C. Fefferman [F1], [F2] has recently given criteria for a function defined on a compact set $E\subset {\mathbb{R}}^n$ to extend to a $C^m$- or $C^{m,\omega }$-function. His criteria involve uniformity of the $C^m$- or $C^{m,\omega }$-norms for extension from finite subsets $S\subset E$ of cardinality at most a large natural number $k^{\#}$ depending only on $m$ and $n$. We prove that one can take $k^{\#}=2^{\dim P}$ in both cases, where $P$ denotes the space of polynomials of degree at most $m$ in $n$ variables. We also show that the geometric $C^m$ “paratangent bundle” of $E$ (see [BMP2]) can be defined using limits of distributions supported on $2^{\dim P-1}$ points

Let $p$ be a prime number, let $L$ be a finite extension of the field ${\mathbb{Q}}_p$ of $p$-adic numbers, let $K$ be a spherically complete extension field of $L$, and let $G$ be the group of $L$-rational points of a split reductive group over $L$. We derive several explicit descriptions of the center of the algebra $D(G,K)$ of locally analytic distributions on $G$ with values in $K$. The main result is a generalization of an isomorphism of Harish-Chandra which connects the center of $D(G,K)$ with the algebra of Weyl-invariant, centrally supported distributions on a maximal torus of G. This isomorphism is supposed to play a role in the theory of locally analytic representations of $G$ as studied by P. Schneider and J. Teitelbaum

We verify, for each odd prime $p<1000$, a conjecture of W. G. McCallum and R. T. Sharifi on the surjectivity of pairings on $p$-units constructed out of the cup product on the first Galois cohomology group of the maximal unramified outside $p$ extension of $Q({\mu }_p)$ with ${\mu }_p$-coefficients. In the course of the proof, we relate several Iwasawa-theoretic and Hida-theoretic objects. In particular, we construct a canonical isomorphism between an Eisenstein ideal modulo its square and the second graded piece in an augmentation filtration of a classical Iwasawa module over an abelian pro-$p$ Kummer extension of the cyclotomic $Z_p$-extension of an abelian field. This Kummer extension arises from the Galois representation on an inverse limit of ordinary parts of first cohomology groups of modular curves which was considered by M. Ohta in order to give another proof of the Iwasawa main conjecture in the spirit of that of B. Mazur and A. Wiles. In turn, we relate the Iwasawa module over the Kummer extension to the quotient of the tensor product of the classical cyclotomic Iwasawa module and the Galois group of the Kummer extension by the image of a certain reciprocity map that is constructed out of an inverse limit of cup products up the cyclotomic tower. We give an application to the structure of the Selmer groups of Ohta's modular representation taken modulo the Eisenstein ideal

We find the primitive integer solutions to $x^2+y^3=z^7$. A nonabelian descent argument involving the simple group of order $168$ reduces the problem to the determination of the set of rational points on a finite set of twists of the Klein quartic curve $X$. To restrict the set of relevant twists, we exploit the isomorphism between $X$ and the modular curve $X(7)$ and use modularity of elliptic curves and level lowering. This leaves $10$ genus $3$ curves, whose rational points are found by a combination of methods

Yau [Y2] has conjectured that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable in the sense of geometric invariant theory. There has been much progress on this conjecture by Tian [T1], [T2], Donaldson [Do1], [Do2], and others. The Mabuchi energy functional plays a central role in these ideas. We study the $E_k$ functionals introduced by X. X. Chen and G. Tian [CT1] which generalize the Mabuchi energy. We show that if a Fano manifold admits a Kähler-Einstein metric, then the functional $E_1$ is bounded from below and, modulo holomorphic vector fields, is proper. This answers affirmatively a question raised by Chen [C2]. In fact, we show that $E_1$ is proper if and only if there exists a Kähler-Einstein metric, giving a new analytic criterion for the existence of this canonical metric, with possible implications for the study of stability. We also show that on a Fano Kähler-Einstein manifold, all of the functionals $E_k$ are bounded below on the space of metrics with nonnegative Ricci curvature