The Gram determinant of type $A$ was introduced by Lickorish in his work on invariants 3 - manifolds. We generalize the theory evaluating, annulus, a bilinear form non-intersecting connections disc. main result provides closed formula for this determinant. conclude paper discussing Chen's conjecture about $Mb$ evaluated Klein bottle.

We study a subgroup of the Shafarevich-Tate group of an abelian variety known as the visible subgroup. We explain the geometric intuition behind this subgroup, prove its finiteness and describe several techniques for exhibiting visible elements. Two important results are proved one what we call the visualization theorem, which asserts that every element of the Shafarevich-Tate group of an abeli...

Let G be a commutative algebraic group defined over a perfect field k. Let k be an algebraic closure of k and Γk be the Galois group of k over k. Following Ribet ([1], [19], see also [7]), we say a point P ∈ G(k) is almost rational over k if whenever σ, τ ∈ Γk are such that σ(P )+ τ(P ) = 2P , then σ(P ) = τ(P ) = P . We denote the almost rational points of G over k by Gar k . Let Gtors denote ...

For any discrete, torsion-free subgroup Γ of Sp(n, 1) (resp. F−20 4 ) with no parabolic elements, we prove that H4n−1(Γ;V ) = 0 (resp. Hi(Γ;V ) = 0 for i = 13, 14, 15) for any Γ–module V . The main technical advance is a new bound on the p–Jacobian of the barycenter map of Besson–Courtois–Gallot. We also apply this estimate to obtain an inequality between the critical exponent and homological d...