Volume 2018, Issue 734

The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field ℚ{\mathbb{Q}}, since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14].

In this paper, we prove that, if Deligne’s “petits camarades conjecture” holds, then a Langlands type correspondence holds also for p -adic coefficients on a smooth curve over a finite field. As an application, we prove that any overconvergent F -isocrystal of rank less than or equal to 2 on a smooth curve is ι-mixed.

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems – under certain sharp conditions – for minimizers of the general least gradient problem infu∈B⁢Vf⁢(Ω)⁡∫Ωφ⁢(x,D⁢u)\inf_{u\in BV_{f}(\Omega)}\int_{\Omega}\varphi(x,Du), where f:∂⁡Ω→ℝf:\partial\Omega\to\mathbb{R} is continuous, B⁢Vf⁢(Ω):={v∈B⁢V⁢(Ω):limr→0⁡ess⁢supy∈Ω,|x-y|<r⁡|f⁢(x)-v⁢(y)|=0⁢ for ⁢x∈∂⁡Ω}BV_{f}(\Omega):=\bigl{\{}v\in BV(\Omega):\lim_{r\to 0}\operatornamewithlimits{% ess\,sup}_{y\in\Omega,|x-y|<r}|f(x)-v(y)|=0\text{ for }x\in\partial\Omega\bigr% {\}} and φ⁢(x,ξ)\varphi(x,\xi) is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular, we prove that if a∈C1,1⁢(Ω)a\in C^{1,1}(\Omega) is bounded away from zero, then minimizers of the weighted least gradient problem infu∈B⁢Vf⁡∫Ωa⁢|D⁢u|\inf_{u\in BV_{f}}\int_{\Omega}a|Du| are unique in B⁢Vf⁢(Ω)BV_{f}(\Omega). We construct counterexamples to show that the regularity assumption a∈C1,1a\in C^{1,1} is sharp, in the sense that it can not be replaced by a∈C1,α⁢(Ω)a\in C^{1,\alpha}(\Omega) with any α<1\alpha<1.

Recently, the theory of currents and the existence theory for Plateau’s problem have been extended to the case of finite-dimensional currents in infinite-dimensional manifolds or even metric spaces; see [Acta Math. 185 (2000), 1–80] (and also [Proc. Lond. Math. Soc. (3) 106 (2013), 1121–1142], [Adv. Calc. Var. 7 (2014), 227–240] for the most recent developments). In this paper, in the case when the ambient space is Hilbert, we provide the first partial regularity result, in a dense open set of the support, for n -dimensional integral currents which locally minimize the mass. Our proof follows with minor variants [Indiana Univ. Math. J. 31 (1982), 415–434], implementing Lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space.

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k . Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p -component of the equivariant Tamagawa number of the pair (h1⁢(A/F)⁢(1),ℤ⁢[Gal⁢(F/k)])(h^{1}(A_{/F})(1),\mathbb{Z}[{\rm Gal}(F/k)]). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p -component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell–Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p . More generally, our approach leads us to the formulation of certain precise families of conjectural p -adic congruences between the values at s=1s=1 of derivatives of the Hasse–Weil L -functions associated to twists of A , normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate–Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

We prove the compactness of solutions of general fourth order elliptic equations which are L1L^{1}-perturbations of the Q -curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of the Q -curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for the Q -curvature problem when the Paneitz operator is positive with trivial kernel.

We prove the K -theoretic Farrell–Jones conjecture for groups with the Haagerup approximation property and coefficient rings and C*C^{*}-algebras which are stable with respect to compact operators. We use this and Higson–Kasparov’s result that the Baum–Connes conjecture holds for such a group G , to show that the algebraic and the C*C^{*}-crossed product of G with a stable separable G -C*C^{*}-algebra have the same K -theory.