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Given a perverse sheaf on the moduli stack of principally polarized abelian varieties or the moduli stack of smooth curves with n marked points over a field of characteristic zero, we prove that the (orbifold) Euler characteristic is nonnegative. For constant coefficients, this follows immediately from formulas of Harer-Zagier and Harder. Our proof is different and in the case of abelian varieties uses log Dubson-Kashiwara plus the fact that Hodge bundles are nef. For curves, we require an additional inequality established using Beilinson's gluing construction. The first main result is shown to be false in positive characteristic., Comment: 20 pages

In this article, we characterize the image of the Brylinski-Radon transform in characteristic $p>0$ via Beilinson's theory of singular supports. We also provide an alternate proof of Brylinski's results over $\mathbb{C}$, which also works for sheaves with finite coefficients. Along the way, we also obtain a microlocal criterion for the descent of perverse sheaves which could be of independent interest., Comment: Minor changes in v2. Comments are welcome!

Let $k$ be a perfect field of characteristic $p>2$, $R := W(k)[\![t_1, \dots, t_d]\!]$ be the power series ring over the Witt vectors, and $X$ be a smooth proper scheme over $R$. The main goal of this article is to extend classical Fontaine-Messing theory to the setting where the base ring is $R$. In particular, we obtain comparison theorems between torsion crystalline cohomology of $X/R$ and torsion \'etale cohomology in this setting.

Let $X \subset \mathbb{P}^{n}$ be a non-empty closed subscheme over an algebraically closed field $k$, and $\mathrm{J}^{[p]}(X) = \mathrm{J}(X,\mathrm{J}(X,\cdots,\mathrm{J}(X,X)\cdots)$ denote the $p$-fold iterated join of $X$ with itself. In this article, we prove that the restriction homomorphism on cohomology $\mathrm{H}^{i}(\mathbb{P}^{N}) \rightarrow \mathrm{H}^{i}(\mathrm{J}^{[p]}(X))$, with $N = (p+1)(n+1)-1$, is an isomorphism for $0 \leq i < p$, and injective for $i=p$, for any good cohomology theory. We also prove this result in the more general setting of relative joins for $X$ over a base scheme $S$, where $S$ is of finite type over $k$. We give several applications of these results including a cohomological version of classical quantifier elimination in the first order theory of algebraically closed fields of arbitrary characteristic, as well as an algebraic version of Toda's theorem in complexity theory valid over algebraically closed fields of arbitrary characteristic. We also apply our results to obtain effective bounds on the Betti numbers of image of projective varieties under projection map., Comment: 40 pages. Final version to appear in Selecta Mathematica

In this largely expository article, we present a Kawamata-Viehweg type formulation of the (logarithmic) Akizuki-Nakano Vanishing Theorem. While the result is likely known to the experts, it does not seem to appear in the existing literature., Comment: 20 pages. A reference added

We prove a tight bound on the number of realized $0/1$ patterns (or equivalently on the Vapnik-Chervonenkis codensity) of definable families in models of the theory of algebraically closed valued fields with a non-archimedean valuation. Our result improves the best known result in this direction proved by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko, who proved a weaker bound in the restricted case where the characteristics of the field $K$ and its residue field are both assumed to be $0$. The bound obtained here is optimal and without any restriction on the characteristics. We obtain the aforementioned bound as a consequence of another result on bounding the Betti numbers of semi-algebraic subsets of certain Berkovich analytic spaces, mirroring similar results known already in the case of o-minimal structures and for real closed, as well as, algebraically closed fields. The latter result is the first result in this direction and is possibly of independent interest. Its proof relies heavily on recent results of Hrushovski and Loeser on the topology of semi-algebraic subsets of Berkovich analytic spaces., Comment: 40 pages. Extensively revised

A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve., Comment: 45 pages, 1 figure; minor revision

Let G be the group of L-rational points of a connected split reductive group over a finite extension L of Q_p. We show that formal models of the algebraic flag variety X of G are D-affine for certain sheaves of arithmetic differential operators. We then introduce the category of coadmissible G-equivariant arithmetic D-modules on the system of formal models of X and prove that it is anti-equivalent to the category of admissible locally L-analytic G-representations with trivial infinitesimal character. We compute the equivariant arithmetic D-modules of certain classes of representations., Comment: 59 pages, revised version, following referees' comments. Added more details in sections three and five, corrected statements 5.2.13 (now 5.2.29) and 5.3.8 (now 5.3.12/5.3.16), some parts of sections two and four now appear in revised form in the paper arXiv:1709.00555

In this paper we study certain sheaves of $p$-adically complete rings of differential operators on semistable models of the projective line over the ring of integers in a finite extension $L$ of ${\mathbb Q}_p$. The global sections of these sheaves can be identified with (central reductions of) analytic distribution algebras of wide open congruence subgroups. It is shown that the global sections functor furnishes an equivalence between the categories of coherent module sheaves and finitely presented modules over the distribution algebras. Using work of M. Emerton, we then describe admissible representations of ${\rm GL}(2,L)$ in terms of sheaves on the projective limit of these formal schemes., Comment: Added acknowledgement of support by the ANR program p-adic Hodge Theory and beyond (Th\'eHopaD). v3: section 6 now contains examples other than the first Drinfeld covering of the p-adic upper half plane, the new section 7 discusses in detail the admissibility of some representations furnished by the first Drinfeld covering of the p-adic upper half plane

In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one., Comment: Added acknowledgement of support by the ANR program p-adic Hodge Theory and beyond (Th\'eHopaD)

Let k be an algebraically closed subfield of the complex numbers, and X a variety defined over k. One version of the Beilinson-Hodge conjecture that seems to survive scrutiny is the statement that the Betti cycle class map cl_{r,m} : H_M^{2r-m}(k(X),Q(r)) -> hom_{MHS}(Q(0),H^{2r-m}(k(X)(C),Q(r))) is surjective, that being equivalent to the Hodge conjecture in the case m=0. Now consider a smooth and proper map \rho : X -> S of smooth quasi-projective varieties over k. We formulate a version of this conjecture for the generic fibre, expecting the corresponding cycle class map to be surjective. We provide some evidence in support of this in the case where X is a product, the map is the projection to one factor, and m=1.

In the first part of the paper we show that the ring of global sections of arithmetic differential operators on the formal projective line over Zp is isomorphic to the analytic distribution algebra of the 'wide open' congruence subgroup of level zero of GL(2,Zp). In the second part we study rings of logarithmic differential operators on semistable integral models of the projective line over Zp and we relate these to analytic distribution algebra of 'wide open' congruence subgroups of higher level.

Let L be a finite field extension of Q_p and let G be the group of L-rational points of a split connected reductive group over L. We view G as a locally L-analytic group with Lie algebra g. We define a functor from admissible locally analytic G-representations with prescribed infinitesimal character to a category of equivariant sheaves on the Bruhat-Tits building of G. For smooth representations, the corresponding sheaves are closely related to the sheaves constructed by S. Schneider and U. Stuhler. The functor is also compatible, in a certain sense, with the localization of g-modules on the flag variety by A. Beilinson and J. Bernstein., Comment: Replaces earlier version. Exposition shortened and improved in several places, and new material and examples added. In particular, section 11 is new."