On an invariant related to a linear inequality

Let A be an m-dimensional vector with positive real entries. Let A_{i,j} be the vector obtained from A on deleting the entries A_i and A_j. We investigate some invariant and near invariants related to the solutions E (m-2 dimensional vectors with entries either +1 or -1) of the linear inequality |A_i-A_j| < denotes the usual inner product. One of our methods relates, by the use of Rademacher functions, integrals involving trigonometric quantities to these quantities.Comment: 9 page

Coleman-Gross height pairings and the pp-adic sigma function

We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis, show that, in particular, its component above $p$ gives, in the special case of an ordinary elliptic curve, the $p$-adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Coleman integrals from tangential basepoints.Comment: AMS-LaTeX 17 page