  * Rahmstorf (2008) Rahmstorf, S., 2008. Anthropogenic climate change: Revisiting the facts. In: Zedillo, E. (Ed.), Global Warming: Looking Beyond Kyoto. Brookings Institution Press, Washington, Ch.3, pp. 34–53.
  * Riahi et al. (2017) Riahi, K., van Vuuren, D.P., Kriegler, E., Edmonds, J., O’Neill, B.C., Fujimori, S., Bauer, N., Calvin, K., Dellink, R., Fricko, O., Lutz, W., Popp, A., Cuaresma, J.C., KC, S., Leimbach, M., Jiang, L., Kram, T., Rao, S., Emmerling, J., Ebi, K., Hasegawa, T., Havlik, P., Humpenöder, F., Silva, L. A.D., Smith, S., Stehfest, E., Bosetti, V., Eom, J., Gernaat, D., Masui, T., Rogelj, J., Strefler, J., Drouet, L., Krey, V., Luderer, G., Harmsen, M., Takahashi, K., Baumstark, L., Doelman, J.C., Kainuma, M., Klimont, Z., Marangoni, G., Lotze-Campen, H., Obersteiner, M., Tabeau, A., Tavoni, M., 2017. The shared socioeconomic pathways and their energy, land use, and greenhouse gas emissions implications: An overview. Global Environmental Change 42, 153–168.
  * Ritchie and Dowlatabadi (2017) Ritchie, J., Dowlatabadi, H., 2017. Why do climate change scenarios return to coal? Energy 140(P1), 1276–1291.
  * Rocque et al. (2021) Rocque, R.J., Beaudoin, C., Ndjaboue, R., Cameron, L., Poirier-Bergeron, L., Poulin-Rheault, R.-A., Fallon, C., Tricco, A.C., Witteman, H.O., 2021. Health effects of climate change: an overview of systematic reviews. BMJ Open 11(6), e046333.
  * Roe (2009) Roe, G., 2009. Feedbacks, timescales, and seeing red. Annual Review of Earth and Planetary Sciences 37(1), 93–115.
  * Rohde and Hausfather (2020) Rohde, R.A., Hausfather, Z., 2020. The Berkeley earth land/ocean temperature record. Earth System Science Data 12(4), 3469–3479.
  * Rugenstein et al. (2023) Rugenstein, M. Zelinka, M., Karnauskas, K.B., Ceppi, P., Andrews, T., 2023. Patterns of surface warming matter for climate sensitivity, Eos, 104, https://doi.org/10.1029/2023EO230411.
  * Santer et al. (2017) Santer, B., Fyfe, J., Pallotta, G., Pallotta, G., Flato, G. M., Meehl, G.A., England, M.H., Hawkins, E., Mann, M.E., Painter, J.F., Bonfils, C., Cvijanovic, I., Mears, C., Wentz, F.J., Po-Chedley, S., Fu, Q., Zou, C.-Z., 2017. Causes of differences in model and satellite tropospheric warming rates. Nature Geoscience 10, 478–485.
  * Scafetta (2010) Scafetta, N., 2010. Empirical evidence for a celestial origin of the climate oscillations and its implications. Journal of Atmospheric and Solar-Terrestrial Physics, 72, 951–970.
  * Scafetta (2012a) Scafetta, N., 2012a. Testing an astronomically based decadal-scale empirical harmonic climate model versus the IPCC (2007) general circulation climate models. Journal of Atmospheric and Solar-Terrestrial Physics, 80, 124–137.
  * Scafetta (2012b) Scafetta, N.: 2012b. Multi-scale harmonic model for solar and climate cyclical variation throughout the Holocene based on Jupiter-Saturn tidal frequencies plus the 11-year solar dynamo cycle. _Journal of Atmospheric and Solar-Terrestrial Physics_ 80, 296–311.
  * Scafetta (2013) Scafetta, N., 2013. Discussion on climate oscillations: CMIP5 general circulation models versus a semi-empirical harmonic model based on astronomical cycles. Earth-Science Reviews 126, 321–357.
  * Scafetta (2014a) Scafetta, N., 2014a. Multi-scale dynamical analysis (MSDA) of sea level records versus PDO, AMO, and NAO indexes.___Climate Dynamics_ 43, 175–192.
  * Scafetta (2014b) Scafetta, N.: 2014b. Discussion on the spectral coherence between planetary, solar and climate oscillations: a reply to some critiques. Astrophysics and Space Science, 354, 275–299.
  * Scafetta (2020) Scafetta, N., 2020. Solar Oscillations and the Orbital Invariant Inequalities of the Solar System. Solar Physics 295(2), 33.
  * Scafetta (2021a) Scafetta, N., 2021a. Detection of non-climatic biases in land surface temperature records by comparing climatic data and their model simulations. Climate Dynamics 56(9-10), 2959–2982.
  * Scafetta (2021b) Scafetta, N., 2021b. Reconstruction of the interannual to millennial scale patterns of the global surface temperature. Atmosphere 12(2), 147.
  * Scafetta (2021c) Scafetta, N., 2021c. Testing the CMIP6 GCM simulations versus surface temperature records from 1980–1990 to 2011–2021: High ECS is not supported. Climate 9(11), 161.
  * Scafetta (2022) Scafetta, N., 2022. Advanced testing of low, medium, and high ECS CMIP6 GCM simulations versus ERA5-T2m. Geophysical Research Letters 49(6), e2022GL097716.
  * Scafetta (2023a) Scafetta, N., 2023a. CMIP6 GCM ensemble members versus global surface temperatures. Climate Dynamics 60, 3091–3120.
  * Scafetta (2023b) Scafetta, N., 2023b. CMIP6 GCM validation based on ECS and TCR ranking for 21st century temperature projections and risk assessment. Atmosphere 14(2), 345.
  * Scafetta (2023c) Scafetta, N., 2023c. Empirical assessment of the role of the sun in climate change using balanced multi-proxy solar records. Geoscience Frontiers 14(6), 101650.
  * Scafetta (2023d) Scafetta, N., 2023d. Reply to “Comment on ‘Advanced testing of low, medium, and high ECS CMIP6 GCM simulations versus ERA5-T2m’ by N. Scafetta (2022)” by Schmidt et al. (2023). Geophysical Research Letters, 50, e2023GL104960.
  * Scafetta et al. (2013) Scafetta, N., Humlum, O., Solheim, J.-E., Stordahl, K.: 2013. Comment on “The influence of planetary attractions on the solar tachocline” by Callebaut, de Jager and Duhau.___Journal of Atmospheric and Solar–Terrestrial Physics_ 102, 368–371.
  * Scafetta and Ouyang (2019) Scafetta, N., Ouyang, S., 2019. Detection of UHI bias in China climate network using Tmin and Tmax surface temperature divergence. Global and Planetary Change 181, 102989.
  * Scafetta et al. (2019) Scafetta, N., Willson, R., Lee, J., Wu, D., 2019. Modeling quiet solar luminosity variability from TSI satellite measurements and proxy models during 1980–2018. Remote Sensing 11, 2569.
  * Scafetta et al. (2016) Scafetta, N., Milani, F., Bianchini, A., Ortolani, S.: 2016. On the astronomical origin of the Hallstatt oscillation found in radiocarbon and climate records throughout the Holocene. _Earth-Science Reviews_ __ 162, 24–43.
  * Scafetta et al. (2020) Scafetta, N., Milani, F., Bianchini, A., 2020. A 60-Year cycle in the meteorite fall frequency suggests a possible interplanetary dust forcing of the Earth’s climate driven by planetary oscillations. Geophysical Research Letters 47(18), e2020GL089954.
  * Scafetta and Bianchini (2023) Scafetta, N., Bianchini, A., 2023. Overview of the spectral coherence between planetary resonances and solar and climate oscillations. Climate 11(4), 77.
  * Schmidt et al. (2023) Schmidt, G.A., Jones, G.S., Kennedy, J.J., 2023. Comment on “Advanced testing of low, medium, and high ECS CMIP6 GCM simulations versus ERA5-T2m” by N. Scafetta (2022). Geophysical Research Letters 50, e2022GL102530.
  * Schmutz (2021) Schmutz, W.K., 2021. Changes in the total solar irradiance and climatic effects. Journal of Space Weather and Space Climate 11, 40.
  * Shaviv (2002) Shaviv, N.J., 2002. Cosmic ray diffusion from the galactic spiral arms, iron meteorites, and a possible climatic connection. Physical Review Letters 89 (5), 051102.
  * Sherwood et al. (2020) Sherwood, S., Webb, M., Annan, J., Armour, K., Forster, P., Hargreaves, J., Hegerl, G., Klein, S., Marvel, K., Rohling, E., Watanabe, M., Andrews, T., Braconnot, P., Bretherton, C., Foster, G., Hausfather, Z., von der Heydt, A., Knutti, R., Mauritsen, T., Norris, J., Proistosescu, C., Rugenstein, M., Schmidt, G., Tokarska, K., Zelinka, M., 2020. An assessment of earth’s climate sensitivity using multiple lines of evidence. Reviews of Geophysics 58, e2019RG000678.
  * Shukla et al. (2022) Shukla, P., Skea, J., Reisinger, A., Slade, R., Fradera, R., Pathak, M., Khourdajie, A., Belkacemi, M., van Diemen, R., Hasija, A., Lisboa, G., Luz, S., Malley, J., McCollum, D., Some, S., (eds.), P.V., 2022. IPCC, 2022: Climate Change 2022: Mitigation of Climate Change. Contribution of Working Group III to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press.
  * Spencer et al. (2017) Spencer, R.W., Christy, J.R., Braswell, W.D., 2017. UAH version 6 global satellite temperature products: Methodology and results. Asia-Pacific Journal of Atmospheric Sciences 53(1), 121–130.
  * Spencer and Christy (2023) Spencer, R.W., Christy, J.R., 2023. Effective climate sensitivity distributions from a 1D model of global ocean and land temperature trends, 1970–2021. Theoretical and Applied Climatology. https://doi.org/10.1007/s00704-023-04634-7
  * Spencer (2023) Spencer, R.W., 2023. Examples from our New UAH Urban Heat Island Dataset. https://www.drroyspencer.com/2023/11/examples-from-our-new-uah-urban-heat-island-dataset/. Accessed on December 7 2023.
  * Soon et al. (2023) Soon, W., Connolly, R., Connolly, M., Akasofu, S.-I., Baliunas, S., Berglund, J., Bianchini, A., Briggs, W.M., Butler, C.J., Cionco, R.G., et al., 2023. The Detection and Attribution of Northern Hemisphere Land Surface Warming (1850–2018) in Terms of Human and Natural Factors: Challenges of Inadequate Data. Climate 11, 179.
  * Stefani (2021) Stefani, F., 2021. Solar and Anthropogenic Influences on Climate: Regression Analysis and Tentative Predictions. Climate 9, 163.
  * Steinhilber et al. (2012) Steinhilber, F., Abreu, J.A., Beer, J., Brunner, I., Christl, M., Fischer, H., Heikkilä, U., Kubik, P.W., Mann, M., McCracken, K.G., Miller, H., Miyahara, H., Oerter, H., Wilhelms, F., 2012. 9,400 years of cosmic radiation and solar activity from ice cores and tree rings. Proceedings of the National Academy of Sciences 109 (16), 5967–5971.
  * Stocker et al. (2013) Stocker, T.F., D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley (eds.). IPCC, 2013: Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.
  * Svensmark et al. (2016) Svensmark, J., Enghoff, M.B., Shaviv, N.J., Svensmark, H., 2016. The response of clouds and aerosols to cosmic ray decreases. Journal of Geophysical Research: Space Physics 121 (9), 8152–8181.
  * Svensmark (2022) Svensmark, H., 2022. Supernova rates and burial of organic matter. Geophysical Research Letters 49(1), e2021GL096376.
  * Tol (2015) Tol, R. S.J., 2015. Who benefits and who loses from climate change? In: Handbook of Climate Change Mitigation and Adaptation. Springer New York, pp. 1–12.
  * Tol (2023) Tol, R., 2023. Costs and benefits of the Paris climate targets. Climate Change Economics 14, 2340003.
  * UNFCCC (2023) UNFCCC, 2023. The Paris Agreement: What is the Paris Agreement? Accessed on June 10, 2023. https://unfccc.int/process-and-meetings/the-paris-agreement
  * van Oldenborgh (2020) van Oldenborgh, G., 2020\. The knmi climate explorer: Knmi (results, code)/world meteorological organization (wmo) (design). https://climexp.knmi.nl/start.cgi.
  * van Vuuren et al. (2011) van Vuuren, D.P., Edmonds, J., Kainuma, M., Riahi, K., Thomson, A., Hibbard, K., Hurtt, G.C., Kram, T., Krey, V., Lamarque, J.-F., Masui, T., Meinshausen, M., Nakicenovic, N., Smith, S.J., Rose, S.K., 2011. The representative concentration pathways: an overview. Climatic Change 109, 5–31.
  * Voosen (2019) Voosen, P., 2019. New climate models forecast a warming surge. Science 364, 222–223.
  * Voosen (2021) Voosen, P., 2021. Climate panel confronts implausibly hot models. Science 373, 474–475.
  * Wyatt and Curry (2013) Wyatt, M.G., Curry, J.A., 2013\. Role for Eurasian arctic shelf sea ice in a secularly varying hemispheric climate signal during the 20th century. Climate Dynamics 42(9-10), 2763–2782.
  * Wu et al. (2018) Wu, C.-J. Krivova, N.A., Solanki, S.K., Usoskin, I.G., 2018. Solar total and spectral irradiance reconstruction over the last 9000 years. Astronomy & Astrophysics 620, A120.
  * Zhang et al. (2019) Zhang, H.-M., Lawrimore, J., Huang, B., Menne, M., Yin, X., Sánchez-Lugo, A., Gleason, B., Vose, R., Arndt, D., Rennie, J., Williams, C., 2019. Updated temperature data give a sharper view of climate trends. EOS 100.
  * Zhu et al. (2016) Zhu, Z., Piao, S., Myneni, R.B., Huang, M., Zeng, Z., Canadell, J.G., Ciais, P., Sitch, S., Friedlingstein, P., Arneth, A., Cao, C., Cheng, L., Kato, E., Koven, C., Li, Y., Lian, X., Liu, Y., Liu, R., Mao, J., Pan, Y., Peng, S., Peñuelas, J., Poulter, B., Pugh, T. A.M., Stocker, B.D., Viovy, N., Wang, X., Wang, Y., Xiao, Z., Yang, H., Zaehle, S., Zeng, N., 2016. Greening of the earth and its drivers. Nature Climate Change 6(8), 791–795.
  * Zou et al. (2023) Zou, C.-Z., Xu, H., Hao, X., Liu, Q., 2023. Mid-tropospheric layer temperature record derived from satellite microwave sounder observations with backward merging approach. Journal of Geophysical Research: Atmospheres 128, e2022JD037472.


