Diagrams in the mod p cohomology of Shimura curves

We prove a local–global compatibility result in the mod $p$ Langlands program for $\mathrm {GL}_2(\mathbf {Q}_{p^f})$ . Namely, given a global residual representation $\bar {r}$ appearing in the mod $p$ cohomology of a Shimura curve that is sufficiently generic at $p$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $p$ completed cohomology is determined by the restrictions of $\bar {r}$ to decomposition groups at $p$ . If these restrictions are moreover semisimple, we show that the $(\varphi ,\Gamma )$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $\bar {r}$ to decomposition groups at $p$ .

CM cycles on Kuga–Sato varieties over Shimura curves and Selmer groups

Given a modular form{{f}}of even weight larger than two and an imaginary quadratic field{{K}}satisfying a relaxed Heegner hypothesis, we construct a collection of CM cycles on a Kuga–Sato variety over a suitable Shimura curve which gives rise to a system of Galois cohomology classes attached to{{f}}enjoying the compatibility properties of an Euler system. Then we use Kolyvagin’s method [21], as adapted by Nekovář [28] to higher weight modular forms, to bound the size of the relevant Selmer group associated to{{f}}and{{K}}and prove the finiteness of the (primary part) of the Shafarevich–Tate group, provided that a suitable cohomology class does not vanish.

Abelian Surfaces with an Automorphism and Quaternionic Multiplication

We construct one-dimensional families of Abelian surfaces with quaternionic multiplication, which also have an automorphism of order three or four. Using Barth's description of the moduli space of (2,4)- polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space that parametrizes Abelian surfaces with real multiplication by .

ON THE NON-TRIVIALITY OF THE -ADIC ABEL–JACOBI IMAGE OF GENERALISED HEEGNER CYCLES MODULO , II: SHIMURA CURVES

Generalised Heegner cycles are associated to a pair of an elliptic newform and a Hecke character over an imaginary quadratic extension $K/\mathbf{Q}$. The cycles live in a middle-dimensional Chow group of a Kuga–Sato variety arising from an indefinite Shimura curve over the rationals and a self-product of a CM abelian surface. Let $p$ be an odd prime split in $K/\mathbf{Q}$. We prove the non-triviality of the $p$-adic Abel–Jacobi image of generalised Heegner cycles modulo $p$ over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$. The result implies the non-triviality of the generalised Heegner cycles in the top graded piece of the coniveau filtration on the Chow group, and proves a higher weight analogue of Mazur’s conjecture. In the case of weight 2, the result provides a refinement of the results of Cornut–Vatsal and Aflalo–Nekovář on the non-triviality of Heegner points over the $\mathbf{Z}_{p}$-anticyclotomic extension of $K$.

THE SHIMURA CURVE OF DISCRIMINANT 15 AND TOPOLOGICAL AUTOMORPHIC FORMS

We find defining equations for the Shimura curve of discriminant 15 over $\mathbb{Z}[1/15]$. We then determine the graded ring of automorphic forms over the 2-adic integers, as well as the higher cohomology. We apply this to calculate the homotopy groups of a spectrum of ‘topological automorphic forms’ associated to this curve, as well as one associated to a quotient by an Atkin–Lehner involution.

Twists of Shimura Curves

Consider a Shimura curve XD0(N) over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on Qp points when p |D and for the first time give criteria for Qp points when p |N. We also give congruence conditions for roots modulo p of Hilbert class polynomials.

Computing fundamental domains for the Bruhat–Tits tree for , -adic automorphic forms, and the canonical embedding of Shimura curves

We describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.