  In this paper we investigate the behaviour of the specific heat around the
critical point of the Ising model in dimension 5 to 7. We find a specific heat
discontinuity, like that for the mean field Ising model, and provide estimates
for the left and right hand limits of the specific heat at the critical point.
We also estimate the singular exponents, describing how the specific heat
approaches those limits. Additionally, we make a smaller scale investigation of
the same properties in dimension 6 and 7, and provide strongly improved
estimates for the critical termperature $K_c$ in $d=5,6,7$ which bring the best
MC-estimate closer to those obtained by long high temperature series expanions.
