  The amplitude of fluctuations in the Ly-a forest on small spatial scales is
sensitive to the temperature of the IGM and its spatial fluctuations. The
temperature of the IGM and its spatial variations contain important information
about hydrogen and helium reionization. We present a new measurement of the
small-scale structure in the Ly-a forest from 40 high resolution, high
signal-to-noise, VLT spectra at z=2.2-4.2. We convolve each Ly-a forest
spectrum with a suitably chosen wavelet filter, which allows us to extract the
amount of small-scale structure in the forest as a function of position across
each spectrum. We compare these measurements with high resolution hydrodynamic
simulations of the Ly-a forest which track more than 2 billion particles. This
comparison suggests that the IGM temperature close to the cosmic mean density
(T_0) peaks near z=3.4, at which point it is greater than 20,000 K at 2-sigma
confidence. The temperature at lower redshift is consistent with the fall-off
expected from adiabatic cooling ($T_0 \propto (1+z)^2$), after the peak
temperature is reached near z=3.4. At z=4.2 our results favor a temperature of
T_0 = 15-20,000 K. However, owing mostly to uncertainties in the mean
transmitted flux at this redshift, a cooler IGM model with T_0 = 10,000 K is
only disfavored at the 2-sigma level here, although such cool IGM models are
strongly discrepant with the z ~ 3-3.4 measurement. We do not detect large
spatial fluctuations in the IGM temperature at any redshift covered by our data
set. The simplest interpretation of our measurements is that HeII reionization
completes sometime near z ~ 3.4, although statistical uncertainties are still
large [Abridged].
