We study the $\overline {\mathbb {F}}_{p}$-points of the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level. We show that if the group is quasi-split, then every isogeny class contains the reduction of a CM point, proving a conjecture of Kisin–Madapusi–Shin. We, furthermore, show that the mod p isogeny classes are of the form predicted by the Langlands–Rapoport conjecture (cf. Conjecture 9.2 of [Rap05]) if either the Shimura variety is proper or if the group at p is unramified. The main ingredient in our work is a global argument that allows us to reduce the conjecture to the case of very special parahoric level. This case is dealt with in the Appendix by Zhou. As a corollary to our arguments, we determine the connected components of Ekedahl–Oort strata.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the ‘weakly adelic Mumford–Tate hypothesis’ and prove the generalised André–Pink–Zannier conjecture under this assumption, using Pila–Zannier strategy.

The attractor conjecture for Calabi–Yau moduli spaces predicts the algebraicity of the moduli values of certain isolated points picked out by Hodge-theoretic conditions. Using tools from transcendence theory, we provide a family of counterexamples to the attractor conjecture in almost all odd dimensions conditional on a specific case of the Zilber–Pink conjecture in unlikely intersection theory; these Calabi–Yau manifolds were first studied by Dolgachev. We also give constructions of new families of Calabi–Yau varieties, analogous to the mirror quintic family, with all middle Hodge numbers equal to one, which would also give counterexamples to the attractor conjecture.

Let $X$ be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring $R$, and $\mathfrak {m}$ a modulus on $X$, given by a closed subscheme of $X$ which is geometrically reduced. The generalized Jacobian $J_\mathfrak {m}$ of $X$ with respect to $\mathfrak {m}$ is then an extension of the Jacobian of $X$ by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of $X$ over $R$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves $X_0(N)$ with moduli supported on the cusps.

We prove the compatibility of local and global Langlands correspondences for $\operatorname {GL}_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne [10] and Scholze [18]. More precisely, let $r_p(\pi )$ denote an n-dimensional p-adic representation of the Galois group of a CM field F attached to a regular algebraic cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n(\mathbb {A}_F)$. We show that the restriction of $r_p(\pi )$ to the decomposition group of a place $v\nmid p$ of F corresponds up to semisimplification to $\operatorname {rec}(\pi _v)$, the image of $\pi _v$ under the local Langlands correspondence. Furthermore, we can show that the monodromy of the associated Weil-Deligne representation of $\left .r_p(\pi )\right |{}_{\operatorname {Gal}_{F_v}}$ is ‘more nilpotent’ than the monodromy of $\operatorname {rec}(\pi _v)$.

We prove the existence of $\mathrm {GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of ${\mathrm {GSO}}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb {H}}$, arising from forms of ${\mathrm {GSO}}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (half) spin L-functions and improve on the construction of ${\mathrm {SO}}_{2n}$-valued Galois representations by removing the outer automorphism ambiguity.

We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $\ell $-adic cohomology is Galois invariant for all $\ell $.

For an unramified reductive group, we determine the connected components of affine Deligne–Lusztig varieties in the affine flag variety. Based on work of Hamacher, Kim, and Zhou, this result allows us to verify, in the unramified group case, the He–Rapoport axioms, the almost product structure of Newton strata, and the precise description of isogeny classes predicted by the Langlands–Rapoport conjecture, for the Kisin–Pappas integral models of Shimura varieties of Hodge type with parahoric level structure.

In this paper, we propose a modified Kudla–Rapoport conjecture for the Krämer model of unitary Rapoport–Zink space at a ramified prime, which is a precise identity relating intersection numbers of special cycles to derivatives of Hermitian local density polynomials. We also introduce the notion of special difference cycles, which has surprisingly simple description. Combining this with induction formulas of Hermitian local density polynomials, we prove the modified Kudla–Rapoport conjecture when $n=3$. Our conjecture, combining with known results at inert and infinite primes, implies the arithmetic Siegel–Weil formula for all non-singular coefficients when the level structure of the corresponding unitary Shimura variety is defined by a self-dual lattice.

We give a new proof of Faltings's $p$-adic Eichler–Shimura decomposition of the modular curves via Bernstein–Gelfand–Gelfand (BGG) methods and the Hodge–Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was used in the original proof. Then we construct overconvergent Eichler–Shimura maps for the modular curves providing ‘the second half’ of the overconvergent Eichler–Shimura map of Andreatta, Iovita and Stevens. We use higher Coleman theory on the modular curve developed by Boxer and Pilloni to show that the small-slope part of the Eichler–Shimura maps interpolates the classical $p$-adic Eichler–Shimura decompositions. Finally, we prove that overconvergent Eichler–Shimura maps are compatible with Poincaré and Serre pairings.

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g>1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$.

We present a Mordell–Weil sieve that can be used to compute points on certain bielliptic modular curves $X_0(N)$ over fixed quadratic fields. We study $X_0(N)(\mathbb {Q}(\sqrt {d}))$ for $N \in \{ 53,61,65,79,83,89,101,131 \}$ and ${\lvert d \rvert < 100}$.

In this paper, we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well-established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit, such as the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM-points.

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 generalize the works of Pappas–Rapoport–Zhu on twisted affine Grassmannians to the wildly ramified case under mild assumptions. This rests on a construction of certain smooth affine $\mathbb {Z}[t]$ -groups with connected fibers of parahoric type, motivated by previous work of Tits. The resulting $\mathbb {F}_p(t)$ -groups are pseudo-reductive and sometimes non-standard in the sense of Conrad–Gabber–Prasad, and their $\mathbb {F}_p [\hspace {-0,5mm}[ {t} ]\hspace {-0,5mm}] $ -models are parahoric in a generalized sense. We study their affine Grassmannians, proving normality of Schubert varieties and Zhu’s coherence theorem.

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.

Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline {K}}$ has infinitely many rational curves or X has infinitely many unirational specialisations.

Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.

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 $X_H$ .

We use the method of Bruinier–Raum to show that symmetric formal Fourier–Jacobi series, in the cases of norm-Euclidean imaginary quadratic fields, are Hermitian modular forms. Consequently, combining a theorem of Yifeng Liu, we deduce Kudla’s conjecture on the modularity of generating series of special cycles of arbitrary codimension for unitary Shimura varieties defined in these cases.