of a given type: a quantity traditionally calculated by means of the Riemann–Roch theorem. Applications The trace formula has applications to arithmetic geometry and number theory. For instance, using the trace theorem, Eichler and Shimura calculated the Hasse–Weil L-functions associated to modular curves; Goro Shimura's methods by-passed the analysis involved in the trace formula. The development of parabolic cohomology (from Eichler cohomology) provided a purely algebraic setting based on group cohomology, taking account of the cusps characteristic of non-compact Riemann surfaces and modular curves. The trace formula also has purely differential-geometric applications. For instance, by a result of Buser, the length spectrum of a Riemann surface is an isospectral invariant, essentially by the trace formula. Later work The general theory of Eisenstein series was largely motivated by the requirement to separate out the continuous spectrum, which is characteristic of the non-compact case. The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroup into an algebraic group over a field which is technically easier to work with. The case of SL2(C) is discussed in and . Gel'fand et al also treat SL2() where is a locally compact topological field with ultrametric norm, so a finite extension of the p-adic numbers Qp or of the formal Laurent series Fq((T)); they also handle the adelic case in characteristic 0, combining all completions R and Qp of the rational numbers Q. Contemporary successors of the theory are the Arthur–Selberg trace formula applying to the case of general semisimple G, and the many studies of the trace formula in the Langlands philosophy (dealing with technical issues such as endoscopy). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort. Selberg trace formula for compact hyperbolic surfaces A compact hyperbolic surface can be written as the space of orbits where is a subgroup of , and is the upper half plane, and acts on by linear fractional transformations. The Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the group has no parabolic or elliptic elements (other than the identity). Then the spectrum for the Laplace–Beltrami operator on is discrete and real, since the Laplace operator is self adjoint with compact resolvent; that is where the eigenvalues correspond to -invariant eigenfunctions in