In the article The third homology of $SL_{2}(\mathbb{Q})$ ([10]), Hutchinson determined the structure of
by expressing it in terms of $K_{3}^{\mathrm{ind}}(\mathbb{Q})\cong
\mathbb{Z}/24$ and the scissor congruence group of the residue field
$\mathbb{F}_{p}$ with $p$ a prime number. In this paper, we develop further
the properties of the refined scissors congruence group in order to
extend this result to the case of imaginary quadratic number fields whose ring of
integers is a Euclidean domain with respect to the norm.

§ INTRODUCTION

In the recent paper The third homology of $SL_{2}(\mathbb{Q})$ ([10]), Hutchinson calculates the structure of $H_{3}(\mathrm{SL}_{2}(\mathbb{Q}),\mathbb{Z})$ by calculating the surjective homomorphism $H_{3}(\mathrm{SL}_{2}(\mathbb{Q}),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(\mathbb{Q})$.
He asked whether the structure theorem proved in his paper generalize to number fields and even global fields. We answer this question in this paper, the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm of the field (see section <ref> below).

Let $F$ be a field, there is a natural induced map from $H_{3}(\mathrm{SL}_{2}(F), \mathbb{Z})$ to the indecomposable quotient,$K_{3}^{\mathrm{ind}}(F)$, of the third Quillen $K$-group of $F$. It can be shown this map is surjective (see [6]). In [4], Johan Dupont and Chih-Han Sah showed that this map is an isomorphism for the case of $F=\mathbb{C}$ (and more generally $F^{\times}=(F^{\times})^{2}$). Furthermore, in [5], Chih-Han Sah proved this result for the case of $\mathbb{R}$. However, for general fields, the map $H_{3}(\mathrm{SL}_{2}(F), \mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(F)$ has a large kernel ([8]). In fact, $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})$ is naturally a module over the integral group ring $R_{F}:=\mathbb{Z}[F^{\times}/(F^{\times})^2]$ of the group of square classes of the field $F$ and the above homomorphism is a homomorphism of $R_{F}$-modules if $K_{3}^{\mathrm{ind}}(F)$ is given the trivial module structure. In ([8]), Hutchinson shows that the action of $R_{F}$ on $H_{3}(\mathrm{SL}_{2}(F), \mathbb{Z})$ is highly nontrivial for fields with (many) discrete valuations. When $F^{\times}$ acts nontrvially on $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})$, the map  $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(F)$ has a nontrivial kernel. Mirzaii has shown ([3]) that the kernel of the induced homomorphism $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})_{F^{\times}}\rightarrow K_{3}^{\mathrm{ind}}(F)$  consists at most of $2$-primary torsion. In other words $\mathrm{Ker}\left(H_{3}\left(\mathrm{SL}_{2}(F), \mathbb{Z}\left[\frac{1}{2}\right]\right)\rightarrow K_{3}^{\mathrm{ind}}(F)\left[\frac{1}{2}\right]\right)$ is $\mathcal{I}_{F}H_{3}(\mathrm{SL}_{2}(F), \mathbb{Z})$ (up to some $2$-torsion), where $\mathcal{I}_{F}$ is the augmentation ideal of $R_{F}$.

For $F=\mathbb{Q}(\sqrt{-m})$  where $m\in \{1,2,3,7,11\}$, the main proposition of this article (see Proposition <ref>) describes the structure of $H_{0}\left(\mathcal{O}^{\times}_{F}, \mathcal{I}_{F}H_{3}\left(\mathrm{SL}_{2}(F), \mathbb{Z}\left[\frac{1}{2}\right]\right)\right)$ as a $R_{\mathbb{Q}(\sqrt{-m})}$-module. The proposition states that this module - via a natural residue homomorphism $S_{p}$ - is isomorphic to the direct sum of the scissor congruence group of the fields $k(\upsilon_{p})$, $\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]$, where the sum is over all the primes of the field $F$(see Section <ref> and Section <ref> for more details). The square class of $-1$ acts trivially on each factor, while the square class of $p$ acts by multiplication by $-1$ on the corresponding factor $\mathcal{P}(k(\upsilon_{p}))$. When $m\not=1$, it follows that as an abelian group

$H_{3}\left(\mathrm{SL}_{2}(\mathbb{Q}(\sqrt{-m})),\mathbb{Z}\left[\frac{1}{2}\right]\right)\cong K_{3}^{\mathrm{ind}}(\mathbb{Q}(\sqrt{-m}))\left[\frac{1}{2}\right]\oplus\left(\displaystyle\bigoplus_{p\in\Pi}\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}\right)$

where $\Pi$ denotes the set of primes (since $\mathcal{O}_{F}^{\times}$ acts trivially in this case) and the structure of indecomposable $K_{3}$ over number fields and the scissor congruence group of finite fields are known (see [2] and [7] respectively).

One of the main tool we use is the description of $H_{3}\left(\mathrm{SL}_{2}(F),\mathbb{Z}\left[\frac{1}{2}\right]\right)$ in terms of the refined scissors congruence group. The scissors congruence group $\mathcal{P}(F)$ of a field $F$ was introduced by Dupont and Sah in their study of the Hilbert's third problem in hyperbolic $3$-space ([4]). It is an abelian group defined by an explicit presentation (see section <ref>) and it was shown for the case $F=\mathbb{C}$ (and more generally when $F$ is algebraically closed) by the authors that $K_{3}^{\mathrm{ind}}(\mathbb{C})\cong H_{3}(\mathrm{SL}_{2}(\mathbb{C}),\mathbb{Z})$. Soon afterwards, Suslin ([11]) proved that, for any infinite field $F$ it is closely connected with $K_{3}^{\mathrm{ind}}(F)$, the indecomposable $K_{3}$ of $F$ (see Theorem <ref>). As noted above, $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})$ is naturally a module over the integral group ring $R_{F}:=\mathbb{Z}[F^{\times}/(F^{\times})^2]$ of the group of square classes of the field $F$. The refined scissors congruence group of the field $F$ - introduced by Hutchinson in [7]- is defined by generators and relations analogously to the scissors congruence group but as  a module over $R_{F}$ (see section <ref>). In [7], it is elucidated the relation of the refined scissors congruence group $\mathcal{RP}(F)$ to $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})$ and $K_{3}^{\mathrm{ind}}(F)$ (see Theorem <ref> for a precise statement). Our starting point in this article is essentially an isomorphism $\mathcal{I}_{F}H_{3}\left(\mathrm{SL}_{2}(F)), \mathbb{Z}\left[\frac{1}{2}\right]\right)\cong \mathcal{I}_{F}\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]$ where $\mathcal{RP}_{+}(F)$ is a certain quotient of $\mathcal{RP}(F)$ (see Proposition <ref>).

§.§ Layout of this article

In Section <ref>, we review some known results about the scissor congruence groups and their relation to the third homology of $\mathrm{SL}_{2}$ of fields.

In Section <ref>, we review the character-theoretic local-global principle for modules over a group ring ([9]). We recall some applications to Scissor congruence group ([10]), and generalize some of these results.

In Section <ref>, we review some algebraic properties of the refined scissor congruence groups over fields $F$ which have a discrete valuation.

In Section <ref>, we review relevant facts about quadratic number fields.

Section <ref> contains the proof of the main proposition (Proposition <ref>) for the field $\mathbb{Q}(\sqrt{-m})$, where $m\in \{1,2,3,7,11\}$.

§.§ Some notation

For a Field $F$, we let $F^{\times}$ denote the group of units of $F$. For $x\in F^{\times}$ we will let $\langle x\rangle\in F^{\times}/(F^{\times})^{2}$ denote the corresponding square class. Let $R_{F}$ denote the integral group ring $\mathbb{Z}\left[F^{\times}/(F^{\times})^{2}\right]$ of the group $F^{\times}/(F^{\times})^{2}$. We will use the notation $\langle\langle x\rangle\rangle$ for the basis elements, $\langle x\rangle-1$ of the augmentation ideal $\mathcal{I}_{F}$ of $R_{F}$. For any $a\in F^{\times}$, we will let $p^{a}_{+}$ and $p^{a}_{-}$ denote the elements $1+\langle a\rangle$ and $1-\langle a\rangle$ in $R_{F}$ respectively. For any abelian group $G$ we will let $G\left[\frac{1}{2}\right]$ denote $G\otimes\mathbb{Z}\left[\frac{1}{2}\right]$.

§.§ $H_{\bullet}(\mathrm{SL}_{2}(F), \mathbb{Z})$ is a $R_{F}$-module

Let us recall the group extension

$\xymatrix{1\ar[r]& \mathrm{SL}_{n}(F)\ar[r]& \mathrm{GL}_{n}(F)\ar[r]&F^{\times}\ar[r]&1}$

induces an action - by conjugation - of $F^{\times}$ on  the homology group $H_{\bullet}(\mathrm{SL}_{n}(F), \mathbb{Z})$. Since the determinant of a scalar matrix is an $n$-th power, the subgroup $(F^{\times})^{n}$ acts trivially. In particular, the groups
$H_{\bullet}(\mathrm{SL}_{2}(F), \mathbb{Z})$ are modules over the integral group ring $R_{F}:=\mathbb{Z}[F^{\times}/(F^{\times})^2]$.

§ BLOCH GROUPS OF FIELDS.

In this section we will recall the definition and applications of the classical pre-Bloch group $\mathcal{P}(F)$ and the refined pre-Bloch group $\mathcal{RP}(F)$.

§.§ Classical Bloch Group $\mathcal{B}(F)$.

For a field $F$, with at least $4$ elements, the pre-Bloch group or Scissors congruence group, $\mathcal{P}(F)$, is the group generated by the elements $[x]$, with $F^{\times}\setminus\{1\}$, subject to the relations

$R_{x,y}:\; 0=[x]-[y]+[y/x]-[(1-x^{-1})/(1-y^{-1})]+[(1-x)/(1-y)]$,    $x\not=y$.

Let $S^{2}_{\mathbb{Z}}(F^{\times})$ denote the group

$\frac{F^{\times}\otimes_{\mathbb{Z}}F^{\times} }{\langle x\otimes y +y\otimes x\; |\;x,y\in F^{\times}\rangle}$

and denote by $x\circ y$ the image of $x\otimes y$ in $S^{2}_{\mathbb{Z}}(F^{\times})$. The map

$\lambda: \mathcal{P}(F)\rightarrow S^{2}_{\mathbb{Z}}(F^{\times})$,     $[x]\mapsto (1-x)\circ x$

is well-defined, and the Bloch group of $F$, $\mathcal{B}(F)\subset\mathcal{P}(F)$, is defined to be the kernel of $\lambda$.

For the fields $\mathbb{F}_{2}$ and $\mathbb{F}_{3}$ the following definitions allows us to include these fields in the statements of some of our results:

$\mathcal{P}(\mathbb{F}_{2})=\mathcal{B}(\mathbb{F}_{2})$ is a cyclic group of order $3$ with generator denoted $C_{\mathbb{F}_{2}}$.

$\mathcal{P}(\mathbb{F}_{3})$ is cyclic of order $4$ with generator $[-1]$. $\mathcal{B}(\mathbb{F}_{3})$ is the subgroup generated by $2[-1]$.

The Bloch group $\mathcal{B}(F)$ of a general field $F$ is of interest because of the following result of Suslin on $K_{3}^{\mathrm{ind}}$:

Let $F$ be an infinite field, then there is a short exact sequence

where $\mathrm{Tor}_{1}^{\mathbb{Z}}(\widetilde{\mu_{F},\mu_{F}})$  is the unique nontrivial extension of $\mathrm{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})$ by $\mathbb{Z}/2$ when $\mathrm{Char}(F)\not=2$ (and $\mathrm{Tor}_{1}^{\mathbb{Z}}(\widetilde{\mu_{F},\mu_{F}})=\mathrm{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})$ if $\mathrm{Char}(F)=2$) .

§.§ The refined Bloch Group $\mathcal{RB}(F)$.

The refined pre-Bloch group $\mathcal{RP}(F)$, of a field $F$ which has at least $4$ elements, is the $R_{F}$-module with generators $[x]$, $x\in F^{\times}$ subject to the relations $[1]=0$ and

$S_{x,y}:\; 0=[x]-[y]+\langle x\rangle[y/x]-\langle x^{-1}-1\rangle[(1-x^{-1})/(1-y^{-1})]+\langle1-x\rangle[(1-x)/(1-y)]$,    $x,y\not=1$.

From the definitions, it follows that $\mathcal{P}(F)=(\mathcal{RP}(F))_{F^{\times}}$. Let $\Lambda=(\lambda_{1},\lambda_{2})$ be the $R_{F}$-module homomorphism

$\mathcal{RP}(F)\rightarrow \mathcal{I}_{F}^{2}\oplus S^{2}_{\mathbb{Z}}(F^{\times})$

where $S^{2}_{\mathbb{Z}}(F^{\times})$ has the trivial $R_{F}$-module structure, $\lambda_{1}:\mathcal{RP}(F)\rightarrow \mathcal{I}^{2}_{F}$ is the map $[x]\mapsto \langle\langle1-x\rangle\rangle\langle\langle x\rangle\rangle$ and $\lambda_{2}$ is the composite

$\xymatrix{ \mathcal{RP}(F)\ar[r]&\mathcal{P}(F)\ar[r]^{\lambda} &S^{2}_{\mathbb{Z}}(F^{\times})}$,

The refined Bloch group of $F$ is the module

$\mathcal{RB}(F):=\mathrm{Ker}\left(\Lambda: \mathcal{RP}(F)\rightarrow \mathcal{I}_{F}^{2}\oplus S^{2}_{\mathbb{Z}}(F^{\times}  )\right)$.

Furthermore, the refined scissor congruence group of $F$ is the $R_{F}$-module

Thus $\mathcal{RB}(F)=\mathrm{Ker}(\lambda_{2}:\mathcal{RP}_{1}(F)\rightarrow S^{2}_{\mathbb{Z}}(F^{\times}))$.

The refined Bloch group is of interest because of the following result on the third homology of $\mathrm{SL}_{2}$ over a general field $F$:

Let $F$ be a field with at least $4$ elements.

