In this paper, we introduce the study of minimal torsion curves within a fixed geometric isogeny class. For a $\overline{{\mathbb{Q}}}$-isogeny class $\mathcal{E}$ of elliptic curves and $N \in {\mathbb{Z}}^+$, we wish to determine the least degree of a point on the modular curve $X_1(N)$ associated to any $E \in \mathcal{E}$. We consider the cases where $\mathcal{E}$ is rational, i.e., contains an elliptic curve with rational j-invariant, or where $\mathcal{E}$ consists of elliptic curves with complex multiplication (CM). If $N=\ell^k$ is a power of a single prime, we give a complete characterisation upon restricting to points of odd degree, and also in the case where $\mathcal{E}$ is CM. We include various partial results in the more general setting.

For a finite extension F of ${\mathbb Q}_p$ and $n \geq 1$, we show that the category of Lubin-Tate bundles on the $(n-1)$-dimensional Drinfeld symmetric space is equivalent to the category of finite-dimensional smooth representations of the group of units of the division algebra of invariant $1/n$ over F.

In this paper, we develop canonical bundle formulas for fibrations of relative dimension one in characteristic $p>0$. For such a fibration from a log pair $f\colon (X, \Delta ) \to S$, if f is separable, we can obtain a formula similar to the one due to Witaszek [36]; if f is inseparable, we treat the case when S is of maximal Albanese dimension. As an application, we prove that for a klt pair $(X,\Delta )$ with $-(K_X+\Delta )$ nef, if the Albanese morphism $a_X\colon X \to A$ is of relative dimension one, then X is a fiber space over A.

In this article, we investigate the p-rank stratification of the moduli space of curves of genus g that admit a double cover to a fixed elliptic curve E in characteristic $p>2$. We show that the closed p-rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of E of all the possible values of the p-rank on this moduli space.

We establish several new properties of the p-adic Jacquet-Langlands functor defined by Scholze in terms of the cohomology of the Lubin-Tate tower. In particular, we reprove Scholze’s basic finiteness theorems, prove a duality theorem, and show a kind of partial Künneth formula. Using these results, we deduce bounds on Gelfand-Kirillov dimension, together with some new vanishing and nonvanishing results.

Our key new tool is the six functor formalism with solid almost $\mathcal {O}^+/p$-coefficients developed recently by the second author [Man22]. One major point of this paper is to extend the domain of validity of the $!$-functor formalism developed in [Man22] to allow certain ‘stacky’ maps. In the language of this extended formalism, we show that if G is a p-adic Lie group, the structure map of the classifying small v-stack $B\underline {G}$ is p-cohomologically smooth.

For any power q of the positive ground field characteristic, a smooth q-bic threefold—the Fermat threefold of degree $q+1$, for example—has a smooth surface S of lines which behaves like the Fano surface of a smooth cubic threefold. I develop projective, moduli-theoretic, and degeneration techniques to study the geometry of S. Using, in addition, the modular representation theory of the finite unitary group and the geometric theory of filtrations, I compute the cohomology of the structure sheaf of S when q is prime.

We show that the moduli spaces of Scholze’s p-adic shtukas with framing satisfy a p-adic rigid analytic version of Borel’s extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data $(G, [b],\{\mu \})$, even for exceptional groups G, and extends work of Oswal-Shankar-Zhu-Patel who proved a p-adic Borel extension property for Rapoport-Zink spaces. As a corollary, we deduce that all these spaces satisfy a p-adic rigid analytic version of Brody hyperbolicity.

The compactly supported $\mathbb {A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an analogue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm {K}_0(\mathrm {Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm {GW}(k)$ of the base field k. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, the first author and Pál construct a power structure on $\mathrm {GW}(k)$ and show that the compactly supported $\mathbb {A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently étale k-algebras. In this paper, we define the class $\mathrm {Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb {A}^1$-Euler characteristic respects the power structures and study the algebraic properties of the subring $\mathrm {K}_0(\mathrm {Sym}_k)$ of symmetrisable varieties. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb {A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.

