Reduction theory and periods of modular forms

Among Manin's most beautiful and influential contributions to number theory was his study of periods of modular forms, in particular the theory of modular symbols and his algebraicity theorem for the periods of cusp forms, both of which are related to the theory of continued fractions. After reviewing this material, I will turn to the inverse problem of determining a cusp form from its periods and will describe a complete solution for the case of the full modular group. This result, which I found more than 25 years ago and had always intended to dedicate to Manin, depends on a simple but surprising lemma about continued fractions. But for the solution of the corresponding result for other Fuchsian groups I needed to establish a rather beautiful statement about reduction theory that I discovered experimentally and checked in many cases, so I kept postponing the publication and dedication until this question could be resolved and I could give the complete result. This still has not happened, and I have decided to belatedly present what I already know.