(1)If $F$ is infinite, there is a natural complex of $R_{F}$-modules

$\xymatrix{0\ar[r]&\mathrm{Tor}_{1}^{\mathbb{Z}}(\mu_{F},\mu_{F})\ar[r]&H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\ar[r]& \mathcal{RB}(F)\ar[r]&0}$,

which is exact except at the middle term where the homology is annihilated by $4$.

(2)If $F$ is finite of odd characteristic, there is a complex

$\xymatrix{0\ar[r]&H_{3}(B,\mathbb{Z})\ar[r]&H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\ar[r]& \mathcal{RB}(F)\ar[r]&0}$,

which is exact except at the middle term, where the homology has order $2$.

(3)If $F$ is finite of characteristic $2$, there is a exact sequence

$\xymatrix{0\ar[r]&H_{3}(B,\mathbb{Z})\ar[r]&H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\ar[r]& \mathcal{RB}(F)\ar[r]&0}$,

Now for $x\in F^{\times}$, we define the following elements of $\mathcal{RP}(F)$

$\psi_{1}(x):=[x]+\langle-1\rangle[x^{-1}]$    and   $\psi_{2}(x):=\left\{
\begin{array}{ll}
\langle x^{-1}-1\rangle[x]+\langle1-x\rangle[x^{-1}], & \hbox{$x\not=1$;} \\
0, & \hbox{$x=1$.}
\end{array}
\right.

From the definitions of the the elements $\psi_{i}(x)$, we get that $\langle-1\rangle\psi_{i}(-1)=\psi_{i}(-1)$ for $i\in\{1,2\}$.
We define $\widetilde{\mathcal{RP}}(F)$ to be $\mathcal{RP}(F)$ modulo the submodule generated by the elements $\psi_{1}(x)$, $x\in F^{\times}$.

In the section $3.2$ of [9], it is shown that the elements

$C(x)=[x]+\langle-1\rangle[1-x]+ \langle\langle1-x\rangle\rangle\psi_{1}(x) \in \mathcal{RP}(F)$

are constant for a field with at least 4 elements; i.e. $C(x)=C(y)$ for all $x,y\in F^{\times}$. Therefore we have the following definition.

Let $F$ be a field with at least $4$ elements. We will denote by $C_{F}$ the common value of the expression $C(x)$ for $x\in F\setminus\{0,1\}$; i.e.

$C_{F}:=[x]+\langle-1\rangle[1-x]+ \langle\langle1-x\rangle\rangle\psi_{1}(x)$   in   $\mathcal{RP}(F)$.

For the field with $2$ and $3$ elements the following definitions allow us to include these fields in the statements of some results:

$\mathcal{RP}(\mathbb{F}_{2})=\mathcal{RP}_{1}(\mathbb{F}_{2})=\mathcal{RB}(\mathbb{F}_{2})=\mathcal{P}(\mathbb{F}_{2})$; i.e. it a cyclic module generated by $C_{\mathbb{F}_{2}}$.

$\mathcal{RP}(\mathbb{F}_{3})$ is a cyclic $R_{\mathbb{F}_{3}}$-module generated by $[-1]$ subject to the relation $[1]=0$ and

$\mathcal{RB}(\mathbb{F}_{3})=\mathcal{RP}_{1}(\mathbb{F}_{3})$ is the submodule generated by $\psi_{1}(-1)=[-1]+\langle-1\rangle[-1]$.

For any field $F$, there is a natural surjective homomorphism of $R_{F}$-modules $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\rightarrow K_{3}^{\mathrm{ind}}(F)$, it can be shown this is an homomorphism of $R_{F}$-modules where $F^{\times}/(F^{\times})^{2}$ acts trivially on $K_{3}^{\mathrm{ind}}(F)$. In [3], we get this map induces an isomorphism

$H_{0}\left(F^{\times}/(F^{\times})^{2},H_{3}\left(\mathrm{SL}_{2}(F),\mathbb{Z}\left[\frac{1}{2}\right]\right)\right)\cong K_{3}^{\mathrm{ind}}(F)\left[\frac{1}{2}\right]$

Now let $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})_{0}$ denote the kernel of the surjective homomorphism $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\rightarrow K^{\mathrm{ind}}_{3}(F)$. Note the isomorphism above implies

Let us recall that when $F$ is a number field the surjective homomorphism $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\rightarrow K^{\mathrm{ind}}_{3}(F)$ is split as a map of abelian groups. In fact $K_{3}^{ind}(F)$ is finitely generated abelian group an it is enough that there is a torsion subgroup of $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})$ mapping isomorphically to the cyclic torsion subgroup of $K_{3}^{ind}(F)$. This latter statement follows from the explicit calculations of Christian Zickert in <cit.> . It follows as an abelian group

$H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\cong K_{3}^{ind}(F)\oplus H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})_{0}$

for any number field $F$.

Furthermore, we have the following proposition:

For any field $F$, the map $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\rightarrow \mathcal{RP}(F)$ induces an isomorphism of $R_{F}$-modules

and furthermore

where $e_{+}^{-1}:=\frac{p_{+}^{-1}}{2}=\frac{1+\langle-1\rangle}{2}\in R_{F}\left[\frac{1}{2}\right]$

Note that from $\mathcal{RB}(F)\subseteq \mathcal{RP}_{1}(F)$ and Theorem <ref>, it follows that the square class $\langle-1\rangle$ acts trivially on $H_{3}\left(\mathrm{SL}_{2}(F),\mathbb{Z}\left[\frac{1}{2}\right]\right)$ (see <cit.> for more details). Now we define $\mathcal{RP}_{+}(F)$ to be $\widetilde{\mathcal{RP}}(F)$ modulo the submodule generated by the elements $(1-\langle-1\rangle)[x]$, $x\in F^{\times}$. Thus $\mathcal{RP}_{+}(F)$ is the $R_{F}$-module generated by the elements $[x]$, $x\in F^{\times}$ subject to the relations

1) $[1]=0$.

2) $S_{x,y}=0$ for $x,y\not=1$.

3) $\langle-1\rangle[x]=[x]$ for all $x\in F^{\times}$.

4) $[x]=-[x^{-1}]$ for all $x\in F^{\times}$.

The proposition above implies that the map $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})\rightarrow \mathcal{RP}(F)$ induces an isomorphism

It will be convinient below to introduce the following notation in $\mathcal{RP}_{+}(F)$:

$[1]:=0$,    $[0]:=C_{F}$    and   $[\infty]:=-C_{F}$.

Thus the symbol $[x]\in \mathcal{RP}_{+}(F)$ is defined for all $x\in \mathbb{P}^{1}(F)$. Furthermore $[x]+[1-x]=C_{F}$ and $[x]=-[x^{-1}]$ in $\mathcal{RP}_{+}(F)$ for all $x\in \mathbb{P}^{1}(F)$.

§ CHARACTERS

In this section we will review the character-theoretic method introduced by Hutchinson ([9]) for calculating with
modules over a group ring with an elementary abelian $2$-group. Furthermore,
we prove some new related results about subrings in number field which will be useful in Section <ref>.

Let $G$ be an abelian group satisfying $g^{2}=1$ for all $g\in G$. Let
$\mathcal{R}$ denote the group ring $\mathbb{Z}[G]$. For a character $\chi\in
\widehat{G}:=\mathrm{Hom}(G,\mu_{2})$, let $\mathcal{R}^{\chi}$ be the ideal
of $\mathcal{R}$ generated by the elements $\{g-\chi(g)\;|\; g\in G\}$; i.e.
$\mathcal{R}^{\chi}$ is the kernel of the ring homomorphism
$\rho(\chi):\mathcal{R}\rightarrow \mathbb{Z}$ sending $g$ to $\chi(g)$ for
any $g\in G$. We let $\mathcal{R}_{\chi}$ denote the associated
$\mathcal{R}$-algebra structure on $\mathbb{Z}$. In other words

If $M$ is a $\mathcal{R}$-module, we let $M^{\chi}=\mathcal{R}^{\chi}M$ and
we let

Thus $M_{\chi}$ is the largest quotient module of $M$ with the property that
$g\cdot m=\chi(g)\cdot m$ for all $g\in G$. In particular, if $\chi=\chi_{0}$
is the trivial character, then $\mathcal{R}^{\chi_{0}}$ is the augmentation
ideal $\mathcal{I}_{G}$, $M^{\chi_{0}}=\mathcal{I}_{G}M$ and

Given $m\in M$, $\chi\in \widehat{G}$, we denote the image of $m$ in
$M_{\chi}$ by $m_{\chi}$. For example for any character $\chi\in
\widehat{F^{\times}/(F^{\times})^{2}}$, we can give a presentation of the
$R_{F}$-module $\mathcal{RP}_{+}(F)_{\chi}$ which is our main object of
study. $\mathcal{RP}_{+}(F)_{\chi}$ is the $R_{F}$-module with generators
$[x]_{\chi}$, $x\in F^{\times}$, subject to the relations

1) $\langle a\rangle\cdot[x]_{\chi}:=\chi(a)\cdot[x]_{\chi}$ for
all $a,x\in F^{\chi}$.

2) $[1]_{\chi}=0$

3) The five term relation

for all $x,y\not=1$

4) $\chi(-1)\cdot[x]_{\chi}=[x]_{\chi}$ for all $x\in F^{\times}$.

5) $[x]_{\chi}=-[x^{-1}]_{\chi}$ for all $x\in F^{\times}$.

Let us recall for $M$ a $\mathcal{R}$-module, $M_{\chi}=\mathcal{R}_{\chi}\otimes_{\mathcal{R}}M$. Hence, for any
$\mathcal{R}$-homomorphism $f:M\rightarrow N$, it follows that we have the
following induced map

\mathcal{R}_{\chi}\otimes_{\mathcal{R}}N=N_{\chi}$

The following the character-theoretic local-global principle is central to
calculations in the following chapter below

(1) For any $\chi\in\widehat{G}$, $M\rightarrow M_{\chi}$ is an
exact functor on the category of

(2) Let $f:M\rightarrow N$ be an
$\mathcal{R}\left[\frac{1}{2}\right]$-module homomorphism. For any
$\chi\in\widehat{G}$, let $f_{\chi}:M_{\chi}\rightarrow N_{\chi}$ be
the induced map. Then $f$ is bijective (resp. injective, surjective)
if and only if $f_{\chi}$ is bijective (resp. injective, surjective)
for all $\chi\in\widehat{G}$.

Let $F$ be a field. Let $\chi\in \widehat{F^{\times}/(F^{\times})^{2}}$.
Suppose that $a\in F^{\times}$ satisfy $\chi(1-a)=-1$ and $\chi(a)=1$ then
$[a]_{\chi}=0$ in $\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi}$

Let $F$ be a field. Let $\chi\in \widehat{F^{\times}/(F^{\times})^{2}}$ with
$\chi(-1)=1$. Suppose that $\ell\in F^{\times}$ satisfy $\chi(\ell)=-1$ and
$\chi(1-\ell)=1$. Then $[a]_{\chi}=[(1-\ell)^{m}a]_{\chi}$ in
$\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi}$ for all $a\in
\mathbb{P}^{1}(F)$ and all $m\in\mathbb{Z}$.

For a field $F$, let $\chi\in \widehat{F^{\times}/(F^{\times})^{2}}$ such
that $\chi(-1)=1$. Suppose that $\ell\in F^{\times}$ satisfy $\chi(\ell)=-1$
and $\chi(1-\ell)=-1$ then $\chi\left(\frac{1}{\ell}\right)=-1$,
$\chi\left(1-\frac{1}{\ell}\right)=1$. Therefore by Corollary
<ref>, we can deduce the following for all $a\in
\mathbb{P}^{1}(F)$:

a) If $x(\ell)=1$ and $\chi(1-\ell)=-1$ then $[a]_{\chi}=[\ell

b) If $\chi(\ell)=-1$ and $\chi(1-\ell)=-1$ then

From Corollary <ref>, we can deduce the following

Let $F$ be a field with at least $4$ elements. Let $\chi\in \widehat{F^{\times}/(F^{\times})^{2}}$ with
$\chi(-1)=1$. Suppose that $\ell\in F^{\times}$ satisfy $\chi(\ell)=-1$ and
$\chi(1-\ell)=1$. Then $[a]_{\chi}=[a+t\ell]_{\chi}$ in
$\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi}$ for all $a\in
F^{\times}$ and all $t\in\mathbb{Z}$.

Now let $R$ be a commutative ring and $x\in R$. Let us recall that $\mathbb{Z}[x]$ is
the subring

$\{a_{0}+a_{1}x+\cdots+a_{n}x^{n}\;|\; n\geq0\;\; a_{i}\in\mathbb{Z}\}\subset

Now $x\mathbb{Z}[x]=\{xp(x)\;|\; p(x)\in \mathbb{Z}[x] \}$; i.e.
$x\mathbb{Z}[x]$ is the ideal in $\mathbb{Z}[x]\subset R$ generated by $x\in

The following proposition is a generalization of the Corollary
<ref> for any field $F$:

Let $F$ be a field with at least $4$ elements. Let $\chi\in \widehat{F^{\times}/(F^{\times})^{2}}$ with
$\chi(-1)=1$. If $\chi(\ell)=-1$, $\chi(1-\ell)=1$ for $\ell\not=1\in F$. Then $[a]_{\chi}=[a+t]_{\chi}$ in $\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi}$ for all $a\in F^{\times}$ and all $t\in \ell\mathbb{Z}[\ell]$

$\textbf{\emph{Proof}}.$ Let $t\in \ell\mathbb{Z}[\ell]$. So there exist
$m\geq1$, $b_{1},\ldots, b_{m}\in\mathbb{Z}$ with
$t=b_{1}\ell+\cdots+b_{m}\ell^{m}$.  We will prove it by induction on $m$. If
$m=1$, the result follows from the Corollary <ref>. Now we
suppose it is valid for any number $m\leq k$. We will prove it for the case
$k+1$. First, note that $\chi\left(\frac{1}{1-\ell}\right)=1$ and

\begin{eqnarray}
\nonumber [a]_{\chi} &= &\left[\frac{a}{1-\ell}\right]_{\chi}  \mbox{(From the Remark \ref{Remark[a]=[la]})}\\
\nonumber &=&\left[\frac{a}{1-\ell}+f_{1}\ell+\cdots +f_{k}\ell^{k}\right]_{\chi} \mbox{ (For any $f_{1},\ldots f_{k}\in\mathbb{Z}$ by the induction hypothesis)}\\
\nonumber &=&\left[a+(1-\ell)(f_{1}\ell+\cdots +f_{k}\ell^{k})\right]_{\chi}\mbox{(From the Corollary \ref{ArticlehomologyQprint(1-l)a})}\\
\nonumber &=&\left[a+(1-\ell)(f_{1}\ell+\cdots +f_{k}\ell^{k})+   c_{1}\ell+\cdots +c_{k}\ell^{k} \right]_{\chi}\mbox{ (For any $c_{1},\ldots c_{k}\in\mathbb{Z}$ by hypothesis)}\\
\nonumber &=&\left[a+ (f_{1}+c_{1})\ell+ (f_{2}-f_{1}+c_{2})\ell^{2}+\cdots +(f_{k}-f_{k-1}+c_{k})\ell^{k-1}+f_{k}\ell^{k+1} \right]_{\chi}.
\end{eqnarray}

Since $f_{1},\ldots f_{k}, c_{1},\ldots, c_{k}$ are chosen arbitrarily
integers. It follows that for any $b_{1},\ldots, b_{k+1}\in\mathbb{Z}$, we can solve the following system of equations:

\begin{eqnarray}
\nonumber f_{1}+c_{1} &=&b_{1}\\
\nonumber f_{2}-f_{1}+c_{2}&=&b_{2}\\
\nonumber &\vdots&\\
\nonumber f_{k}-f_{k-1}+c_{k}&=&b_{k}\\
\nonumber f_{k}&=&b_{k+1}
\end{eqnarray}

Therefore $[a]_{\chi}=\left[a+b_{1}\ell+\cdots
+b_{k+1}\ell^{k+1}\right]_{\chi}$ for any $b_{1},\ldots,

Let $F$ be a number field such that $[F:\mathbb{Q}]=d$ and $\mathcal{O}_{F}$
its ring of algebraic integers. Let $\ell\in \mathcal{O}_{F}$ of degree $d$
and $N=N_{F/\mathbb{Q}}(\ell)\in \mathbb{Z}$. Then

\mathcal{O}_{F}$.