Let X be a smooth proper rigid analytic space over a complete algebraically closed field extension K of $\mathbb {Q}_p$. We establish a Hodge–Tate decomposition for X with G-coefficients, where G is any commutative locally p-divisible rigid group. This generalizes the Hodge–Tate decomposition of Faltings and Scholze, which is the case $G=\mathbb {G}_a$. For this, we introduce geometric analogs of the Hodge–Tate spectral sequence with general locally p-divisible coefficients. We prove that these spectral sequences degenerate at $E_2$. Our results apply more generally to a class of smooth families of commutative adic groups over X and in the relative setting of smooth proper morphisms $X\rightarrow S$ of seminormal rigid spaces. We deduce applications to analytic Brauer groups and the geometric p-adic Simpson correspondence.

In this article, we classify irregular threefolds with numerically trivial canonical divisors in positive characteristic. For a threefold, if its Albanese dimension is not maximal, then the Albanese morphism will induce a fibration which either maps to a curve or is fibered by curves. In practice, we treat arbitrary dimensional irregular varieties with either one-dimensional Albanese fiber or one-dimensional Albanese image. We prove that such a variety carries another fibration transversal to its Albanese morphism (a “bi-fibration” structure), which is an analog structure of bielliptic or quasi-bielliptic surfaces. In turn, we give an explicit description of irregular threefolds with trivial canonical divisors.

For a complete discrete valuation field K, we show that one may always glue a separated formal algebraic space $\mathfrak {X}$ over $\mathcal {O}_K$ to a separated algebraic space U over K along an open immersion of rigid spaces $j\colon \mathfrak {X}^{\mathrm {rig}}\to U^{\mathrm {an}}$, producing a separated algebraic space X over $\mathcal {O}_K$. This process gives rise to an equivalence between such ‘gluing triples’ $(U,\mathfrak {X},j)$ and separated algebraic spaces X over $\mathcal {O}_K$, which one might interpret as a version of the Beauville–Laszlo theorem for algebraic spaces rather than coherent sheaves. Moreover, an analogous equivalence exists over any excellent base. Examples due to Matsumoto imply that the result of such a gluing might be a genuine algebraic space (not a scheme) even if U and the special fibre of $\mathfrak {X}$ are projective. The proof is a combination of the Nagata compactification theorem for algebraic spaces and of Artin’s contraction theorem. We give multiple examples and applications of this idea.

This work concerns generators for the bounded derived category of coherent sheaves over a noetherian scheme X of prime characteristic. The main result is that when the Frobenius map on X is finite, for any compact generator G of $\mathsf {D}(X)$ the Frobenius pushforward $F ^e_*G$ generates the bounded derived category whenever $p^e$ is larger than the codepth of X, an invariant that is a measure of the singularity of X. The conclusion holds for all positive integers e when X is locally complete intersection. The question of when one can take $G=\mathcal {O}_X$ is also investigated. For smooth projective complete intersections it reduces to a question of generation of the Kuznetsov component.

Inspired by Bhatt–Scholze [BS22], in this article, we introduce prismatic cohomology for rigid analytic spaces with l.c.i. singularities, with coefficients over Fontaine’s de Rham period ring ${\mathrm {B_{dR}^+}}$.

We prove an André–Oort-type result for a family of hypersurfaces in ${\mathbb{C}}^n$ that is both uniform and effective. Let $K_*$ denote the single exceptional imaginary quadratic field which occurs in the Siegel–Tatuzawa lower bound for the class number. We prove that, for $m, n \in {\mathbb{Z}}_{\gt0}$, there exists an effective constant $c(m, n)\gt0$ with the following property: if pairwise distinct singular moduli $x_1, \ldots, x_n$ with respective discriminants $\Delta_1, \ldots, \Delta_n$ are such that $a_1 x_1^m + \cdots + a_n x_n^m \in {\mathbb{Q}}$ for some $a_1, \ldots, a_n \in {\mathbb{Q}} \setminus \{0\}$ and $\# \{ \Delta_i \;:\; {\mathbb{Q}}(\sqrt{\Delta_i}) = K_*\} \leq 1$, then $\max_i \lvert \Delta_i \rvert \leq c(m, n)$. In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples $(x_1, x_2, x_3)$ of singular moduli such that $a_1 x_1 + a_2 x_2 + a_3 x_3 \in {\mathbb{Q}}$ for some $a_1, a_2, a_3 \in {\mathbb{Q}} \setminus \{0\}$.

