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We establish a Harder–Narasimhan formalism for modifications of $G$-bundles on the Fargues–Fontaine curve. The semi-stable stratum of the associated stratification of the ${B^+_{{\rm dR}}}$-Grassmannian coincides with the variant of the weakly admissible locus defined by Viehmann, and its classical points agree with those of the basic Newton stratum. When restricted to minuscule affine Schubert cells, the stratification corresponds to the Harder–Narasimhan stratification of Dat, Orlik and Rapoport. We also study basic geometric properties of the strata, and the relation to the Hodge–Newton decomposition.
We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack
$\mathcal {X}$
, which specializes to the Batyrev–Manin conjecture when
$\mathcal {X}$
is a scheme and to Malle’s conjecture when
$\mathcal {X}$
is the classifying stack of a finite group.
We prove some
$\ell $
-independence results on local constancy of étale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighbourhood in the analytic topology such that, for every prime number
$\ell $
different from the residue characteristic, the closed subscheme and the open neighbourhood have the same étale cohomology with
${\mathbb Z}/\ell {\mathbb Z}$
-coefficients. The existence of such an open neighbourhood for each
$\ell $
was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.
Erdős considered the second moment of the gap-counting function of prime divisors in 1946 and proved an upper bound that is not of the right order of magnitude. We prove asymptotics for all moments. Furthermore, we prove a generalisation stating that the gaps between primes p for which there is no
$\mathbb{Q}_p$
-point on a random variety are Poisson distributed.
We study fundamental properties of analytic K-theory of Tate rings such as homotopy invariance, Bass fundamental theorem, Milnor excision, and descent for admissible coverings.
We prove an analogue of Lang's conjecture on divisible groups for polynomial dynamical systems over number fields. In our setting, the role of the divisible group is taken by the small orbit of a point $\alpha$ where the small orbit by a polynomial $f$ is given by
\begin{align*} \mathcal{S}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ n}(\alpha) \text{ for some } n \in \mathbb{Z}_{\geq 0}\}. \end{align*}
Our main theorem is a classification of the algebraic relations that hold between infinitely many pairs of points in $\mathcal {S}_\alpha$ when everything is defined over the algebraic numbers and the degree $d$ of $f$ is at least 2. Our proof relies on a careful study of localisations of the dynamical system and follows an entirely different approach than previous proofs in this area. In particular, we introduce transcendence theory and Mahler functions into this field. Our methods also allow us to classify all algebraic relations that hold for infinitely many pairs of points in the grand orbit
\begin{align*} \mathcal{G}_\alpha = \{\beta \in \mathbb{C}; f^{\circ n}(\beta) = f^{\circ m}(\alpha) \text{ for some } n ,m\in \mathbb{Z}_{\geq 0}\} \end{align*}
of $\alpha$ if $|f^{\circ n}(\alpha )|_v \rightarrow \infty$ at a finite place $v$ of good reduction co-prime to $d$.
We transfer several elementary geometric properties of rigid-analytic spaces to the world of adic spaces, more precisely to the category of adic spaces which are locally of (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and a Stein factorization theorem. Most notably, we show that a finite morphism in our category of adic spaces is automatically open if the target is normal and both source and target are of the same pure dimension. Moreover, our version of the Stein factorization theorem includes a statement about the geometric connectedness of fibers which we have not found in the literature of rigid-analytic or Berkovich spaces.
We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$-adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.
In this note, we prove the semiampleness conjecture for Kawamata log terminal Calabi–Yau (CY) surface pairs over an excellent base ring. As applications, we deduce that generalized abundance and Serrano’s conjecture hold for surfaces. Finally, we study the semiampleness conjecture for CY threefolds over a mixed characteristic DVR.