$\textbf{\emph{Proof}}.$  Let $p(t)=N+b_{1}t+\cdots + b_{d-1}t^{d-1}+t^{d}$
the minimal polynomial of $\ell$. Then

$0=p(\ell)=N+b_{1}\ell+\cdots +b_{d-1}\ell^{d-1}+\ell^{d}$.

Since $b_{1}\ell+\cdots b_{d-1}\ell^{d-1}+\ell^{d}\in \ell\mathbb{Z}[\ell]$,
it follows that $N\in \ell\mathbb{Z}[\ell]$. Therefore

\ell\mathbb{Z}[\ell]$.

Now note that $\ell^{k}\in
\mathbb{Z}N+\mathbb{Z}\ell+\cdots\mathbb{Z}\ell^{d-1}$ with
$k\in\{1,2,\ldots,d-1\}$ and from the equation $0=p(\ell)$ we get that
$\ell^{d}\in \mathbb{Z}N+\mathbb{Z}\ell+\cdots\mathbb{Z}\ell^{d-1}$.

Now, multiplying the equation $0=p(\ell)$ by $\ell$ we obtain that

$\ell^{d+1}=-N-b_{1}\ell-\cdots -b_{d-1}\ell^{d}$.

Since $\ell^{d}\in \mathbb{Z}N+\mathbb{Z}\ell+\cdots\mathbb{Z}\ell^{d-1}$ ,
it follows that
\ell\mathbb{Z}[\ell]$. Hence following a similar recursive way, we get that
\ell\mathbb{Z}[\ell]$ with $m\geq d+1$. Therefore it follows that
\subseteq\mathbb{Z}N+\mathbb{Z}\ell+\cdots\mathbb{Z}\ell^{d-1}$.

Let $F$ be a number field such that $[F:\mathbb{Q}]=d$ and $\mathcal{O}_{F}$
its ring of algebraic integers. Let $\ell\in \mathcal{O}_{F}$ of degree $d$
and $N=N_{F/\mathbb{Q}}(\ell)\in \mathbb{Z}$. Then $\mathbb{Z}[\ell]
\subset\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$.

$\textbf{\emph{Proof}}.$  Let $p(t)=N+b_{1}t+\cdots b_{d-1}t^{d-1}+t^{d}$ the
minimal polynomial of $\ell$. Then

$0=p(\ell)=N+b_{1}\ell+\cdots b_{d-1}\ell^{d-1}+\ell^{d}$.

Multiplying the last equation by $\ell^{-d}$, we get

\frac{b_{d-1}}{\ell}+1$.

Since $\frac{N}{\ell^{d}}+\frac{b_{1}}{\ell^{d-1}}+\cdots
\frac{b_{d-1}}{\ell}\in \frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$.
It follows that $1\in \frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$
whence we obtain that $\mathbb{Z}\subset
\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$.

Now we can finally prove $\mathbb{Z}[\ell] \subset\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$. For that, it is enough to prove that $\ell^{m}\in \frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$ where $m\geq1$. We will prove it by induction on $m$. If $m=1$, note that since $1\in \frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$, we get that $1=\frac{a_{1}}{\ell}+\frac{a_{2}}{\ell^{2}}+\cdots +\frac{a_{d-1}}{\ell^{d-1}}$ with $a_{1},a_{2},\ldots,a_{d-1}\in\mathbb{Z}$, then multiplying the last equation by $\ell$ we get the following equation:

$\ell=a_{1}+\frac{a_{2}}{\ell}+\cdots +
\frac{a_{d-2}}{\ell^{d-3}}+\frac{a_{d-1}}{\ell^{d-2}}$.

Since $a_{1}\in \frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$, it
follows that $\ell\in \frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$.
Now we suppose that $\ell^{m}\in \mathbb{Z}\left[\frac{1}{\ell}\right]$ for all $m\leq k$. We will prove the case
$m=k+1$. Multiplying the last equation by $\ell^{k}$, we get the equation

$\ell^{k+1}=a_{1}\ell^{k}+a_{2}\ell^{k-1}\cdots + a_{d-1}\ell^{k-d+2}$.

By the induction hypothesis, it follows that $\ell^{k+1}\in
\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$. Therefore as required  $\mathbb{Z}[\ell]
\subset\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$.

From the proof of Lemma <ref>, we have that
$\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$ is closed under
multiplication by $\ell$.

§ FIELDS WITH A VALUATION

Given a field $F$ and a surjective valuation $\upsilon:F^{\times}\rightarrow \Gamma$, where $\Gamma$ is a totally ordered additive abelian group, we let $\mathcal{O}_{\upsilon}:=\{x\in F^{\times}\;|\;\upsilon(x)\geq0\}\cup\{0\}$ be the associated valuation ring, with maximal ideal $\mathcal{M}_{\upsilon}=\{x\in \mathcal{O}_{\upsilon}\;|\;\upsilon(x)\not=0\}$, group of units $U_{\upsilon}=U:=\mathcal{O}_{\upsilon}\backslash\mathcal{M}_{\upsilon}$ and residue field $k=k(\upsilon):=\mathcal{O}_{\upsilon}/\mathcal{M}_{\upsilon}$.

Since $\Gamma$ is a torsion-free group, we have a short exact sequence of $\mathbb{F}_{2}$-vector spaces

We have a homomorphism of commutative rings

$\xymatrix{\mathbb{Z}\left[\frac{U}{U^{2}}\right]\ar@{^{(}->}[r] \ar@{>>}[d]&R_{F}\\

Therefore given an $R_{k}$-module $M$, we denote by $\mathrm{Ind}_{k}^{F}M$ the $R_{F}$-module $R_{F}\otimes_{\mathbb{Z}[U_{\upsilon}]}M$.
We have the following result <cit.>:

There is a natural homomorphism of $R_{F}$-modules $S_{\upsilon}:\widetilde{\mathcal{RP}}(F)\rightarrow \mathrm{Ind}_{k}^{F}\widetilde{\mathcal{RP}}(k)$ given by

\begin{array}{ll}
1\otimes[\overline{x}], & \hbox{$\upsilon(x)=0$;} \\
1\otimes C_{k}, & \hbox{$\upsilon(x)>0$;} \\
-(1\otimes C_{k}), & \hbox{$\upsilon(x)<0$.}
\end{array}
\right.

§.§ Discrete valuations

Suppose that $\upsilon:F^{\times}\rightarrow \mathbb{Z}$ is a discrete valuation on the field $F$ with residue field $k=k(\upsilon)$. Let $\chi_{\upsilon}:F^{\times}/(F^{\times})^{2}\rightarrow \mu_{2}$ denote the associated character defined by $\chi_{\upsilon}(a)=(-1)^{\upsilon(a)}$. For an abelian group $M$, we let $M\{\upsilon\}$ denote the $R_{F}$-module $R_{\chi_{\upsilon}}\otimes_{\mathbb{Z}}M$. Equivalently we equip $M$ with the $R_{F}$-module structure $\langle a\rangle m:=(-1)^{\upsilon(a)}m$ for all $a\in F^{\times}$ and $m\in M$. We have the following result:

Let $F$ be a field with discrete valuation $\upsilon:F^{\times}\rightarrow \mathbb{Z}$ and residue field $k$. Then we have a natural isomorphism

$\xymatrix{\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi_{\upsilon}}\ar[r]^{S_{\upsilon}}&\left(\mathrm{Ind}^{F}_{k}\mathcal{RP}_{+}(k)\left[\frac{1}{2}\right]\right)_{\chi_{\upsilon}}\ar[r]^{\cong}& \mathcal{P}(k)\left[\frac{1}{2}\right]\{\upsilon\} }$

For any field $F$ with discrete valuation $\upsilon$, we let $\overline{S}_{\upsilon}$ denote the composite $R_{F}$-module homomorphism

$H_{3}\left(\mathrm{SL}_{2}(F), \mathbb{Z}\left[\frac{1}{2}\right]\right)\rightarrow \left(\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]\right)_{\chi_{\upsilon}}\cong\mathcal{P}(k(\upsilon))\left[\frac{1}{2}\right]\{\upsilon\}$

By abuse of notation, we will use the same symbol to denote $\overline{S}_{\upsilon}$ restricted to $H_{3}\left(\mathrm{SL}_{2}(F), \mathbb{Z}\left[\frac{1}{2}\right]\right)_{0}$.

Let us recall that if $\pi$ is a uniformizer for the valuation $\upsilon$,
then $\upsilon(\pi)=1$. Therefore the square class of $\pi$ acts as $-1$ on
the factor $\mathcal{P}(k(\upsilon))\{\upsilon\}$ on the right.

Give a family $\mathcal{V}$ of discrete valuations of $F$, we obtain a map

$\xymatrix{H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z} )\ar[r]& \mathcal{RP}_{+}(F)\ar[r]& \displaystyle\prod_{\upsilon\in \mathcal{V}}\mathcal{P}(k(\upsilon))\{\upsilon\}}$.

Note that when we restrict to $H_{3}(\mathrm{SL}_{2}(F),\mathbb{Z})_{0}$ and tensor with $\mathbb{Z}\left[\frac{1}{2}\right]$, the image lies in the direct sum instead (For more details see <cit.> ).

Let $\mathcal{V}$ be a set of discrete valuations, we get a homomorphism $\overline{S}=\{\overline{S}_{\upsilon}\}_{\upsilon\in\mathcal{V}}$  of $R_{F}$-modules

$\overline{S}:H_{3}\left(\mathrm{SL}_{2}(F),\mathbb{Z}\left[\frac{1}{2}\right]\right)_{0}\rightarrow \displaystyle\bigoplus_{v\in\mathcal{V}}\mathcal{P}(k(\upsilon))\{\upsilon\}$.

If $\upsilon(x)$ is odd, then $\langle x\rangle$ acts as $-1$ on $M\{\upsilon\}$.

§ QUADRATIC FIELDS.

Let us recall that if $a,b\in R$ then $a$ is associated to $b$ ($a\sim b$), if there exist $u\in R^{\times}$ such
that $a=ub$.

For the field $\mathbb{Q}(\sqrt{m})$, we have the ring of algebraic
a,b\in\mathbb{Z}\}$ where

\begin{array}{ll}
\sqrt{m}, & \hbox{$m\;\equiv\;2,3\;(\mathrm{mod}\;4)$;} \\
\frac{1+\sqrt{m}}{2}, & \hbox{$m\;\equiv\;1\;(\mathrm{mod}\;4)$.}
\end{array}
\right.

For $\alpha\in \mathbb{Q}(\sqrt{m})$, let $\overline{\alpha}$ denote its conjugate. Then if $\alpha=a+\omega_{m}b\in \mathcal{O}_{\mathbb{Q}(\sqrt{m})}$ , we get that:

\begin{array}{ll}
a-\omega_{m}b, & \hbox{$m\;\equiv\;2,3\;(\mathrm{mod}\;4)$;} \\
(a+b)-\omega_{m}b, & \hbox{$m\;\equiv\;1\;(\mathrm{mod}\;4)$.}
\end{array}
\right.$

Therefore in any quadratic field $\mathbb{Q}(\sqrt{m})$, the norm of
$\alpha\in\mathcal{O}_{\mathbb{Q}(\sqrt{m})}$ is given by

\begin{array}{ll}
a^{2}-mb^{2}, & \hbox{$m\;\equiv\;2,3\;(\mathrm{mod}\;4)$;} \\
a^{2}+ab+\left(\frac{1-m}{4}\right)b^{2}, & \hbox{$m\;\equiv\;1\;(\mathrm{mod}\;4)$.}
\end{array}
\right.

Let us recall that the discriminant $\delta$ of any quadratic field $\mathbb{Q}(\sqrt{m})$

\begin{array}{ll}
m, & \hbox{$m\;\equiv\;1\;(\mathrm{mod}\;4)$;} \\
4m, & \hbox{$otherwise$.}
\end{array}
\right.

Now, the prime elements in $\mathcal{O}_{\mathbb{Q}(\sqrt{m})}$ are found by
implementing the three following primality criteria

* If $N(\alpha)$ is a rational prime, then $\alpha$ is a prime.

* An odd rational prime $p$ is a prime in
$\mathcal{O}_{\mathbb{Q}(\sqrt{m})}$ if and only if $p$ does not
divide $\delta$ and $\left(\frac{\delta}{p}\right)=-1$.