We consider the countably many families $\mathcal {L}_d$, $d\in \mathbb {N}_{\geq 2}$, of K3 surfaces admitting an elliptic fibration with positive Mordell–Weil rank. We prove that the elliptic fibrations on the very general member of these families have the potential Mordell–Weil rank jump property for $d\neq 2,3$ and moreover the Mordell–Weil rank jump property for $d\equiv 3\ \mod 4$, $d\neq 3$. We provide explicit examples and discuss some extensions to subfamilies. The result is based on the geometric interaction between the (potential) Mordell–Weil rank jump property and the presence of special multisections of the fibration.

In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at hyperspecial level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in an earlier work. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via p-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at p. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers.

For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let $\pi :D \to \Gamma \backslash D = S$ realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of $\Gamma < G$, an arithmetic subgroup. Let $S' \subseteq S$ be a special subvariety of S realized as $\pi (D')$ for $D' \subseteq D$ a homogeneous space for an algebraic subgroup of G. Let $X \subseteq S$ be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with $\pi (gD')$ is persistently likely as g ranges through G with $\pi (gD')$ a special subvariety of S, meaning that whenever $\zeta :S_1 \to S$ and $\xi :S_1 \to S_2$ are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with $\zeta $ finite, $\dim \xi \zeta ^{-1} X + \dim \xi \zeta ^{-1} \pi (gD') \geq \dim \xi S_1$. Then $X \cap \bigcup _{g \in G, \pi (g D') \text { is special }} \pi (g D')$ is dense in X for the Euclidean topology.

Considérons un espace de Berkovich sur un bon anneau de Banach et la droite projective relative sur celui-ci. (C’est un espace dont les fibres sont des droites projectives sur différents corps valués complets.) Pour tout endomorphisme polarisé de cette droite, nous montrons que la famille des mesures d’équilibre associées aux restrictions de l’endomorphisme aux fibres est continue. Le résultat vaut, par exemple, lorsque l’anneau de Banach est un corps valué complet, un corps hybride, un anneau de valuation discrète complet ou un anneau d’entiers de corps de nombres.

The descent method is one of the approaches to study the Brauer–Manin obstruction to the local–global principle and to weak approximation on varieties over number fields, by reducing the problem to ‘descent varieties’. In recent lecture notes by Wittenberg, he formulated a ‘descent conjecture’ for torsors under linear algebraic groups. The present article gives a proof of this conjecture in the case of connected groups, generalizing the toric case from the previous work of Harpaz–Wittenberg. As an application, we deduce directly from Sansuc’s work the theorem of Borovoi for homogeneous spaces of connected linear algebraic groups with connected stabilizers. We are also able to reduce the general case to the case of finite (étale) torsors. When the set of rational points is replaced by the Chow group of zero-cycles, an analogue of the above conjecture for arbitrary linear algebraic groups is proved.

We give a construction of integral local Shimura varieties which are formal schemes that generalise the well-known integral models of the Drinfeld p-adic upper half spaces. The construction applies to all classical groups, at least for odd p. These formal schemes also generalise the formal schemes defined by Rapoport-Zink via moduli of p-divisible groups, and are characterised purely in group-theoretic terms.

More precisely, for a local p-adic Shimura datum $(G, b, \mu)$ and a quasi-parahoric group scheme ${\mathcal {G}} $ for G, Scholze has defined a functor on perfectoid spaces which parametrises p-adic shtukas. He conjectured that this functor is representable by a normal formal scheme which is locally formally of finite type and flat over $O_{\breve E}$. Scholze-Weinstein proved this conjecture when $(G, b, \mu)$ is of (P)EL type by using Rapoport-Zink formal schemes. We prove this conjecture for any $(G, \mu)$ of abelian type when $p\neq 2$, and when $p=2$ and G is of type A or C. We also relate the generic fibre of this formal scheme to the local Shimura variety, a rigid-analytic space attached by Scholze to $(G, b, \mu , {\mathcal {G}})$.