Assuming the generalised Riemann hypothesis (GRH) for CM-fields, we prove a conjecture of Yves André's asserting that a curve contained in a Shimura variety and containing an infinite set of special points is of Hodge type

We prove a motivic analogue of Steenbrink's conjecture [25, Conjecture 2.2] on the Hodge spectrum (proved by M. Saito in [21]). To achieve this, we construct and compute motivic iterated vanishing cycles associated with two functions. We are also led to introduce a more general version of the convolution operator appearing in the motivic Thom-Sebastiani formula. Throughout the article we use the framework of relative equivariant Grothendieck rings of varieties endowed with an algebraic torus action

We describe the cohomology groups of a homogeneous vector bundle $E$ on any Hermitian symmetric variety $X=G/P$ of ADE-type as the cohomology of a complex explicitly described. The main tool is the equivalence (introduced by Bondal, Kapranov, and Hille) between the category of homogeneous bundles and the category of representations of a certain quiver ${\mathcal{Q}}_X$ with relations. We prove that the relations are the commutative ones on projective spaces, but they involve additional scalars on general Grassmannians. In addition, we introduce moduli spaces of homogeneous bundles

We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma in conjunction with a result of Buzzard [Bu, Theorem 5.2] to give a proof of (a generalization of) Coleman's theorem, which states that overconvergent modular forms of small slope are classical. The proof is geometric in nature and is suitable for generalization to other Shimura varieties

In this article we show that if a stable-stationary harmonic map $u$ from a domain in ${\mathbb{R}}^4$ into ${\mathbb{S}}^3$ has an isolated singular point $\xi$, then the mapping degree $\mathrm{deg}(u,\xi )$ of $u$ at $\xi$ is $+1$ or $-1$. Furthermore, the complete characterization of stable-stationary tangent maps from ${\mathbb{B}}^4$ to ${\mathbb{S}}^3$ is given

For any $N\ge 2$, let $Z\subset {\mathbb{P}}^N$ be a geometrically integral algebraic variety of degree $d$. This article is concerned with the number $N_Z(B)$ of $\mathbb{Q}$-rational points on $Z$ which have height at most $B$. For any $\epsilon >0$, we establish the estimate ${\displaystyle N_Z(B)=O_{d,\epsilon ,N}(B^{\dim Z+\epsilon })}$, provided that $d,\epsilon$. As indicated, the implied constant depends at most on $d,\epsilon$, and $N$