* $2$ is prime in $\mathcal{O}_{\mathbb{Q}(\sqrt{m})}$ if and only if

We introduce the following notation. For $\alpha=a+b\omega_{m}\in
\mathbb{Z}[\omega_{m}]$, we let $a:=R(\alpha)$ and $b:=I(\alpha)$. Then
with this notation we have:

i) If $m\;\equiv\;2,3\;(\mathrm{mod}\;4)$ then

$R(\overline{\alpha})=R(\alpha)$    and

ii)  If $m\;\equiv\;1\;(\mathrm{mod}\;4)$ then

$R(\overline{\alpha})=R(\alpha)+I(\alpha)$    and

\begin{array}{ll}
R(\alpha)-\omega_{m}I(\alpha), & \hbox{$m\;\equiv\;2,3\;(\mathrm{mod}\;4)$;} \\
(R(\alpha)+I(\alpha))-\omega_{m}I(\alpha), & \hbox{$m\;\equiv\;1\;(\mathrm{mod}\;4)$.}
\end{array}
\right.$

We will require the following observation below.

Let $\mathbb{Q}(\sqrt{-m})$ with $m>0$ such that $m\;\equiv\;1,2\;(\mathrm{mod}\;4)$ and let
$\alpha=R(\alpha)+\omega_{m}I(\alpha)\in \mathcal{O}_{\mathbb{Q}(\sqrt{-m})}$ a prime
element such that $N(\alpha)=p$ is a rational prime. If $p$ divides
$R(\alpha)$ or $m$ in $\mathbb{Z}$ then $m=p$. In particular $\alpha\sim\sqrt{-p}$.

$\textbf{\emph{Proof}}.$ If $p|R(\alpha)$, we have $R(\alpha)=pn$ for some
$n\in\mathbb{Z}$. Thus

From this the last equation, we get $m=p$ and $R(\alpha)=0$,
$I(\alpha)=\pm1$ as otherwise $N(\alpha)=(R(\alpha))^{2}+m(I(\alpha))^{2}>p$.

If $p|m$ then $m=p\ell$  for some $\ell\in\mathbb{Z}$. Thus $p=N(\alpha)=(R(\alpha))^{2}+p\ell(I(\alpha))^{2}$. It follows that $p|R(\alpha)$ and from above, we get our result.

§ THE MAIN THEOREM

Let $F$ be a number field. We consider $\Pi$, the set of discrete valuations on $F$. As we seen, the maps $S_{p}$ induce

\displaystyle\bigoplus_{p\in\Pi}\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}$

In [10], Hutchinson proved this is an isomorphism for $F=\mathbb{Q}$. Our main theorem (see Proposition <ref>) is that it is an isomorphism for $F=\mathbb{Q}(\sqrt{-m})$ for $m\in \{2,3,,7,11\}$.
Furthermore we will prove a slightly weaker statement for the case $m=-1$. In all cases, we will use the local-global
principle for characters; i.e. we will prove that

$\overline{S}_{\chi}:H_{0}\left(\mathcal{O}^{\times}_{F}, \mathcal{I}_{F}\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]\right)_{\chi}\rightarrow
\displaystyle\bigoplus_{p\in\Pi}\left(\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}\right)_{\chi}$

is an isomorphism for all $\chi\in \widehat{F^{\times}/(F^{\times})^{2}}$.

We will use the following observation repeatedly below:

Let $M$ be an $R_{F}$-module. Let $\chi\in\widehat{F^{\times}/(F^{\times})^{2}}$, let $\varepsilon\in\{-1, 1\}$. Suppose there exist $a\in F^{\times}$ such that $\langle a\rangle$ acts as $\varepsilon$ on $M$ and $\chi(\langle a\rangle)=-\varepsilon$. Then $M\left[\frac{1}{2}\right]_{\chi}=0$

$\textbf{\emph{Proof}}.$ Let $m\in M$. In $M\left[\frac{1}{2}\right]_{\chi}$ we get

$-\varepsilon m_{\chi}=\chi(\langle a\rangle) m_{\chi}=\langle a\rangle m_{\chi}=\varepsilon m_{\chi}$.

Then $2\varepsilon m_{\chi}=0$. Therefore $m_{\chi}=0$ for all $m_{\chi}\in M\left[\frac{1}{2}\right]_{\chi}$.

To start with, we consider the trivial character $\chi_{0}$. By definition $\mathcal{I}_{F}\mathcal{RP}_{+}(F)$ is generated by elements of the form
$\langle\langle x\rangle\rangle[y]$. Note that the square class of $x$ acts as multiplication by $-1$  on such an element (Since $\langle
x\rangle\langle\langle x\rangle\rangle=-\langle\langle x\rangle\rangle$). On the other hand, $\langle x\rangle$
acts as $1$ on any element of $\mathcal{I}_{F}\mathcal{RP}_{+}(F)_{\chi_{0}}$. By Lemma <ref> we get $\left(\mathcal{I}_{F}\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]\right)_{\chi_{0}}=0$.

Similarly for $\mathcal{P}(k(\upsilon_{p}))\{p\}$, the element
$\langle p\rangle$ acts as $-1$ (see Remark <ref>) and
since $\chi_{0}(p)=1$, by Lemma <ref> we deduce that

Next, we consider characters $\chi$ satisfying $\chi(-1)=-1$. Let us recall that $\langle-1\rangle$ acts as multiplication by $1$ on
$\mathcal{I}_{F}\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]$ by Proposition
<ref> and also $\langle-1\rangle$ acts trivially
on $\mathcal{P}(k(\upsilon_{p}))$, since $-1$ is a unit for all $p\in\Pi$. By Lemma <ref>, it follows that

\displaystyle\bigoplus_{p\in\Pi}\left(\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}\right)_{\chi}$.

Hence we only need to consider $\chi\in
\widehat{F^{\times}/(F^{\times})^{2}}$ such that $\chi\not=\chi_{0}$ and

Let $F=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic number field with $m>0$ and
$m\not=1$. Also let $\mathcal{O}_{F}$ be the ring of integers of $F$ then
$\langle u\rangle$ acts trivially on
$H_{3}\left(\mathrm{SL}_{2}(F),\mathbb{Z}\left[\frac{1}{2}\right]\right)$ for
all $u\in U=(\mathcal{O}_{F})^{\times}$.

$\textbf{\emph{Proof}}.$ We have the following cases:

a) If $m\not=-3$, then $U=(\mathcal{O}_{F})^{\times}=\{1,-1\}$.
Since $\langle -1\rangle$ acts trivially on
the result follows.

b) If $m=-3$ , then
$U=(\mathcal{O}_{F})^{\times}=\{\pm1,\pm\omega,\pm\omega^{2}\}$ where
$\omega^{3}=1$. It follows that $\omega=\omega^{4}$ and therefore
$\langle\omega\rangle=\langle1\rangle$. Clearly we have that
$\langle\omega^{2}\rangle=\langle1\rangle$ and therefore we get our

Now let $F=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic number field with
$m\not=1$ and let $\chi\in \widehat{F^{\times}/(F^{\times})^{2}}$. If
$\chi(u)=-1$ for some $u\in (\mathcal{O}_{F})^{\times}$, then by Lemma <ref> and Lemma <ref>, we get $\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi}=0$. Furthermore, by Lemma <ref> we also deduce $\left(\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}\right)_{\chi}=0$.

If the class number of $F$ is $1$ then each discrete valuation $\upsilon$ is associated to a prime $p$ and

(In fact, if $F$ has odd class number this latter isomorphism still holds, if for each prime ideal $\mathcal{P}$
we choose $\mathcal{P}^{m}=<p>$ for some odd $m$.)

It follows that

$\widehat{\frac{F^{\times}}{(F^{\times})^{2}(\mathcal{O}_{F})^{\times}}}\cong\widehat{\displaystyle\bigoplus_{p\in\Pi}p^{\mathbb{Z}/2}}=\mathrm{Hom}_{\mathbb{Z}}\left(\displaystyle\bigoplus_{p\in\Pi}p^{\mathbb{Z}/2}, \mu_{2}\right)=\mathrm{Hom}_{\mathbf{Sets}}(\Pi,\mu_{2})$.

$\widehat{\frac{F^{\times}}{(F^{\times})^{2}(\mathcal{O}_{F})^{\times}}}$ is
naturally parameterized by the subsets of the set $\Pi$: If $S\in\Pi$ then
the corresponding character $\chi_{S}$ is defined by

\begin{array}{ll}
-1, & \hbox{$p\in S$;} \\
1, & \hbox{$p\not\in S$.}
\end{array}
\right.

for all $p\in \Pi$ or, equivalently, for $x\in F^{\times}$, letting $\mathcal{L}(x):=\displaystyle\sum_{p\in S}\upsilon_{p}(x)$

for all $x\in F^{\times}$. Conversely, the character $\chi$ is equal to $\chi_{S}$ where

$S:=\mathrm{Supp}(\chi):=\{p\in \Pi \;|\; \chi(p)=-1\}$.

Thus, for a prime element $p$ in $\mathcal{O}_{F}$, $\chi_{p}$ is the unique
character satisfying $\mathrm{Supp}(\chi_{p})=\{p\}$.

We consider three kinds of characters:

* $\chi=\chi_{0}$ ($\mathrm{Supp}(\chi)=\emptyset$).

* $\chi=\chi_{p}$ for some $p\in \mathcal{O}_{F}$; i.e.

* $|\mathrm{Supp}(\chi)|\geq2$.

The following lemmas are immediate from the definition of the
$R_{F}$-structure on $\mathcal{P}(k(\upsilon_{p}))\{p\}$.

Let $\chi\in\widehat{F^{\times}/(F^{\times})^{2}}$. Let $p$ be a prime element. Then

\begin{array}{ll}
\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}, & \hbox{$\chi=\chi_{p}$;} \\
0, & \hbox{$otherwise$.}
\end{array}
\right.

$\textbf{\emph{Proof}}.$ If $\chi\not=\chi_{p}$, then there exist $\langle
a\rangle\in F^{\times}/(F^{\times})^{2}$ such that $\chi(\langle
a\rangle)\not=\chi_{p}(\langle a\rangle)$; i.e. $\chi(\langle
a\rangle)=-\chi_{p}(\langle a\rangle)$. Therefore, the element $\langle
a\rangle$ acts in
as multiplication by $\chi(\langle a\rangle)$ and multiplication by
$\chi_{p}(\langle a\rangle)=-\chi(\langle a\rangle)$ . By Lemma <ref> we get

On other hand, when $\chi=\chi_{p}$, we have $\mathcal{P}(k(\upsilon_{p}))\{p\}=\mathcal{R}_{\chi_{p}}\otimes\mathcal{P}(k(\upsilon_{p}))= \mathcal{P}(k(\upsilon_{p}))_{\chi_{p}}$  by definition.

For $\chi\in\widehat{F^{\times}/(F^{\times})^{2}}$, we get

\begin{array}{ll}
\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}, & \hbox{$\chi=\chi_{p}\; for\; some\; prime\; element \; p$;} \\
0, & \hbox{$otherwise$.}
\end{array}
\right.

The following lemmas will allow us to prove that $\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi}=0$ when $|\mathrm{Supp}(\chi)|\geq2$
(see Proposition <ref>)

Let $F=\mathbb{Q}(\sqrt{m})$ be a quadratic number field, let $\alpha\not\sim\beta$ be prime elements of
$\mathcal{O}_{F}=\mathbb{Z}[\omega_{m}]$. Also let
$x\in\{\alpha, \alpha^{-1}\}$, $y\in\{\omega_{m}\alpha,
(\omega_{m}\alpha)^{-1}\}$, $z\in\{\beta,\beta^{-1}\}$ and
$w\in\{\omega_{m}\beta,(\omega_{m}\beta)^{-1}\}$. Then:

$\textbf{\emph{Proof}}.$ Note by Lemma <ref> and
Remark <ref>, $\ell\mathbb{Z}[\ell]\subseteq
\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$  for all non zero
$\ell\in \mathbb{Z}[\omega]$. So it is enough to prove the result for the
case when $x=\alpha$, $y=\omega_{m}\alpha$, $z=\beta$ and
$w=\omega_{m}\beta$. We have the following cases:

a)  $\alpha=p\in \mathbb{Z}$ and $\beta=q\in \mathbb{Z}$; i.e.
$\alpha$ and $\beta$ are primes in $\mathbb{Z}$ that inert in

Note that $p\mathbb{Z}[p]=p\mathbb{Z}$. Thus

\begin{eqnarray}
\nonumber p\mathbb{Z}[p]+ q\mathbb{Z}[q] &=&p\mathbb{Z}+q\mathbb{Z}\\
\nonumber &=&\mathbb{Z}.
\end{eqnarray}

Also note that
from Lemma <ref>. Therefore

\begin{eqnarray}
\nonumber \omega_{m}p\mathbb{Z}[\omega_{m}p]+ \omega_{m}q\mathbb{Z}[\omega_{m}q]&=&p^{2}N(\omega_{m})\mathbb{Z}+p\omega_{m}\mathbb{Z}+q^{2}N(\omega_{m})\mathbb{Z}+q\omega_{m}\mathbb{Z}\\
\nonumber &\supset& p\omega_{m}\mathbb{Z}+ q\omega_{m}\mathbb{Z}\\
\nonumber &=&\omega_{m}\mathbb{Z}
\end{eqnarray}

\begin{eqnarray}
\nonumber p\mathbb{Z}[p]+ q\mathbb{Z}[q]+\omega_{m}p\mathbb{Z}[\omega_{m}p]+ \omega_{m}q\mathbb{Z}[\omega_{m}q] &\supset& \mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

b) $N(\alpha)$ is a prime number and $\beta=p\in \mathbb{Z}$; i.e.
$\beta$ is a prime in $\mathbb{Z}$ that inert in

Let $\alpha=R(\alpha)+\omega_{m}I(\alpha)$. Note that

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]&=&N(\alpha)\mathbb{Z}+\alpha\mathbb{Z}\\
\nonumber &=&\alpha(\overline{\alpha}\mathbb{Z}+\mathbb{Z})\\
\nonumber &=&\alpha(\mathbb{Z}+I(\overline{\alpha})\omega_{m}\mathbb{Z})\\
\nonumber &\supset&\alpha\mathbb{Z}.
\end{eqnarray}

Changing $\alpha$ by $\omega_{m}\alpha$, we get that

\begin{eqnarray}
\nonumber \omega_{m}\alpha\mathbb{Z}[\omega_{m}\alpha]&=&N(\omega_{m})N(\overline{\alpha})\mathbb{Z}+\alpha\omega_{m}\mathbb{Z}\\
\nonumber &=&\alpha(N(\omega_{m})\overline{\alpha}\mathbb{Z}+\omega_{m}\mathbb{Z})\\
\nonumber &=&\alpha(\mathrm{R}(\overline{\alpha})N(\omega_{m})\mathbb{Z}+I(\overline{\alpha})N(\omega_{m})\omega_{m}\mathbb{Z}+\omega_{m}\mathbb{Z})\\
\nonumber &\supset&\alpha\omega_{m}\mathbb{Z}. %\;\;\;\; \mbox{(Since $\omega_{m}\mathbb{Z}\subset I(\overline{\alpha})N(\omega_{m})\omega_{m}\mathbb{Z}$)}
\end{eqnarray}

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\omega_{m}\alpha\mathbb{Z}[\omega_{m}\alpha]&\supset&\alpha\mathbb{Z}+\alpha\omega_{m}\mathbb{Z}\\
\nonumber &=&\alpha\mathcal{O}_{F}
\end{eqnarray}

Since $p\mathbb{Z}[p]=p\mathbb{Z}$. We get that

\begin{eqnarray}
\nonumber p\mathbb{Z}[p]+ \omega_{m}p\mathbb{Z}[\omega_{m}p]&=&p\mathbb{Z}+ p^{2}N(\omega_{m})\mathbb{Z}+\omega_{m}p\mathbb{Z}\\
\nonumber &\supset&p\mathbb{Z}+\omega_{m}p\mathbb{Z}\\
\nonumber &=&p(\mathbb{Z}+\omega_{m}\mathbb{Z})\\
\nonumber &=&p\mathcal{O}_{F}.
\end{eqnarray}

\begin{eqnarray}
\nonumber p\mathbb{Z}[p]+ \omega_{m}p\mathbb{Z}[\omega_{m}p]+\alpha\mathbb{Z}[\alpha]+\omega_{m}\alpha\mathbb{Z}[\omega_{m}\alpha]&\supset&p\mathcal{O}_{F}+\alpha\mathcal{O}_{F}\\
\nonumber &=&\langle p, \alpha\rangle\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

c) $N(\alpha), N(\beta)$ are prime integers.

