| Given a Weierstrass equation y('2) = 4x('3) - G(,2)x - G(,3) over a function field over (//C) with non-constant J invariant, we associate an elliptic surface with base a compact Riemann surface (')X, a subset X of (')X such that (')X - X is a finite set and J behaves nicely on it, the universal cover of X and the monodromy representation of the fundamental group (pi)(,1)(X) of X. Then we define the space T of generalized cusp forms of the second kind and of weight three on the universal cover of X with cusps, and a parabolic cohomology group H(,par)('1) via the composite representation of (pi)(,1)(X) on the space of polynomials through monodromy representation.;In (SECTION)2, we compute the dimensions of subspaces T('1) and T('2) of T by constructing divisors using results of (SECTION)1 and the Riemann-Roch theorem.;In (SECTION)3, we compute the dimensions of H(,par)('1) to show that dim T('1) + dim T('2) = dim H(,par)('1).;In (SECTION)4, we define a bilinear form on Pd(T) x Pd(T) by defining the residues of differentials and showing that poles of a differential can be moved by adding an exact form of T.;Our main result is the surjectivity of the period map Pd:T (--->) H(,par)('1). We prove this by showing the existence of dim(H(,par)('1)) < (INFIN) linearly independent linear functionals on Pd(T).;In (SECTION)5, we exhibit a basis of the linear space Pd(T) and compute the matrix of the bilinear form on the basis to show the bilinear form is non-degenerate and that T('1) and T('2) are orthogonal to each other relative to the bilinear form, up to exact forms of H(,par)('1) in (SECTION)6.;In (SECTION)7, we prove the equivalence of the bilinear form and the cup product on certain cohomology groups associated with the elliptic surface. |
