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Let G be a semi-simple algebraic group without G2-factors over an algebraically closed field k of characteristic p ≠ 2, 3, and suppose B is a Borel subgroup, T ⊂ B is a maximal toms, and P is a parabolic in G containing B. In an earlier paper, the authors classified the singular T-fixed points x of an arbitrary irreducible T-stable subvariety X in G/P in all characteristics, the key to this being the notion of a Peterson translate. In particular, we showed that if X is Cohen-Macaulay, then X is smooth at x if and only if there exists a T-invariant curve in X through x which contains a smooth point of X and dim Θx(X) = dim X, where Θx(X) is the linear span of the reduced tangent cone to X at x. The purpose of this paper is to describe Θx(X) when X is a Schubert variety in G/P and x is a maximal singular T-fixed point of X. In fact, we give two characterizations. We first show that in all characteristics, Θx(X) is the sum of all the Peterson translates at x. The second characterization involves further study of the Peterson translates, along the good T-invariant curves at x, for which the assumption char(k) ≠ 2.3 is needed. This leads to the following consequence: if x is a maximal singularity of X which is rationally smooth, then either the span of the tangent lines to the T-stable curves is not a module for the isotropy subgroup of B at x, or there exist a pair of orthogonal T-invariant curves at x which determine what we call a B2-pair. This characterization gives a nonrecursive algorithm for finding the singular locus of an arbitrary Schubert variety in G/P in terms of its Bruhat graph. [ABSTRACT FROM AUTHOR]

The specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as Eisenstein classes. In this paper, we prove that the syntomic realization of the Eisenstein classes restricted to the ordinary locus of the modular curve may be expressed using p-adic Eisenstein series of negative weight, which are p-adic modular forms defined using the two-variable p-adic measure with values in p-adic modular forms constructed by Katz. The motivation of our research is the p-adic Beilinson conjecture formulated by Perrin-Riou. [ABSTRACT FROM AUTHOR]

We study the Gauss-Martin connection for the meduli space of an arrangement of complex hyperplanes in the cohomology of a nontesonant complex rank one local system. Aomoto and Kita determined this Gauss-Martin connection for arrangements in general position. We use their results and an algorithm constructed in this paper to determine this Gauss-Martin connection for all arrangements. [ABSTRACT FROM AUTHOR]