From case $b)$ we get that:

\alpha\mathcal{O}_{F}$

\beta\mathcal{O}_{F}$

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\omega_{m}\alpha\mathbb{Z}[\omega_{m}\alpha]+ \beta\mathbb{Z}[\beta]+\omega_{m}\beta\mathbb{Z}[\omega_{m}\beta]&\supset&\alpha\mathcal{O}_{F}+\beta\mathcal{O}_{F}\\
\nonumber &=&\langle\alpha,\beta\rangle\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

Let $F$ be an imaginary quadratic number field, let $\alpha\not\sim\beta$ be prime elements of
$\mathcal{O}_{F}$. Also let
$x\in\{\alpha,\alpha^{-1}\}$, $y\in\{\beta,\beta^{-1}\}$ and
$z\in\{\omega_{m},\omega_{m}^{-1}\}$. Then:

$\textbf{\emph{Proof}}.$ Similarly to Lemma <ref>,
by Lemma <ref> and Remark
<ref>, $\ell\mathbb{Z}[\ell]\subseteq
\frac{1}{\ell}\mathbb{Z}\left[\frac{1}{\ell}\right]$  for all non zero
$\ell\in \mathbb{Z}[\omega]$. So it is enough to prove the result for the
case when $x=\alpha$, $y=\beta$ and $z=\omega_{m}$. We have the following

a)  $\alpha=p\in \mathbb{Z}$ and $\beta=q\in \mathbb{Z}$; i.e.
$\alpha$ and $\beta$ are primes in $\mathbb{Z}$ that inert in

Note that $p\mathbb{Z}[p]=p\mathbb{Z}$. It follows that
$p\mathbb{Z}[p]+q\mathbb{Z}[q]=p\mathbb{Z}+q\mathbb{Z}=\mathbb{Z}$. Hence

\begin{eqnarray}
\nonumber p\mathbb{Z}[p]+q\mathbb{Z}[q]+ \omega_{m}\mathbb{Z}[\omega_{m}]&=&\mathbb{Z}+N(\omega_{m})\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &\supset&\mathbb{Z}+ \omega_{m}\mathbb{Z}\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

b) $N(\alpha)$ is a prime number and $\beta=p\in \mathbb{Z}$; i.e.
$\beta$ is a prime in $\mathbb{Z}$ that inert in

Let $\alpha=R(\alpha)+\omega_{m}I(\alpha)$. Since $\alpha\not\sim
\beta=p$, it follows that $N(\alpha)\not=p$ and thus
$N(\alpha)\mathbb{Z}+p\mathbb{Z}=\mathbb{Z}$. Therefore

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\beta\mathbb{Z}[\beta]+\omega_{m}\mathbb{Z}[\omega_{m}]&=&N(\alpha)\mathbb{Z}+\alpha\mathbb{Z}+p\mathbb{Z}+N(\omega_{m})\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &=&\left(N(\alpha)\mathbb{Z}+p\mathbb{Z}\right)+N(\omega_{m})\mathbb{Z}+\alpha\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &=&\mathbb{Z}+N(\omega_{m})\mathbb{Z}+\alpha\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &\supset&\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

c) $N(\alpha), N(\beta)$ are prime integers.

We will consider the following cases:

1) $N(\alpha)\not=N(\beta)$.

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\beta\mathbb{Z}[\beta]&=& N(\alpha)\mathbb{Z}+\alpha
\mathbb{Z}+N(\beta)\mathbb{Z}+\beta\mathbb{Z}\\
\nonumber &=&\mathbb{Z}+ \alpha
\mathbb{Z}+\beta\mathbb{Z}\\
\nonumber &\supset&\mathbb{Z}
\end{eqnarray}

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\beta\mathbb{Z}[\beta]+ \omega_{m}\mathbb{Z}[\omega_{m}]&\supset&\mathbb{Z}+ N(\omega_{m})\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &\supset&\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

2) $N(\alpha)=N(\beta)=p$.

Then for $\alpha=R(\alpha)+I(\alpha)\omega_{m}$, we have
$\beta=\overline{\alpha}$. We consider the following subcases:

a) $m\;\equiv\;1\;(\mathrm{mod}\;4)$.

Then $\beta=(R(\alpha)+I(\alpha))-I(\alpha)\omega_{m}$. Thus

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\beta\mathbb{Z}[\beta]&=&N(\alpha)\mathbb{Z}+\alpha\mathbb{Z}+N(\beta)\mathbb{Z}+\beta\mathbb{Z}\\
\nonumber &=& N(\alpha)\mathbb{Z}+\alpha\mathbb{Z}+\beta\mathbb{Z}  \;\;\;\mbox{(Since $N(\alpha)=N(\beta)$)}\\
\nonumber &=&N(\alpha)\mathbb{Z}+ R(\alpha)\mathbb{Z}+ I(\alpha)\mathbb{Z}+ I(\alpha)\omega_{m}\mathbb{Z}\\
\nonumber &\supset&\mathbb{Z} + I(\alpha)\omega_{m}\mathbb{Z}\;\;\; \mbox{(Since $p\not|R(\alpha)$ or $p\not|I(\alpha)$ )}\\
\nonumber &\supset&\mathbb{Z}.
\end{eqnarray}

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\beta\mathbb{Z}[\beta]+ \omega_{m}\mathbb{Z}[\omega_{m}]&\supset&\mathbb{Z}+ N(\omega_{m})\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &\supset&\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

b) $m\;\equiv\;2,3\;(\mathrm{mod}\;4)$.

Then $\beta=R(\alpha)-I(\alpha)\omega_{m}$. Hence

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\beta\mathbb{Z}[\beta]&=&N(\alpha)\mathbb{Z}+\alpha\mathbb{Z}+N(\beta)\mathbb{Z}+\beta\mathbb{Z}\\
\nonumber &=& N(\alpha)\mathbb{Z}+\alpha\mathbb{Z}+\beta\mathbb{Z}  \;\;\;\mbox{(Since $N(\alpha)=N(\beta)$)}\\
\nonumber &=&N(\alpha)\mathbb{Z}+ R(\alpha)\mathbb{Z}+ I(\alpha)\omega_{m}\mathbb{Z}.
\end{eqnarray}

Note that either $N(\alpha)$ and $R(\alpha)$ are coprimes or $N(\alpha)$ and $N(\omega_{m})$ are coprimes otherwise
by Lemma <ref>, we get $\alpha\sim\beta$. Hence it follows:

\begin{eqnarray}
\nonumber \alpha\mathbb{Z}[\alpha]+\beta\mathbb{Z}[\beta]+ \omega_{m}\mathbb{Z}[\omega_{m}]&=&N(\alpha)\mathbb{Z}+ R(\alpha)\mathbb{Z}+ I(\alpha)\omega_{m}\mathbb{Z} + N(\omega_{m})\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &\supset&\mathbb{Z}+\omega_{m}\mathbb{Z}\\
\nonumber &=&\mathcal{O}_{F}.
\end{eqnarray}

Let $F$ be an imaginary quadratic number field of class number $1$
and let $\chi\in\widehat{\frac{F^{\times}}{(F^{\times})^{2}(\mathcal{O}_{F})^{\times}}}$.
Suppose that $|\mathrm{Supp}(\chi)|\geq2$. Then

in $\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]_{\chi}$
for all $a\in F$  and $ t\in\mathcal{O}_{F}$.

$\textbf{\emph{Proof}}.$ Let $p,q\in \mathrm{Supp}(\chi)$. We consider the
following cases:

a) $\chi(\omega_{m})=1$.

Then $\chi(\omega_{m}p)=\chi(\omega_{m}q)=-1$. Therefore, depending of the values of $\chi(1-p)$, $\chi(1-q)$, $\chi(1-\omega_{m}p)$ and
$\chi(1-\omega_{m}q)$, by Remark <ref>, there is a choice
of $\epsilon_{p},\epsilon_{q},\epsilon'_{p},\epsilon'_{q}\in \{\pm 1\}$
such that $\ell=p^{\epsilon_{p}}, q^{\epsilon_{q}},
(\omega_{m}p)^{\epsilon'_{p}},(\omega_{m}q)^{\epsilon'_{q}}$ satisfy
$\chi(\ell)=-1$ and $\chi(1-\ell)=1$. Therefore the result follows from
the Proposition <ref> and the Lemma

b) $\chi(\omega_{m})=-1$.

Then depending of the values of $\chi(1-p)$, $\chi(1-q)$ and
$\chi(1-\omega_{m})$, by the Remark <ref> there is a
choice of $\epsilon_{p},\epsilon_{q},\epsilon_{\omega_{m}}\in \{\pm 1\}$ such that
$\ell=p^{\epsilon_{p}}, q^{\epsilon_{q}},
(\omega_{m})^{\epsilon_{\omega_{m}}}$ satisfy $\chi(\ell)=-1$ and
$\chi(1-\ell)=1$. Therefore from the Proposition
<ref> and the Lemma
<ref>, we get our result.

Let $F=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic number field with $m=1,2,3,7,11$ and
Suppose that $|\mathrm{Supp}(\chi)|\geq2$. Then

$\textbf{\emph{Proof}}.$ First, let us recall, for these fields, the ring of algebraic integer $\mathcal{O}_{F}$ is a Euclidean domain with respect the norm of $F$.

From the Proposition <ref>, we get
$[x]_{\chi}=[x+a]_{\chi}$  for all $x\in F$ and $a\in \mathcal{O}_{F}$. Then
it follows that $[a]_{\chi}=[1]_{\chi}=0$ for all $a\in \mathcal{O}_{F}$ and
therefore $\left[\frac{1}{a}\right]_{\chi}=0$ for all $a\in \mathcal{O}_{F}$.

Now for $x\in F$ we have that $x=\frac{a}{b}$ with $a,b \in \mathcal{O}_{F}$, we will prove that $[x]_{\chi}=0$ by induction on $\mathrm{min}(N(a),N(b))=m$. If $m=1$, then $a\in \mathcal{O}^{\times}_{F}$ or $b\in \mathcal{O}^{\times}_{F}$. Thus either $x\in \mathcal{O}_{F}$ or $x^{-1}\in \mathcal{O}_{F}$. Therefore, it follows from the previous paragraph that $[x]_{\chi}=0$.

Since $[x]_{\chi}=-\left[\frac{1}{x}\right]_{\chi}$, we can suppose that
$N(b)<N(a)$. Now we suppose the results holds if norm $m=N(b)\leq n$. We will
prove it for $\mathrm{min}(N(a),N(b))=N(b)=n+1$. Note that there exist
$q,r\in \mathcal{O}_{F}$ such that $a=qb+r$ and $N(r)<N(b)$. Taking $t=q$
from Proposition <ref>, we obtain that
$[x]_{\chi}=\left[\frac{r}{b}\right]_{\chi}$. Note that
$\mathrm{min}(N(r),N(b))=N(r)\leq n$. Hence by induction hypothesis we have
that $[x]_{\chi}=0$.

If $f:M\rightarrow N$ is a map of $R_{F}$-modules and if $f_{\chi}:M_{\chi}\rightarrow N_{\chi}$ is an isomorphism for all $\chi\in\widehat{\frac{F^{\times}}{(F^{\times})^{2}(\mathcal{O}_{F})^{\times}}}$, then the induced map

$H_{0}\left(\mathcal{O}^{\times}_{F}, M\left[\frac{1}{2}\right]\right)\rightarrow H_{0}\left(\mathcal{O}^{\times}_{F}, N\left[\frac{1}{2}\right]\right)$

is an isomorphism.

$\textbf{\emph{Proof}}.$ Let $\chi\in\widehat{\frac{F^{\times}}{(F^{\times})^{2}}}$. If $\chi\in \widehat{\frac{F^{\times}}{(F^{\times})^{2}(\mathcal{O}_{F})^{\times}}}$, then $H_{0}\left(\mathcal{O}^{\times}_{F}, M\left[\frac{1}{2}\right]\right)_{\chi}=M\left[\frac{1}{2}\right]_{\chi}\cong N\left[\frac{1}{2}\right]_{\chi} =H_{0}\left(\mathcal{O}^{\times}_{F}, N\left[\frac{1}{2}\right]\right)_{\chi}$.

Otherwise, $\chi(u)=-1$ for some $u\in \mathcal{O}^{\times}$. But the element $\langle u\rangle$ acts trivially on $H_{0}\left(\mathcal{O}^{\times}_{F}, M\left[\frac{1}{2}\right]\right)$ and $H_{0}\left(\mathcal{O}^{\times}_{F}, N\left[\frac{1}{2}\right]\right)$. By Lemma <ref>

$H_{0}\left(\mathcal{O}^{\times}_{F}, M\left[\frac{1}{2}\right]\right)_{\chi}=0=H_{0}\left(\mathcal{O}^{\times}_{F}, N\left[\frac{1}{2}\right]\right)_{\chi}$.

