Consider a Shimura curve $X^D_0(N)$ over the rational numbers. We determine criteria for the twist by an Atkin-Lehner involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan-Livné on $\mathbf{Q}_p$ points when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$ points when $p\mid N$. We also give congruence conditions for roots modulo $p$ of Hilbert class polynomials.

The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.

We say that an abelian variety over a $p$-adic field $K$ has anisotropic reduction (AR) if the special fiber of its N\'eron minimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the $K$-rational torsion subgroup of a $g$-dimensional AR variety depending only on $g$ and the numerical invariants of $K$ (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of $g$, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

We determine which isogeny classes of supersingular abelian surfaces over a finite field $k$ of characteristic $2$ contain jacobians. We deal with this problem in a direct way by computing explicitly the zeta function of all supersingular curves of genus $2$. Our procedure is constructive, so that we are able to exhibit curves with prescribed zeta function and find formulas for the number of curves, up to $k$-isomorphism, leading to the same zeta function.

Generically, one can attach to a $\mathbf{Q}$-curve $C$ octahedral representations $\rho\colon\Gal(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow\GL_2(\bar\mathbf{F}_3)$ coming from the Galois action on the $3$-torsion of those abelian varieties of $\GL_2$-type whose building block is $C$. When $C$ is defined over a quadratic field and has an isogeny of degree $2$ to its Galois conjugate, there exist such representations $\rho$ having image into $\GL_2(\mathbf{F}_9)$. Going the other way, we can ask which $\mod 3$ octahedral representations $\rho$ of $\Gal(\bar\mathbf{Q}/\mathbf{Q})$ arise from $\mathbf{Q}$-curves in the above sense. We characterize those arising from quadratic $\mathbf{Q}$-curves of degree $2$. The approach makes use of Galois embedding techniques in $\GL_2(\mathbf{F}_9)$, and the characterization can be given in terms of a quartic polynomial defining the $\mathcal{S}_4$-extension of $\mathbf{Q}$ corresponding to the projective representation $\bar{\rho}$.