On the cohomology groups of certain quotients of products of upper half planes and upper half spaces
Abstract A theorem of Matsushima–Shimura shows that the space of harmonic differential forms on the quotient of products of upper half planes under the action of certain groups, when the quotient is compact, is the direct sum of two subspaces called the universal and cuspidal subspaces. We generalize this result to compact quotients of products of upper half planes and upper half spaces (hyperbolic three spaces) under the action of certain groups to obtain a similar decomposition.
Hilbert Modular Forms
A discrete subgroup Γ ⊂ SL, (2ℝ) acts discontinuously on the upper half-plane H. The parabolic elements of Γ give rise to a natural extension of H/Γ by the so-called cusp classes. We are mainly interested in the case where this extension is compact. Our basic example is Γ = SL(2, Z). The method of construction is such that it can easily be generalized to the case of several variables, i.e. discrete subgroups of SL(2, ℝ) n acting on the product of n upper half-planes. This will be done in the next section (§2).
Augusta Klumpke: pioneering female neurologist.
The latter part of the 19th century was an exciting period when women at last were succeeding in their battle to obtain their medical training and qualifications to practice in the UK, continental Europe and in the USA. Moreover, and perhaps not surprisingly, many of these determined and doughty pioneers showed themselves to be exceptionally able in their chosen specialities. Perhaps the best example of this was Augusta Klumpke (Figure 1), who became an outstanding neurologist in Paris. This year marks the 150th anniversary of her birth.