Uniform bounds on sup-norms of holomorphic forms of real weight

We establish uniform bounds for the sup-norms of modular forms of arbitrary real weight k with respect to a finite index subgroup Gamma of SL2(Z). We also prove corresponding bounds for the supremum over a compact set. We achieve this by extending to a sum over an orthonormal basis Sigma(j) y(k)vertical bar f(j)(z)vertical bar(2) and analyzing this sum by means of a Bergman kernel and the Fourier coefficients of Poincare series. Under some weak assumptions, we further prove the right order of magnitude of sup(z is an element of H) Sigma(j) y(k) vertical bar f(j)(z)vertical bar(2). Our results are valid without any assumption that the forms are Hecke eigenfunctions.