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We consider Willmore surfaces in with an isolated singularity of finite density at the origin. We show that locally, the surface is a union of finitely many multivalued graphs, each with a unique tangent plane at zero and with second fundamental form satisfying where is the maximal multiplicity. Examples of branched minimal surfaces show that this estimate is optimal up to the error
We classify joinings between a fairly general class of higher-rank diagonalizable actions on locally homogeneous spaces. In particular, we classify joinings of the action of a maximal -split torus on with a simple Lie group of -rank at least and a lattice. We deduce from this a classification of measurable factors of such actions as well as certain equidistribution properties
We use the recent proof of Jacquet's conjecture due to Harris and Kudla [HK] and the Burger-Sarnak principle (see [BS]) to give a proof of the relationship between the existence of trilinear forms on representations of for a non-Archimedean local field and local epsilon factors which was earlier proved only in the odd residue characteristic by this author in [P1, Theorem 1.4]. The method used is very flexible and gives a global proof of a theorem of Saito and Tunnell about characters of using a theorem of Waldspurger [W, Theorem 2] about period integrals for and also an extension of the theorem of Saito and Tunnell by this author in [P3, Theorem 1.2] which was earlier proved only in odd residue characteristic. In the appendix to this article, H. Saito gives a local proof of Lemma 4 which plays an important role in the article
In this article, we consider the problem of finding upper bounds on the minimum norm of representatives in residue classes in quotient , where is an integral ideal in the maximal order of a number field . In particular, we answer affirmatively a question of Konyagin and Shparlinski [KS], stating that an upper bound holds for most ideals , denoting the norm of . More precise statements are obtained, especially when is prime. We use the method of exponential sums over multiplicative groups, essentially exploiting some new bounds obtained by the authors