In my thesis, I worked on compact quaternionic Shimura surfaces. Ideveloped techniques which made it possible to study particular casesof these surfaces and place them into the Kodaira classification of algebraic surfaces. In this proposal, I describe how these investigations can be extended in several directions and broadened toinvolve other research areas, in particular modular forms. With the techniques which are now available, it should be possible tosolve some types of problems which have previously been considered forHubert modular surfaces. For example, to classify all cases when the surface is rational, and to, in some particular cases, describe the surface up to isomorphism The connections with modular forms goes in several directions. One is to explore how the intersection numbers of cycles on the surfaces are related to coefficients of modular forms. Another direction is to study modular forms on compact Shimura varieties themselves. Progressing this area will lead to new results about the arithmetic of abeliansurfaces. Working at the Max Plank institute, with Prof. Harder and the other experts, will be a favourable environment to attain the new skills needed to fulfil these goals.
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