The theorems in the title refer to some generalizations and consequences of the quadratic reciprocity law.  These all play on how the various finite rings Z/nZ, which are simple to understand by themselves, are tied together by the more complicated ring Z.  I will give an introduction to these types of theorems, and then indicate how a certain homotopy-theoretic construction relating tori and spheres can be used to provide proofs of some of them, starting with the quadratic reciprocity law itself.