The result follows by the local-global principle for characters (Proposition <ref>).

Let $F=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic number field with $m=1, 2, 3, 7, 11$ , then the map

$\overline{S}:H_{0}\left(\mathcal{O}_{F}^{\times}, H_{3}\left(\mathrm{SL}_{2}(F),\mathbb{Z}\left[\frac{1}{2}\right]\right)_{0}\right)\cong H_{0}\left(\mathcal{O}_{F}^{\times},\mathcal{I}_{F}\mathcal{RP}_{+}(F)\left[\frac{1}{2}\right]\right)\rightarrow
\displaystyle\bigoplus_{p\in\Pi}\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}$

is an isomorphism of $R_{F}$-modules.

$\textbf{\emph{Proof}}.$ By the local-global principle for characters
(Proposition <ref>) and by
Corollary <ref>, we must show

\begin{array}{ll}
\mathcal{P}(k(\upsilon_{p}))\left[\frac{1}{2}\right]\{p\}, & \hbox{$if \; \chi=\chi_{p}$;} \\
0, & \hbox{$if\; \chi\not=\chi_{p}\; for\; any\; p$.}
\end{array}
\right.

If $\chi=\chi_{p}$, this is from Proposition <ref>.
If $\chi=\chi_{0}$, or $\chi(u)=-1$ for some $u\in(\mathcal{O}_{F})^{\times}$
this follows from above. Otherwise
and  $\mathrm{Supp}(\chi)$ contain at least two non-associated distinct prime
elements of $\mathcal{O}_{F}$ and hence from Proposition
<ref> we get our result.

From the Proposition <ref> and the definition of
we get:

Let $F=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic number field with $m=1,2,3,7,11$ , then we have a short exact sequence of $R_{F}$-modules

Let $F=\mathbb{Q}(\sqrt{-m})$ be an imaginary quadratic number field with $m=2,3,7,11$ ,then we have a short exact sequence of $R_{F}$-modules

$\textbf{\emph{Proof}}.$ By Lemma <ref>, we have that $\mathcal{O}^{\times}_{F}$ acts trivially on $H_{3}\left(\mathrm{SL}_{2}(F),\mathbb{Z}\left[\frac{1}{2}\right]\right)$.

[1]
Brown K.S., Cohomology of groups, Graduate text in Mathematics 87,
Springer Verlag, New York $(1982)$.

[2]
Charles Weibel, The $K$-book: an introduction to algebraic $K$-theory
Graduate Studies in Math. vol. $145$, AMS, $2013$

[3]
Behrooz Mirzaii.
Third homology of general linear groups.
J. Algebra, 320(5):1851–1877, 2008.

[4]
Johan L. Dupont and Chih Han Sah. Scissors congruences. II.
J. Pure Appl. Algebra, 25(2):159–195, 1982.

[5]
Chin-Han Sah. Homology of classical Lie groups made discrete, III. J. Pure Appl. Algebra,
$56(3)$:$269$-$312$, $1989$.

[6]
Hutchinson Kevin and Liqun Tao The third homology of the special linear group of a fieldJournal of Pure and Applied Algebra 213 $1665-1680$ $(2009)$.

[7]
Hutchinson Kevin. A Bloch Wigner Complex for $SL_{2}$
J.K. Theory 12 $(2013)$ $15-68$.

[8]
Hutchinson Kevin. A refined Bloch group and the third homology  of $\mathrm{SL}_{2}$ of a fieldJournal of Pure and Applied Algebra 217 $(2013)$ $2003-2035$.

[9]
Hutchinson Kevin. The third homology of $SL_{2}$ of fields with discrete valuation
Journal of Pure and Applied Algebra 221(5), $(2017)$ $1076-1111$.

[10]
Hutchinson Kevin.  The third homology of $SL_{2}(\mathbb{Q})$, Journal of Algebra, Volume 570, 2021, Pages 366-396, ISSN 0021-8693, https://doi.org/10.1016/j.jalgebra.2020.12.006.

[11]
Suslin, A. A.: $K_{3}$ of a field and the Bloch group. Proc. Steklov Inst. Math. $183$, no $4$, $217-239$ (1991)

[14]
Christian K. Zickert, The extendedBloch group and algebraic $K$-theory, J. Reine Angew. Math. $704$, $21-54$ (2015)


