Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.

https://faculty.missouri.edu/~hofmanns/papers/HLMMY_final.pdf

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds, $H^3_\nr(X,\Q/\Z(2))$ actually vanishes. It is an open question whether this holds true for arbitrary threefolds. For a threefold $X$ equipped with a fibration onto a curve $C$, the generic fibre of which is a smooth projective surface $V$ over the global field $\F(C)$, the vanishing of $H^3_\nr(X,\Q/\Z(2))$ together with the Tate conjecture for divisors on $X$ implies a local-global principle of Brauer--Manin type for the Chow group of zero-cycles on $V$. This sheds a new light on work started thirty years ago. 
----- 
Soit $X$ une vari\'et\'e projective et lisse sur un corps fini $\F$. On s'int\'eresse au groupe de cohomologie non ramifi\'ee $H^3_\nr(X,\Q/\Z(2))$. Un faisceau de conjectures implique que ce groupe est fini. Pour certaines classes de solides, on a $H^3_\nr(X,\Q/\Z(2))=0$. Savoir si c'est le cas pour tout solide est un probl\`eme ouvert. Lorsqu'un solide $X$ est fibr\'e au-dessus d'une courbe $C$, de fibre g\'en\'erique une surface projective et lisse $V$ sur le corps global $\F(C)$, la combinaison de $H^3_\nr(X,\Q/\Z(2))=0$ et de la conjecture de Tate pour $X$ a pour cons\'equence un principe local-global de type Brauer--Manin pour le groupe de Chow des z\'ero-cycles de la fibre g\'en\'erique $V$. Ceci \'eclaire d'un jour nouveau des investigations commenc\'ees il y a trente ans.

/pdf/cycles-de-codimension-2-et-h-3-non-ramifi-e-pour-les-vari-et-3qlwp0y3vs.pdf

Cycles de codimension 2 et H^3 non ramifi\'e pour les vari\'et\'es sur les corps finis

We consider a motivic analogue of the height zeta function for integral points of equivariant partial compactifications of affine spaces. We establish its rationality and determine its largest pole. A major problem of Diophantine geometry is to understand the distribution of rational or integral points of algebraic varieties defined over number fields. For example, a well-studied question put forward by Manin in [1] is that of an asymptotic expansion for the number of rational/integral points of bounded height. A basic tool is the height zeta function which is a Dirichlet series. Around 2000, E. Peyre suggested to consider the analogous problem over function fields, which has then an even more geometric flavor since it translates as a problem of enumerative geometry, namely counting algebraic curves of given degree and establishing properties of the corresponding generating series. In view of the developments of motivic integration by Kontsevich, Denef-Loeser [11], etc., it is natural to look at a more general generating series which not only counts the number of such algebraic curves, but takes into account the space they constitute in a suitable Hilbert scheme. A natural coefficient ring for the generating series is the Grothendieck ring of varieties KVark: if k is a field, this ring is generated as a group by the isomorphism classes of k-schemes of finite type, the addition being subject to obvious cut-and-paste relations, and the product is induced by the product of k-varieties. One interest of this generalization is that it also makes sense in a purely geometric context, where no counting is available. Such a situation has been first studied in a paper by Kapranov in [15], where the analogy with the mentioned diophantine problem is not pointed out. Later, Bourqui made some progress on the motivic analogue of Manin’s problem, see [4], as well as his survey report [5]. Manuscript received April 14, 2014; revised March 3, 2015. Research of the first author supported by the Institut Universitaire de France, by the National Science Foundation under agreement No. DMS-0635607, as well as by the Positive project of Agence nationale de la recherche, under agreement ANR-2010-BLAN-0119-01; research of the second author supported in part by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement nr. 246903 NMNAG. Also partially supported by ValCoMO (ANR-13-BS01-0006). American Journal of Mathematics 138 (2016), 1–59. c © 2016 by Johns Hopkins University Press.

/pdf/motivic-height-zeta-functions-snjld5tcq4.pdf

Motivic height zeta functions

A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan's motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser's motivic vanishing cycles.

Motivic Euler products and motivic height zeta functions

We give an asymptotic formula for the number of weak Campana points of bounded height on a family of orbifolds associated to norm forms for Galois extensions of number fields. From this formula we derive an asymptotic for the number of elements with $m$-full norm over a given Galois extension of $\mathbb{Q}$. We also provide an asymptotic for Campana points on these orbifolds which illustrates the vast difference between the two notions, and we compare this to the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Varilly-Alvarado.

Campana points and powerful values of norm forms

The author investigates the anticanonical height zeta function of a (not necessarily split) toric variety defined over a global field of positive characteristic, drawing inspiration from the method used by Batyrev and Tschinkel to deal with the analogous problem over a number field. The author includes a detailed account of their method.

/pdf/fonction-zeta-des-hauteurs-des-varietes-toriques-non-51cunv6z9e.pdf

Fonction zêta des hauteurs des variétés toriques non déployées

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Fonction zêta des hauteurs des variétés toriques non déployées