The core of the global warming theory proposed by the IPCC is that man-made carbon dioxide (and to a lesser extent other gases) leads to warming of the atmosphere, via a so-called "radiative forcing" effect. So you might think that justifying and quantifying this effect would occupy a major part of the AR4 WG1 report.  But it is very difficult to find any details on this in AR4. 

The SPM describes at some length the increase in carbon dioxide and other gases, but only briefly states that the net effect of all the man-made influences is a radiative forcing (RF) of 1.6 Watts/m^2. (Recall that this number is wrong, about 0.3 too large, because of a simple arithmetic error that the IPCC does not acknowledge). The dominant contribution to this is apparently from carbon dioxide, with an effect also of around 1.6, according to Figure SPM2. 

Searching for support for these numbers, we turn to chapter 2 of the main WG1 report. Section 2.3.1 discusses atmospheric carbon dioxide. Here there is again more discussion of how carbon dioxide has increased, from 0.028% to a massive 0.038%. But on the crucial question of how this leads to an RF, there is only this on page 140:

"The simple formulae for RF of the LLGHG quoted in Ramaswamy et al. (2001) are still valid. These formulae are based on global RF calculations where clouds, stratospheric adjustment and solar absorption are included, and give an RF of +3.7 W m–2 for a doubling in the CO2 mixing ratio. (The formula used for the CO2 RF calculation in this chapter is the IPCC (1990) expression as revised in the TAR. Note that for CO2, RF increases logarithmically with mixing ratio.)"

Note that Ramaswamy et al. (2001) is not a scientific paper - it is the previous IPCC report (the TAR)!  Astonishingly, all the IPCC can come up with to justify the key step in its argument is the blunt statement that its previous formulas are valid. The following sentence mentions a paper by Collins et al (2006), "Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the IPCC AR4".  This paper gives estimated RF values at the Earth's surface of 0.38 - 0.57 for present carbon dioxide levels, or 1.12 - 1.64 for a doubling of CO2 (obtained by computer models).  Note that these values are far lower than those claimed by the IPCC.  A key feature of these computations is that the RF depends very strongly on height - for example the value 0.38 increases to 1.82 at the tropopause.  The IPCC AR4 chapter 2 does not mention this, because it would bring up the embarrassment of the missing "greenhouse signature" - warming due to CO2 should be greater in the atmosphere than at the surface, but measurements show that it is not.  More on this story later...

The 'simple formulae' in Ramaswamy et al (the TAR) can be seen in Table 6.2 of the TAR, in section 6.3.5. This table has three different formulae, that give different answers for the RF due to doubling of CO2. The simplest one is RF = 5.35 ln (C/C0), and it is probably this one that is used in AR4, but this is not made clear. This is a further example of the astonishingly sloppy and careless work of the IPCC on this crucial issue.