In order to study integral points of bounded log-anticanonical height on weak del Pezzo surfaces, we classify weak del Pezzo pairs. As a representative example, we consider a quartic del Pezzo surface of singularity type $\mathbf {A}_1+\mathbf {A}_3$ and prove an analogue of Manin’s conjecture for integral points with respect to its singularities and its lines.

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate–Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.

Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of F at fundamental matrices S (i.e., with $-4\det (S)$ equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with $\det (S) \asymp X$ , the sequence $a(F,S)$ has at least $X^{1-\varepsilon }$ sign changes and takes at least $X^{1-\varepsilon }$ ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound $\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$ for fundamental matrices S.

We establish that the existence of a winning strategy in certain topological games, closely related to a strong game of Choquet, played in a topological space $X$ and its hyperspace $K(X)$ of all nonempty compact subsets of $X$ equipped with the Vietoris topology, is equivalent for one of the players. For a separable metrizable space $X$, we identify a game-theoretic condition equivalent to $K(X)$ being hereditarily Baire. It implies quite easily a recent result of Gartside, Medini and Zdomskyy that characterizes hereditary Baire property of hyperspaces $K(X)$ over separable metrizable spaces $X$ via the Menger property of the remainder of a compactification of $X$. Subsequently, we use topological games to study hereditary Baire property in spaces of probability measures and in hyperspaces over filters on natural numbers. To this end, we introduce a notion of strong $P$-filter ${\mathcal{F}}$ and prove that it is equivalent to $K({\mathcal{F}})$ being hereditarily Baire. We also show that if $X$ is separable metrizable and $K(X)$ is hereditarily Baire, then the space $P_{r}(X)$ of Borel probability Radon measures on $X$ is hereditarily Baire too. It follows that there exists (in ZFC) a separable metrizable space $X$, which is not completely metrizable with $P_{r}(X)$ hereditarily Baire. As far as we know, this is the first example of this kind.

Let $E$ be a (right) Hilbert module over a $C^{\ast }$-algebra $A$. If $E$ is equipped with a left action of a second $C^{\ast }$-algebra $B$, then tensor product with $E$ gives rise to a functor from the category of Hilbert $B$-modules to the category of Hilbert $A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules, Compos. Math.FirstView (2016), 1–33, 2].

For a finite extension $F/\mathbb{Q}_p$ we construct and study a class of locally analytic representations of GL$_2(F)$ and related groups such as the quaternion algebra over $F$. The construction is based on inducing a locally analytic character of a maximal torus. We show that for a generic character the resulting representation is topologically irreducible, and not isomorphic to a locally analytic principal series, when the torus is non-split.

Local models are schemes, defined in terms of linear-algebraic moduli problems, which are used to model the étale-local structure of integral models of certain $p$-adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at ${ \mathbb{Q} }_{p} $ is ramified, quasi-split $G{U}_{n} $, Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper, we proved that their new local models are topologically flat when $n$ is odd. In the present paper, we prove topological flatness when $n$ is even. Along the way, we characterize the $\mu $-admissible set for certain cocharacters $\mu $ in types $B$ and $D$, and we show that for these cocharacters admissibility can be characterized in a vertexwise way, confirming a conjecture of Pappas and Rapoport.

Following Jacquet, Lapid and Rogawski, we regularize trilinear periods. We use the regularized trilinear periods to compute Fourier–Jacobi periods of residues of Eisenstein series on metaplectic groups, which has an application to the Gan–Gross–Prasad conjecture.

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

Given a perfect valuation ring $R$ of characteristic $p$ that is complete with respect to a rank-1 nondiscrete valuation, we show that the ring $\mathbb{A}_{\inf }$ of Witt vectors of $R$ has infinite Krull dimension.

In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.

We explain which Weierstrass ${\wp}$-functions are locally definable from other ${\wp}$-functions and exponentiation in the context of o-minimal structures. The proofs make use of the predimension method from model theory to exploit functional transcendence theorems in a systematic way.

Let ${\mathcal{X}}$ be a regular variety, flat and proper over a complete regular curve over a finite field such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of ${\mathcal{X}}$ is finite if and only Tate’s conjecture for divisors on $X$ holds and the Tate–Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the known formula for a surface.

We study $l$-adic analogues of the classical polylogarithms. Using an explicit arithmetic formula we show that they satisfy the same distribution equations as classical polylogarithms. We also identify $l$-adic polylogarithms evaluated at roots of unity with Soulé classes.

AMS 2000 Mathematics subject classification: Primary 11G55; 11D88; 17B70

In this paper, we relate Lie algebroids to Costello’s version of derived geometry. For instance, we show that each Lie algebroid – and the natural generalization to dg Lie algebroids – provides an (essentially unique) $L_{\infty }$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_{\infty }$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_{\infty }$ space. Indeed, this functor sends the adjoint complex of $L$ to the tangent bundle of the $L_{\infty }$ space. Finally, we show that a shifted symplectic structure on a dg Lie algebroid produces a shifted symplectic structure on the associated $L_{\infty }$ space.

Ill-posedness for the compressible Navier–Stokes equations has been proved by Chen et al. [On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces, Revista Mat. Iberoam.31 (2015), 1375–1402] in critical Besov space $L^{p}$$(p>6)$ framework. In this paper, we prove ill-posedness with the initial data satisfying

$$\begin{eqnarray}\displaystyle \Vert \unicode[STIX]{x1D70C}_{0}-\bar{\unicode[STIX]{x1D70C}}\Vert _{{\dot{B}}_{p,1}^{\frac{3}{p}}}\leqslant \unicode[STIX]{x1D6FF},\quad \Vert u_{0}\Vert _{{\dot{B}}_{6,1}^{-\frac{1}{2}}}\leqslant \unicode[STIX]{x1D6FF}. & & \displaystyle \nonumber\end{eqnarray}$$

$L(a)\unicode[STIX]{x1D6E5}u$

$u\cdot \unicode[STIX]{x1D6FB}u$

A quantum Frobenius map a la Lusztig for $\mathfrak{s}\mathfrak{l}_{2}$ is categorified at a prime root of unity.

The base surface $B$ of a Lagrangian fibration of a projective, irreducible symplectic fourfold $X$ is shown to be isomorphic to $\mathbb{P}^{2}$.

Given a quasi-hereditary algebra , we present conditions which guarantee that the algebra obtained by grading by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that might possess. The method involves working with a pair consisting of a quasi-hereditary algebra and a (positively) graded subalgebra . The algebra arises as a quotient of by a defining ideal of . Along the way, we also show that the standard (Weyl) modules for have a structure as graded modules for . These results are applied to obtain new information about the finite dimensional algebras (e.g., the -Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated with semisimple algebraic groups in positive characteristic . These results require, at least at present, considerable restrictions on the size of .