ON A CERTAIN TRIPLE SYSTEM, ELLIPTIC CURVES AND GAUSS THEORY OF QUADRATIC FORMS

Inspired by the important result that the space of cusp forms is generated by Poincaré sums, a triple system (G, (G,M)) is proposed in general and we are interested in whether analogous results hold in other circumstances. First I consider the case (G, (SLm(Z), slm(Z))) with G a cyclic group. In order to compute it, I generalize the core theorem in Gauss the-ory of quadratic forms from quadratic extensions to n-dimensional algebraic extensions and from over the integers to over any ring of integers which is a PID. By that generalization, I get analogous results to that of cusps forms for good cocycles. Second I consider the case (G, (Γ(N),M)) with G a cyclic group and the congruence subgroup Γ(N) in SLm(Z). In order to compute it and others which might occur in the future, I generalize that core theorem in Gauss theory of quadratic forms even more to what I call polynomial cohomol-ogy. Third I consider the case (G, (O×,O)) to explore more about the circumstance with G infinite, where G is the universal covering group for the elliptic curve Eτ associated with an element τ in the upper half plane and O is the ring of holomorphic functions over C. From the results there for the elliptic curve case and those for the original modular form case, I try to formulate a good definition for a triple system when G is countable. Under that definition, in that elliptic curve case, we get analogous results to that of cusps forms for all the cocycles. Advisor: Professor Takashi Ono. i