We investigate a novel geometric Iwasawa theory for
${\mathbf Z}_p$
-extensions of function fields over a perfect field k of characteristic
$p>0$
by replacing the usual study of p-torsion in class groups with the study of p-torsion class group schemes. That is, if
$\cdots \to X_2 \to X_1 \to X_0$
is the tower of curves over k associated with a
${\mathbf Z}_p$
-extension of function fields totally ramified over a finite nonempty set of places, we investigate the growth of the p-torsion group scheme in the Jacobian of
$X_n$
as
$n\rightarrow \infty $
. By Dieudonné theory, this amounts to studying the first de Rham cohomology groups of
$X_n$
equipped with natural actions of Frobenius and of the Cartier operator V. We formulate and test a number of conjectures which predict striking regularity in the
$k[V]$
-module structure of the space
$M_n:=H^0(X_n, \Omega ^1_{X_n/k})$
of global regular differential forms as
$n\rightarrow \infty .$
For example, for each tower in a basic class of
${\mathbf Z}_p$
-towers, we conjecture that the dimension of the kernel of
$V^r$
on
$M_n$
is given by
$a_r p^{2n} + \lambda _r n + c_r(n)$
for all n sufficiently large, where
$a_r, \lambda _r$
are rational constants and
$c_r : {\mathbf Z}/m_r {\mathbf Z} \to {\mathbf Q}$
is a periodic function, depending on r and the tower. To provide evidence for these conjectures, we collect extensive experimental data based on new and more efficient algorithms for working with differentials on
${\mathbf Z}_p$
-towers of curves, and we prove our conjectures in the case
$p=2$
and
$r=1$
.
Motivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their “group-theoretic models”—generalized affine Deligne–Lusztig varieties—in cases where they have a particularly nice description. Continuing the work of Görtz and He (2015, Cambridge Journal of Mathematics 3, 323–353) and Görtz, He, and Nie (2019, Peking Mathematical Journal 2, 99–154), we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new, and open cases from the point of view of Shimura varieties/Rapoport–Zink spaces.
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 prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $\pi (X(k))$ is disjoint from $C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the inverse Galois problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties.
Let A be an abelian scheme of dimension at least four over a
$\mathbb {Z}$
-finitely generated integral domain R of characteristic zero, and let L be an ample line bundle on A. We prove that the set of smooth hypersurfaces D in A representing L is finite by showing that the moduli stack of such hypersurfaces has only finitely many R-points. We accomplish this by using level structures to interpolate finiteness results between this moduli stack and the stack of canonically polarized varieties.
We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general setup for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley–Weil theorem for stacks.
We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser’s stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group
$\Gamma _{\infty }$
, where
$\Gamma $
denotes the value group of K. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of
$\Gamma _{\infty }$
. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance.
We improve to nearly optimal the known asymptotic and explicit bounds for the number of
$\mathbb {F}_q$
-rational points on a geometrically irreducible hypersurface over a (large) finite field. The proof involves a Bertini-type probabilistic combinatorial technique. Namely, we slice the given hypersurface with a random plane.
We discuss the
$\ell $
-adic case of Mazur’s ‘Program B’ over
$\mathbb {Q}$
: the problem of classifying the possible images of
$\ell $
-adic Galois representations attached to elliptic curves E over
$\mathbb {Q}$
, equivalently, classifying the rational points on the corresponding modular curves. The primes
$\ell =2$
and
$\ell \ge 13$
are addressed by prior work, so we focus on the remaining primes
$\ell = 3, 5, 7, 11$
. For each of these
$\ell $
, we compute the directed graph of arithmetically maximal
$\ell $
-power level modular curves
$X_H$
, compute explicit equations for all but three of them and classify the rational points on all of them except
$X_{\mathrm {ns}}^{+}(N)$
, for
$N = 27, 25, 49, 121$
and two-level
$49$
curves of genus
$9$
whose Jacobians have analytic rank
$9$
.
Aside from the
$\ell $
-adic images that are known to arise for infinitely many
${\overline {\mathbb {Q}}}$
-isomorphism classes of elliptic curves
$E/\mathbb {Q}$
, we find only 22 exceptional images that arise for any prime
$\ell $
and any
$E/\mathbb {Q}$
without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on
$X_{\mathrm {ns}}^+(\ell )$
with
$\ell \ge 19$
, or one of the six modular curves noted above. This yields a very efficient algorithm to compute the
$\ell $
-adic images of Galois for any elliptic curve over
$\mathbb {Q}$
.
In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on
$\Gamma _1(N)$
are of
$\operatorname {GL}_2$
-type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of
$X_H$
.