Let $R\subset F$ be an extension of real closed fields, and let ${\mathcal{S}}(M,R)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^{n}$. We prove that every $R$-homomorphism ${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$ is essentially the evaluation homomorphism at a certain point $p\in F^{n}$ adjacent to the extended semialgebraic set $M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when $M$ is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.

Given a field $k$ of characteristic zero and $n\geqslant 0$, we prove that $H_{0}(\mathbb{Z}F(\unicode[STIX]{x1D6E5}_{k}^{\bullet },\mathbb{G}_{m}^{\wedge n}))=K_{n}^{MW}(k)$, where $\mathbb{Z}F_{\ast }(k)$ is the category of linear framed correspondences of algebraic $k$-varieties, introduced by Garkusha and Panin [The triangulated category of linear framed motives $DM_{fr}^{eff}(k)$, in preparation] (see [Garkusha and Panin, Framed motives of algebraic varieties (after V. Voevodsky), Preprint, 2014, arXiv:1409.4372], § 7 as well), and $K_{\ast }^{MW}(k)$ is the Milnor–Witt $K$-theory of the base field $k$.

We prove the following theorems. Theorem 1: for any E-field with cyclic kernel, in particular ℂ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: for the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.

In this article, we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal {L}(E, F)$ . By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set K of $\mathcal {L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm-closed convex hull, then we can guarantee the existence of an operator that does not attain its norm. This allows us to provide the following generalisation of results due to Holub and Mujica. If E is a reflexive space, F is an arbitrary Banach space and the pair $(E, F)$ has the (pointwise-)bounded compact approximation property, then the following are equivalent:

1. (i) $\mathcal {K}(E, F) = \mathcal {L}(E, F)$ ;

2. (ii) Every operator from E into F attains its norm;

3. (iii) $(\mathcal {L}(E,F), \tau _c)^* = (\mathcal {L}(E, F), \left \Vert \cdot \right \Vert )^*$ ,

where $\tau _c$ denotes the topology of compact convergence. We conclude the article by presenting a characterisation of the Schur property in terms of norm-attaining operators.

Nous considérons l’intersection d’une sous-variété $X$ d’une variété abélienne $A$ avec l’union de tous les sous-groupes de $A$ obtenus comme somme d’un sous-groupe donné $\varGamma$ de rang fini et d’un sous-groupe algébrique de dimension donnée. Nous montrons que, quitte à retirer à $X$ un certain ensemble exceptionnel, l’intersection est de hauteur bornée. L’énoncé est optimal pour une courbe $X$. Nous étudions également les épaississements $\varGamma_{\varepsilon}$ introduits par Poonen. La démonstration repose sur une généralisation uniforme de la méthode de Vojta et sur des calculs de nombres d’intersection de cycles réels sur $A$.

We study the intersection of a subvariety $X$ of an abelian variety $A$ over $\bar{\mathbb{Q}}$ with the union of all the subgroups obtained as a sum of a given finite rank subgroup $\varGamma$ and an algebraic subgroup of $A$ of given dimension $d$. Our main result asserts that if we remove a suitable exceptional subset from $X$ then the intersection is a set of bounded height. In some cases, this combines with the output of Part I to yield finiteness. In terms of boundedness of the height, we get an optimal statement for a curve $X$ with $d=2$. We also deal with the fattenings $\varGamma_\varepsilon$ introduced by Poonen. The proof rests on a suitably uniform generalization of the method of Vojta and on computations of intersection numbers of real cycles on $A$.

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.

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.

In this note, we prove that the Drinfeld–Grinberg–Kazhdan theorem on the structure of formal neighborhoods of arc schemes at a nonsingular arc does not extend to the case of singular arcs.

We prove that n-hypergraphs can be interpreted in e-free perfect PAC fields in particular in pseudofinite fields. We use methods of function field arithmetic, more precisely we construct generic polynomials with alternating groups as Galois groups over a function field.

We introduce certain energy functionals to complex Monge–Ampère equations over bounded domains with inhomogeneous boundary conditions, and use these functionals to show the convergence of solutions to certain parabolic Monge–Ampère equations.

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 $\mathrm {GL}_n$ .

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree $3$ and level $1$ .

We consider the equivariant Kasparov category associated to an étale groupoid, and by leveraging its triangulated structure we study its localization at the ‘weakly contractible’ objects, extending previous work by R. Meyer and R. Nest. We prove the subcategory of weakly contractible objects is complementary to the localizing subcategory of projective objects, which are defined in terms of ‘compactly induced’ algebras with respect to certain proper subgroupoids related to isotropy. The resulting ‘strong’ Baum–Connes conjecture implies the classical one, and its formulation clarifies several permanence properties and other functorial statements. We present multiple applications, including consequences for the Universal Coefficient Theorem, a generalized ‘going-down’ principle, injectivity results for groupoids that are amenable at infinity, the Baum–Connes conjecture for group bundles, and a result about the invariance of K-groups of twisted groupoid $C^*$-algebras under homotopy of twists.

This paper concerns arithmetic families of $\varphi $-modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons are lower semicontinuous around any rigid point. We further prove that if the slope polygons are locally constant around a rigid point, then around this point, the family has a global slope filtration after base change to some extended Robba ring.

We derive a quadratic recursion relation for the linear Hodge integrals of the form $\langle \tau _{2}^{n}\lambda _{k}\rangle $ . These numbers are used in a formula for Masur-Veech volumes of moduli spaces of quadratic differentials discovered by Chen, Möller and Sauvaget. Therefore, our recursion provides an efficient way of computing these volumes.

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

We study the consequences of stationary and semi-stationary set reflection. We show that the semi-stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of the weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss, and prove that they follow from stationary and semi-stationary set reflection augmented with a weak form of Martin’s Axiom. We also show that there are some differences between the two reflection principles, which suggests that stationary set reflection is analogous to supercompactness, whereas semi-stationary set reflection is analogous to strong compactness.

We study characters of an $n$-fold cover $\widetilde{SL} (n,\mathbb{F})$ of $SL(n,\mathbb{F})$ over a non-Archimedean local field. We compute the character of an irreducible representation of $\widetilde{SL}(n,\mathbb{F})$ in terms of the character of an irreducible representation of a cover $\widetilde{GL}(n,\mathbb{F})$ of $GL(n,\mathbb{F})$. We define an analogue of L-packets for $\widetilde{SL}(n,\mathbb{F})$, such that the character of a linear combination of the representations in such a packet is computed in terms of the character of an irreducible representation of $PGL(n,\mathbb{F})$. This is analogous to stable endoscopic lifting for linear groups. We also prove an ‘inversion’ formula expressing the character of a genuine irreducible representation of $\widetilde{SL}(n,\mathbb{F})$ as a linear combination of virtual characters, each of which is obtained from $PGL(n,\mathbb{F})$.

AMS 2000 Mathematics subject classification: Primary 22E50. Secondary 11F70

Keller proved in 1999 that the Gerstenhaber algebra structure on the Hochschild cohomology of an algebra is an invariant of the derived category. In this paper, we adapt his approach to show that the Gerstenhaber algebra structure on the Tate–Hochschild cohomology of an algebra is preserved under singular equivalences of Morita type with level, a notion introduced by the author in previous work.