We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$,

\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]

$\operatorname {Tr}_{L/k}$

$\operatorname {GW}(L) \to \operatorname {GW}(k)$

${\mathbf {C}}$

${\mathbf {R}}$

$\mathbf {A}^1$

Building on work of Segre and Kollár on cubic hypersurfaces, we construct over imperfect fields of characteristic $p\geq 3$ particular hypersurfaces of degree p, which show that geometrically rational schemes that are regular and whose rational points are Zariski dense are not necessarily unirational. A likewise behavior holds for certain cubic surfaces in characteristic $p=2$.

An effective lower bound on the entropy of some explicit quadratic plane Cremona transformations is given. The motivation is that such transformations (Hénon maps, or Feistel ciphers) are used in symmetric key cryptography. Moreover, a hyperbolic plane Cremona transformation g is rigid, in the sense of [5], and under further explicit conditions some power of g is tight.

We construct examples of smooth proper rigid-analytic varieties admitting formal models with projective special fibers and violating Hodge symmetry for cohomology in degrees ${\geq }3$. This answers negatively the question raised by Hansen and Li.

We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties ${{\mathcal A}}$ of dimension g over a finite field ${{\mathbb F}}_q$, when $q\ge 4$ and $2g=\rho ^{b-1}(\rho -1)$, for some prime $\rho \ge 5$ with $b\ge 1$. Moreover, we show that ${{\mathcal A}}$ is absolutely simple if $b=1$ and g is prime, but ${{\mathcal A}}$ is not absolutely simple for any prime $\rho \ge 5$ with $b>1$.

Nous développons dans cet article des techniques d'aplatissement des faisceaux cohérents en géométrie de Berkovich, en nous inspirant de la stratégie générale que Raynaud et Gruson ont mise en œuvre pour traiter le problème analogue en théorie des schémas. Nous donnons ensuite quelques applications à l’étude des morphismes entre espaces analytiques compacts, et obtenons notamment une description de l'image d'un tel morphisme.

We study the l-adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of $K3$ surfaces over finite fields. We prove that every $K3$ surface of finite height over a finite field admits a characteristic $0$ lifting whose generic fibre is a $K3$ surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a $K3$ surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a $K3$ surface of finite height and construct characteristic $0$ liftings of the $K3$ surface preserving the action of tori in the algebraic group. We obtain these results for $K3$ surfaces over finite fields of any characteristics, including those of characteristic $2$ or $3$.

We construct a family of fibred threefolds $X_m \to (S , \Delta )$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold $(S,\Delta )$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potentially dense set of rational points. We prove that if m is big enough, the threefolds $X_m$ present behaviours that contradict the function field and analytic analogue of the Weak Specialness Conjecture. We prove our results by adapting the recent method of Ru and Vojta. We also formulate some generalisations of known conjectures on exceptional loci that fit into Campana’s program and prove some cases over function fields.

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.

We provide a direct proof of a Bogomolov-type statement for affine varieties V defined over function fields K of finite transcendence degree over an arbitrary field k, generalising a previous result (obtained through a different approach) of the first author in the special case when K is a function field of transcendence degree $1$. Furthermore, we obtain sharp lower bounds for the Weil height of the points in $V(\overline {K})$, which are not contained in the largest subvariety $W\subseteq V$ defined over the constant field $\overline {k}$.

Rapoport–Zink spaces are deformation spaces for $p$-divisible groups with additional structure. At infinite level, they become preperfectoid spaces. Let ${{\mathscr M}}_{\infty }$ be an infinite-level Rapoport–Zink space of EL type, and let ${{\mathscr M}}_{\infty }^{\circ }$ be one connected component of its geometric fiber. We show that ${{\mathscr M}}_{\infty }^{\circ }$ contains a dense open subset which is cohomologically smooth in the sense of Scholze. This is the locus of $p$-divisible groups which do not have any extra endomorphisms. As a corollary, we find that the cohomologically smooth locus in the infinite-level modular curve $X(p^{\infty })^{\circ }$ is exactly the locus of elliptic curves $E$ with supersingular reduction, such that the formal group of $E$ has no extra endomorphisms.

We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António, $p$ -adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302] to prove the existence representability of the derived Hilbert space $\mathbf{R}\text{Hilb}(X)$ for a separated $k$ -analytic space $X$ . Such representability results rely on a localization theorem stating that if $\mathfrak{X}$ is a quasi-compact and quasi-separated formal scheme, then the $\infty$ -category $\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$ of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the $\infty$ -category $\text{Coh}^{-}(\mathfrak{X})$ . Along the way, we prove several results concerning the $\infty$ -categories of formal models for almost perfect modules on derived $k$ -analytic spaces.

Let S be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb {Z}_{S}$ -points on integral models of Hilbert modular varieties, extending a result of D. Helm and F. Voloch about modular curves. Let L be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre’s conjecture for mod $\ell $ representations of the absolute Galois group of L, we prove that the same holds also for the $\mathcal {O}_{L,S}$ -points.

Let f and g be two cuspidal modular forms and let ${\mathcal {F}}$ be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space $\mathcal {W}$ . Using ideas of Pottharst, under certain hypotheses on f and g, we construct a coherent sheaf over $V \times \mathcal {W}$ that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function $L_p$ interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$ .

We study intersections of orbits in polynomial semigroup dynamics with lines on the affine plane over a number field, extending previous work of D. Ghioca, T. Tucker, and M. Zieve (2008).

We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$ -adic local fields with $p\geqslant 5$ . In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.