* Garrison-Kimmel et al. (2013) Garrison-Kimmel, S.; Rocha, M.; Boylan-Kolchin, M.; Bullock, J.S.; Lally, J.  Can feedback solve the too-big-to-fail problem?  Mon. Not. Roy. Astro. Soc. 2013, 433, 3539–3546, [arXiv:astro-ph.CO/1301.3137].
  * Wadsley et al. (2004) Wadsley, J.W.; Stadel, J.; Quinn, T.  Gasoline: a flexible, parallel implementation of TreeSPH.  New Astron. 2004, 9, 137–158, [astro-ph/0303521].
  * Katz and White (1993) Katz, N.; White, S.D.M.  Hierarchical galaxy formation - Overmerging and the formation of an X-ray cluster.  Astrophys. J. 1993, 412, 455–478.
  * Stinson et al. (2006) Stinson, G.; Seth, A.; Katz, N.; Wadsley, J.; Governato, F.; Quinn, T.  Star formation and feedback in smoothed particle hydrodynamic simulations - I. Isolated galaxies.  Mon. Not. Roy. Astro. Soc. 2006, 373, 1074–1090, [astro-ph/0602350].
  * Stinson et al. (2013) Stinson, G.S.; Brook, C.; Macciò, A.V.; Wadsley, J.; Quinn, T.R.; Couchman, H.M.P.  Making Galaxies In a Cosmological Context: the need for early stellar feedback.  Mon. Not. Roy. Astro. Soc. 2013, 428, 129–140, [1208.0002].
  * Ceverino and Klypin (2009) Ceverino, D.; Klypin, A.  The Role of Stellar Feedback in the Formation of Galaxies.  Astrophys. J. 2009, 695, 292–309, [0712.3285].
  * Pontzen and Governato (2012) Pontzen, A.; Governato, F.  How supernova feedback turns dark matter cusps into cores.  Mon. Not. Roy. Astro. Soc. 2012, 421, 3464–3471, [1106.0499].
  * Oñorbe et al. (2015) Oñorbe, J.; Boylan-Kolchin, M.; Bullock, J.S.; Hopkins, P.F.; Kereš, D.; Faucher-Giguère, C.A.; Quataert, E.; Murray, N.  Forged in FIRE: cusps, cores and baryons in low-mass dwarf galaxies.  Mon. Not. Roy. Astro. Soc. 2015, 454, 2092–2106, [1502.02036].
  * González-Samaniego et al. (2014) González-Samaniego, A.; Colín, P.; Avila-Reese, V.; Rodríguez-Puebla, A.; Valenzuela, O.  Simulations of Isolated Dwarf Galaxies Formed in Dark Matter Halos with Different Mass Assembly Histories.  Astrophys. J. 2014, 785, 58, [1308.4753].
  * Schaller et al. (2015) Schaller, M.; Frenk, C.S.; Bower, R.G.; Theuns, T.; Jenkins, A.; Schaye, J.; Crain, R.A.; Furlong, M.; Dalla Vecchia, C.; McCarthy, I.G.  Baryon effects on the internal structure of $\Lambda$CDM haloes in the EAGLE simulations.  Mon. Not. Roy. Astro. Soc. 2015, 451, 1247–1267, [1409.8617].
  * Pontzen and Governato (2011) Pontzen, A.; Governato, F.  Flattening dark matter cusps with supernova feedback: a physical model.  Galaxy Formation, 2011, p. 12.
  * Peñarrubia et al. (2012) Peñarrubia, J.; Pontzen, A.; Walker, M.G.; Koposov, S.E.  The Coupling between the Core/Cusp and Missing Satellite Problems.  Astrophys. J. Lett. 2012, 759, L42, [1207.2772].
  * Koposov et al. (2009) Koposov, S.E.; Yoo, J.; Rix, H.W.; Weinberg, D.H.; Macciò, A.V.; Escudé, J.M.  A Quantitative Explanation of the Observed Population of Milky Way Satellite Galaxies.  Astrophys. J. 2009, 696, 2179–2194, [0901.2116].
  * Kravtsov (2010) Kravtsov, A.  Dark Matter Substructure and Dwarf Galactic Satellites.  Advances in Astronomy 2010, 2010, 281913, [0906.3295].
  * Maxwell et al. (2015) Maxwell, A.J.; Wadsley, J.; Couchman, H.M.P.  The Energetics of Cusp Destruction.  Astrophys. J. 2015, 806, 229, [1505.00825].
  * Behroozi et al. (2010) Behroozi, P.S.; Conroy, C.; Wechsler, R.H.  A Comprehensive Analysis of Uncertainties Affecting the Stellar Mass-Halo Mass Relation for 0 $<$ z $<$ 4\.  Astrophys. J. 2010, 717, 379–403, [arXiv:astro-ph.CO/1001.0015].
  * Moster et al. (2010) Moster, B.P.; Somerville, R.S.; Maulbetsch, C.; van den Bosch, F.C.; Macciò, A.V.; Naab, T.; Oser, L.  Constraints on the Relationship between Stellar Mass and Halo Mass at Low and High Redshift.  Astrophys. J. 2010, 710, 903–923, [arXiv:astro-ph.CO/0903.4682].
  * Revaz and Jablonka (2012) Revaz, Y.; Jablonka, P.  The dynamical and chemical evolution of dwarf spheroidal galaxies with GEAR.  Astron. Astrophys. 2012, 538, A82, [1109.0989].
  * (234) Sawala, T.; Frenk, C.S.; Fattahi, A.; Navarro, J.F.; Bower, R.G.; Crain, R.A.; Dalla Vecchia, C.; Furlong, M.; Helly, J.C.; Jenkins, A.; Oman, K.A.; Schaller, M.; Schaye, J.; Theuns, T.; Trayford, J.; White, S.D.M.  Local Group galaxies emerge from the dark.  ArXiv e-prints.
  * Ferrero et al. (2012) Ferrero, I.; Abadi, M.G.; Navarro, J.F.; Sales, L.V.; Gurovich, S.  The dark matter haloes of dwarf galaxies: a challenge for the $\Lambda$ cold dark matter paradigm?  Mon. Not. Roy. Astro. Soc. 2012, 425, 2817–2823, [1111.6609].
  * Papastergis et al. (2015) Papastergis, E.; Giovanelli, R.; Haynes, M.P.; Shankar, F.  Is there a ”too big to fail” problem in the field?  Astron. Astrophys. 2015, 574, A113, [1407.4665].
  * Choi et al. (2015) Choi, E.; Ostriker, J.P.; Naab, T.; Oser, L.; Moster, B.P.  The impact of mechanical AGN feedback on the formation of massive early-type galaxies.  Mon. Not. Roy. Astro. Soc. 2015, 449, 4105–4116, [1403.1257].
  * Laporte and Peñarrubia (2015) Laporte, C.F.P.; Peñarrubia, J.  Under the sword of Damocles: plausible regeneration of dark matter cusps at the smallest galactic scales.  Mon. Not. Roy. Astro. Soc. 2015, 449, L90–L94, [1409.3848].
  * Laporte and White (2015) Laporte, C.F.P.; White, S.D.M.  The redistribution of matter in the cores of galaxy clusters.  Mon. Not. Roy. Astro. Soc. 2015, 451, 1177–1189, [1409.1924].
  * Marinacci et al. (2014) Marinacci, F.; Pakmor, R.; Springel, V.  The formation of disc galaxies in high-resolution moving-mesh cosmological simulations.  Mon. Not. Roy. Astro. Soc. 2014, 437, 1750–1775, [1305.5360].
  * Madau et al. (2014) Madau, P.; Shen, S.; Governato, F.  Dark Matter Heating and Early Core Formation in Dwarf Galaxies.  Astrophys. J. Lett. 2014, 789, L17, [1405.2577].
  * Noguchi (1998) Noguchi, M.  Clumpy star-forming regions as the origin of the peculiar morphology of high-redshift galaxies.  Nature 1998, 392, 253.
  * Noguchi (1999) Noguchi, M.  Early Evolution of Disk Galaxies: Formation of Bulges in Clumpy Young Galactic Disks.  Astrophys. J. 1999, 514, 77–95, [astro-ph/9806355].
  * Immeli et al. (2004a) Immeli, A.; Samland, M.; Westera, P.; Gerhard, O.  Subgalactic Clumps at High Redshift: A Fragmentation Origin?  Astrophys. J. 2004, 611, 20–25, [astro-ph/0406135].
  * Immeli et al. (2004b) Immeli, A.; Samland, M.; Gerhard, O.; Westera, P.  Gas physics, disk fragmentation, and bulge formation in young galaxies.  Astron. Astrophys. 2004, 413, 547–561, [astro-ph/0312139].
  * Bournaud et al. (2007) Bournaud, F.; Elmegreen, B.G.; Elmegreen, D.M.  Rapid Formation of Exponential Disks and Bulges at High Redshift from the Dynamical Evolution of Clump-Cluster and Chain Galaxies.  Astrophys. J. 2007, 670, 237–248, [0708.0306].
  * Agertz et al. (2009) Agertz, O.; Teyssier, R.; Moore, B.  Disc formation and the origin of clumpy galaxies at high redshift.  Mon. Not. Roy. Astro. Soc. 2009, 397, L64–L68, [0901.2536].
  * Aumer et al. (2010) Aumer, M.; Burkert, A.; Johansson, P.H.; Genzel, R.  The Structure of Gravitationally Unstable Gas-rich Disk Galaxies.  Astrophys. J. 2010, 719, 1230–1243, [1007.0169].
  * Ceverino et al. (2010) Ceverino, D.; Dekel, A.; Bournaud, F.  High-redshift clumpy discs and bulges in cosmological simulations.  Mon. Not. Roy. Astro. Soc. 2010, 404, 2151–2169, [0907.3271].
  * Ceverino et al. (2012) Ceverino, D.; Dekel, A.; Mandelker, N.; Bournaud, F.; Burkert, A.; Genzel, R.; Primack, J.  Rotational support of giant clumps in high-z disc galaxies.  Mon. Not. Roy. Astro. Soc. 2012, 420, 3490–3520, [1106.5587].
  * Toomre (1964) Toomre, A.  On the gravitational stability of a disk of stars.  Astrophys. J. 1964, 139, 1217–1238.
  * Krumholz and Dekel (2010) Krumholz, M.R.; Dekel, A.  Survival of star-forming giant clumps in high-redshift galaxies.  Mon. Not. Roy. Astro. Soc. 2010, 406, 112–120, [1001.0765].
  * Macciò et al. (2012) Macciò, A.V.; Stinson, G.; Brook, C.B.; Wadsley, J.; Couchman, H.M.P.; Shen, S.; Gibson, B.K.; Quinn, T.  Halo Expansion in Cosmological Hydro Simulations: Toward a Baryonic Solution of the Cusp/Core Problem in Massive Spirals.  Astrophys. J. Lett. 2012, 744, L9, [1111.5620].
  * Del Popolo and Pace (2016) Del Popolo, A.; Pace, F.  The Cusp/Core problem: supernovae feedback versus the baryonic clumps and dynamical friction model.  Astrophys. Space Sci.  2016, 361, 162, [1502.01947].
  * Del Popolo (2016) Del Popolo, A.  On the dark matter haloes inner structure and galaxy morphology.  Astrophys. Space Sci. 2016, 361, 222, [arXiv:astro-ph.GA/1607.07408].
  * Weidner et al. (2013) Weidner, C.; Kroupa, P.; Pflamm-Altenburg, J.; Vazdekis, A.  The galaxy-wide IMF of dwarf late-type to massive early-type galaxies.  Mon. Not. Roy. Astron. Soc. 2013, 436, 3309, [arXiv:astro-ph.CO/1309.6634].
  * Lazar et al. (2020) Lazar, A.; Bullock, J.S.; Boylan-Kolchin, M.; Chan, T.K.; Hopkins, P.F.; Graus, A.S.; Wetzel, A.; El-Badry, K.; Wheeler, C.; Straight, M.C.; Kereš, D.; Faucher-Giguère, C.A.; Fitts, A.; Garrison-Kimmel, S.  A dark matter profile to model diverse feedback-induced core sizes of $\Lambda$CDM haloes.  Mon. Not. Roy. Astro. Soc. 2020, 497, 2393–2417, [arXiv:astro-ph.GA/2004.10817].
  * Freundlich et al. (2020) Freundlich, J.; Jiang, F.; Dekel, A.; Cornuault, N.; Ginzburg, O.; Koskas, R.; Lapiner, S.; Dutton, A.; Macciò, A.V.  The Dekel-Zhao profile: a mass-dependent dark-matter density profile with flexible inner slope and analytic potential, velocity dispersion, and lensing properties.  Mon. Not. Roy. Astro. Soc. 2020, 499, 2912–2933, [arXiv:astro-ph.GA/2004.08395].
  * Zhao (1996) Zhao, H.  Analytical models for galactic nuclei.  Mon. Not. Roy. Astro. Soc. 1996, 278, 488–496, [astro-ph/9509122].
  * Di Cintio et al. (2014) Di Cintio, A.; Brook, C.B.; Dutton, A.A.; Macciò, A.V.; Stinson, G.S.; Knebe, A.  A mass-dependent density profile for dark matter haloes including the influence of galaxy formation.  Mon. Not. Roy. Astro. Soc. 2014, 441, 2986–2995, [1404.5959].
  * Blumenthal et al. (1986) Blumenthal, G.R.; Faber, S.M.; Flores, R.; Primack, J.R.  Contraction of dark matter galactic halos due to baryonic infall.  Astrophys. J. 1986, 301, 27–34.
  * Gnedin et al. (2004) Gnedin, O.Y.; Kravtsov, A.V.; Klypin, A.A.; Nagai, D.  Response of Dark Matter Halos to Condensation of Baryons: Cosmological Simulations and Improved Adiabatic Contraction Model.  Astrophys. J. 2004, 616, 16–26, [astro-ph/0406247].
  * Wang et al. (2015) Wang, L.; Dutton, A.A.; Stinson, G.S.; Macciò, A.V.; Penzo, C.; Kang, X.; Keller, B.W.; Wadsley, J.  NIHAO project - I. Reproducing the inefficiency of galaxy formation across cosmic time with a large sample of cosmological hydrodynamical simulations.  Mon. Not. Roy. Astro. Soc. 2015, 454, 83–94, [1503.04818].
  * Dekel et al. (2017) Dekel, A.; Ishai, G.; Dutton, A.A.; Maccio, A.V.  Dark-matter halo profiles of a general cusp/core with analytic velocity and potential.  Mon. Not. Roy. Astro. Soc. 2017, 468, 1005–1022, [1610.00916].
  * Read et al. (2016) Read, J.I.; Agertz, O.; Collins, M.L.M.  Dark matter cores all the way down.  Mon. Not. Roy. Astro. Soc. 2016, 459, 2573–2590, [arXiv:astro-ph.GA/1508.04143].
  * Dutton and Macciò (2014) Dutton, A.A.; Macciò, A.V.  Cold dark matter haloes in the Planck era: evolution of structural parameters for Einasto and NFW profiles.  Mon. Not. Roy. Astron. Soc. 2014, 441, 3359–3374, [arXiv:astro-ph.CO/1402.7073].
  * Colín et al. (2000) Colín, P.; Avila-Reese, V.; Valenzuela, O.  Substructure and Halo Density Profiles in a Warm Dark Matter Cosmology.  Astrophys. J. 2000, 542, 622–630, [astro-ph/0004115].
  * Polisensky and Ricotti (2011) Polisensky, E.; Ricotti, M.  Constraints on the Dark Matter Particle Mass from the Number of Milky Way Satellites.  Phys. Rev. 2011, D83, 043506, [arXiv:astro-ph.CO/1004.1459].
  * Lovell et al. (2012) Lovell, M.R.; Eke, V.; Frenk, C.S.; Gao, L.; Jenkins, A.; Theuns, T.; Wang, J.; White, S.D.M.; Boyarsky, A.; Ruchayskiy, O.  The haloes of bright satellite galaxies in a warm dark matter universe.  Mon. Not. Roy. Astro. Soc. 2012, 420, 2318–2324, [1104.2929].
  * Angulo et al. (2013) Angulo, R.E.; Hahn, O.; Abel, T.  The warm dark matter halo mass function below the cut-off scale.  Mon. Not. Roy. Astro. Soc. 2013, 434, 3337–3347, [1304.2406].
  * Kuzio de Naray et al. (2010) Kuzio de Naray, R.; Martinez, G.D.; Bullock, J.S.; Kaplinghat, M.  The Case Against Warm or Self-Interacting Dark Matter as Explanations for Cores in Low Surface Brightness Galaxies.  Astrophys. J. Lett. 2010, 710, L161–L166, [0912.3518].
  * Wu and Kroupa (2015) Wu, X.; Kroupa, P.  Galactic rotation curves, the baryon-to-dark-halo-mass relation and space???time scale invariance.  Mon. Not. Roy. Astron. Soc. 2015, 446, 330–344, [arXiv:astro-ph.GA/1410.2256].
  * Dalal and Kochanek (2002) Dalal, N.; Kochanek, C.S.  Direct Detection of Cold Dark Matter Substructure.  Astrophys. J. 2002, 572, 25–33, [astro-ph/0111456].
  * Fadely and Keeton (2011) Fadely, R.; Keeton, C.R.  Near-infrared K and L’ Flux Ratios in Six Lensed Quasars.  Astro. J. 2011, 141, 101, [1101.1917].
  * Fadely and Keeton (2012) Fadely, R.; Keeton, C.R.  Substructure in the lens HE 0435-1223.  Mon. Not. Roy. Astro. Soc. 2012, 419, 936–951, [1109.0548].
  * Macciò et al. (2012) Macciò, A.V.; Paduroiu, S.; Anderhalden, D.; Schneider, A.; Moore, B.  Cores in warm dark matter haloes: a Catch 22 problem.  Mon. Not. Roy. Astro. Soc. 2012, 424, 1105–1112, [1202.1282].
  * Schneider et al. (2014) Schneider, A.; Anderhalden, D.; Macciò, A.V.; Diemand, J.  Warm dark matter does not do better than cold dark matter in solving small-scale inconsistencies.  Mon. Not. Roy. Astro. Soc. 2014, 441, L6–L10, [1309.5960].
  * Narayanan et al. (2000) Narayanan, V.K.; Spergel, D.N.; Davé, R.; Ma, C.P.  Constraints on the Mass of Warm Dark Matter Particles and the Shape of the Linear Power Spectrum from the Ly$\alpha$ Forest.  Astrophys. J. Lett. 2000, 543, L103–L106, [astro-ph/0005095].
  * Burkert (2000) Burkert, A.  The Structure and Evolution of Weakly Self-interacting Cold Dark Matter Halos.  Astrophys. J. Lett. 2000, 534, L143–L146, [astro-ph/0002409].
  * Newman et al. (2013) Newman, A.B.; Treu, T.; Ellis, R.S.; Sand, D.J.; Nipoti, C.; Richard, J.; Jullo, E.  The Density Profiles of Massive, Relaxed Galaxy Clusters. I. The Total Density Over Three Decades in Radius.  Astrophys. J. 2013, 765, 24, [1209.1391].
  * Rocha et al. (2013) Rocha, M.; Peter, A.H.G.; Bullock, J.S.; Kaplinghat, M.; Garrison-Kimmel, S.; Oñorbe, J.; Moustakas, L.A.  Cosmological simulations with self-interacting dark matter - I. Constant-density cores and substructure.  Mon. Not. Roy. Astro. Soc. 2013, 430, 81–104, [1208.3025].
  * Peter et al. (2013) Peter, A.H.G.; Rocha, M.; Bullock, J.S.; Kaplinghat, M.  Cosmological simulations with self-interacting dark matter - II. Halo shapes versus observations.  Mon. Not. Roy. Astro. Soc. 2013, 430, 105–120, [1208.3026].
  * Clowe et al. (2006) Clowe, D.; Bradač, M.; Gonzalez, A.H.; Markevitch, M.; Randall, S.W.; Jones, C.; Zaritsky, D.  A Direct Empirical Proof of the Existence of Dark Matter.  Astrophys. J. Lett. 2006, 648, L109–L113, [astro-ph/0608407].
  * Randall et al. (2008) Randall, S.W.; Markevitch, M.; Clowe, D.; Gonzalez, A.H.; Bradač, M.  Constraints on the Self-Interaction Cross Section of Dark Matter from Numerical Simulations of the Merging Galaxy Cluster 1E 0657-56.  Astrophys. J. 2008, 679, 1173–1180, [0704.0261].
  * Dawson et al. (2012) Dawson, W.A.; Wittman, D.; Jee, M.J.; Gee, P.; Hughes, J.P.; Tyson, J.A.; Schmidt, S.; Thorman, P.; Bradač, M.; Miyazaki, S.; Lemaux, B.; Utsumi, Y.; Margoniner, V.E.  Discovery of a Dissociative Galaxy Cluster Merger with Large Physical Separation.  Astrophys. J. Lett. 2012, 747, L42, [arXiv:astro-ph.CO/1110.4391].
  * Yoshida et al. (2000) Yoshida, N.; Springel, V.; White, S.D.M.; Tormen, G.  Weakly Self-interacting Dark Matter and the Structure of Dark Halos.  Astrophys. J. Lett. 2000, 544, L87–L90, [astro-ph/0006134].
  * Davé et al. (2001) Davé, R.; Spergel, D.N.; Steinhardt, P.J.; Wandelt, B.D.  Halo Properties in Cosmological Simulations of Self-interacting Cold Dark Matter.  Astrophys. J. 2001, 547, 574–589, [astro-ph/0006218].
  * Harko (2011) Harko, T.  Bose-Einstein condensation of dark matter solves the core/cusp problem.  J. Cosmo. Astrop. Phys 2011, 5, 022, [arXiv:astro-ph.CO/1105.2996].
  * Schive et al. (2014) Schive, H.Y.; Chiueh, T.; Broadhurst, T.  Cosmic structure as the quantum interference of a coherent dark wave.  Nature Physics 2014, 10, 496–499, [1406.6586].
  * Robles and Matos (2012) Robles, V.H.; Matos, T.  Flat central density profile and constant dark matter surface density in galaxies from scalar field dark matter.  Mon. Not. Roy. Astro. Soc. 2012, 422, 282–289, [arXiv:astro-ph.CO/1201.3032].
  * Öpik (1915) Öpik, E.  Selective absorption of light in space, and the dynamics of the Universe.  Bull. de la Soc. Astr. de Russie 1915, 21, 150.
  * Strigari et al. (2014) Strigari, L.E.; Frenk, C.S.; White, S.D.M.  Dynamical models for the Sculptor dwarf spheroidal in a Lambda CDM universe.  ArXiv e-prints 2014, [1406.6079].
  * Takada (2010) Takada, M.  Subaru Hyper Suprime-Cam Project.  American Institute of Physics Conference Series; Kawai, N.; Nagataki, S., Eds., 2010, Vol. 1279, American Institute of Physics Conference Series, pp. 120–127.
  * de Bruijne (2012) de Bruijne, J.H.J.  Science performance of Gaia, ESA’s space-astrometry mission.  Astrophys. Space Sci.  2012, 341, 31–41, [arXiv:astro-ph.IM/1201.3238].
  * Richardson et al. (2014) Richardson, T.D.; Spolyar, D.; Lehnert, M.D.  Plan $\beta$: core or cusp?  Mon. Not. Roy. Astro. Soc. 2014, 440, 1680–1689, [arXiv:astro-ph.GA/1311.1522].
  * Strigari et al. (2007) Strigari, L.E.; Bullock, J.S.; Kaplinghat, M.; Diemand, J.; Kuhlen, M.; Madau, P.  Redefining the Missing Satellites Problem.  Astrophys. J. 2007, 669, 676–683, [0704.1817].
  * Battaglia et al. (2013) Battaglia, G.; Helmi, A.; Breddels, M.  Internal kinematics and dynamical models of dwarf spheroidal galaxies around the Milky Way.  New Astron. Rev. 2013, 57, 52–79, [1305.5965].
  * Metcalf and Amara (2012) Metcalf, R.B.; Amara, A.  Small-scale structures of dark matter and flux anomalies in quasar gravitational lenses.  Mon. Not. Roy. Astro. Soc. 2012, 419, 3414–3425, [1007.1599].
  * Carlberg and Grillmair (2013) Carlberg, R.G.; Grillmair, C.J.  Gaps in the GD-1 Star Stream.  Astrophys. J. 2013, 768, 171, [1303.4342].