We show that the image of the Abel–Jacobi map admits functorially a model over the field of definition, with the property that the Abel–Jacobi map is equivariant with respect to this model. The cohomology of this abelian variety over the base field is isomorphic as a Galois representation to the deepest part of the coniveau filtration of the cohomology of the projective variety. Moreover, we show that this model over the base field is dominated by the Albanese variety of a product of components of the Hilbert scheme of the projective variety, and thus we answer a question of Mazur. We also recover a result of Deligne on complete intersections of Hodge level 1.

Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

This paper sets up the foundations for derived algebraic geometry, Goerss–Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially) arbitrary model category. We show that one can do derived algebraic geometry a la Toën–Vezzosi in an abstract category of spectra. We also answer in the affirmative a question of Goerss and Hopkins by showing that the obstruction theory for operadic algebras in spectra can be done in the generality of spectra in an (essentially) arbitrary model category. We construct strictly commutative simplicial ring spectra representing a given cohomology theory and illustrate this with a strictly commutative motivic ring spectrum representing higher order products on Deligne cohomology. These results are obtained by first establishing Smith’s stable positive model structure for abstract spectra and then showing that this category of spectra possesses excellent model-theoretic properties: we show that all colored symmetric operads in symmetric spectra valued in a symmetric monoidal model category are admissible, i.e., algebras over such operads carry a model structure. This generalizes the known model structures on commutative ring spectra and $\text{E}_{\infty }$-ring spectra in simplicial sets or motivic spaces. We also show that any weak equivalence of operads in spectra gives rise to a Quillen equivalence of their categories of algebras. For example, this extends the familiar strictification of $\text{E}_{\infty }$-rings to commutative rings in a broad class of spectra, including motivic spectra. We finally show that operadic algebras in Quillen equivalent categories of spectra are again Quillen equivalent. This paper is also available at arXiv:1410.5699v2.

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then

$$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$

$Z$

$D$

$n$

$X$

$\mathscr{O}_{X}(\ast D)$

$n-3$

We construct a weak Hilbert space that is a twisted Hilbert space.

In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups to $\text{NTP}_{2}$ theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded pseudo real closed fields.

Let $\unicode[STIX]{x1D707}$ be the projection on $[0,1]$ of a Gibbs measure on $\unicode[STIX]{x1D6F4}=\{0,1\}^{\mathbb{N}}$ (or more generally a Gibbs capacity) associated with a Hölder potential. The thermodynamic and multifractal properties of $\unicode[STIX]{x1D707}$ are well known to be linked via the multifractal formalism. We study the impact of a random sampling procedure on this structure. More precisely, let $\{{I_{w}\}}_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ stand for the collection of dyadic subintervals of $[0,1]$ naturally indexed by the finite dyadic words. Fix $\unicode[STIX]{x1D702}\in (0,1)$, and a sequence $(p_{w})_{w\in \unicode[STIX]{x1D6F4}^{\ast }}$ of independent Bernoulli variables of parameters $2^{-|w|(1-\unicode[STIX]{x1D702})}$. We consider the (very sparse) remaining values $\widetilde{\unicode[STIX]{x1D707}}=\{\unicode[STIX]{x1D707}(I_{w}):w\in \unicode[STIX]{x1D6F4}^{\ast },p_{w}=1\}$. We study the geometric and statistical information associated with $\widetilde{\unicode[STIX]{x1D707}}$, and the relation between $\widetilde{\unicode[STIX]{x1D707}}$ and $\unicode[STIX]{x1D707}$. To do so, we construct a random capacity $\mathsf{M}_{\unicode[STIX]{x1D707}}$ from $\widetilde{\unicode[STIX]{x1D707}}$. This new object fulfills the multifractal formalism, and its free energy is closely related to that of $\unicode[STIX]{x1D707}$. Moreover, the free energy of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ generically exhibits one first order and one second order phase transition, while that of $\unicode[STIX]{x1D707}$ is analytic. The geometry of $\mathsf{M}_{\unicode[STIX]{x1D707}}$ is deeply related to the combination of approximation by dyadic numbers with geometric properties of Gibbs measures. The possibility to reconstruct $\unicode[STIX]{x1D707}$ from $\widetilde{\unicode[STIX]{x1D707}}$ by using the almost multiplicativity of $\unicode[STIX]{x1D707}$ and concatenation of words is discussed as well.

In this paper we construct a $\mathbb{Q}$-linear tannakian category $\mathsf{MEM}_{1}$ of universal mixed elliptic motives over the moduli space ${\mathcal{M}}_{1,1}$ of elliptic curves. It contains $\mathsf{MTM}$, the category of mixed Tate motives unramified over the integers. Each object of $\mathsf{MEM}_{1}$ is an object of $\mathsf{MTM}$ endowed with an action of $\text{SL}_{2}(\mathbb{Z})$ that is compatible with its structure. Universal mixed elliptic motives can be thought of as motivic local systems over ${\mathcal{M}}_{1,1}$ whose fiber over the tangential base point $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}q$ at the cusp is a mixed Tate motive. The basic structure of the tannakian fundamental group of $\mathsf{MEM}$ is determined and the lowest order terms of a set (conjecturally, a minimal generating set) of relations are deduced from computations of Brown. This set of relations includes the arithmetic relations, which describe the ‘infinitesimal Galois action’. We use the presentation to give a new and more conceptual proof of the Ihara–Takao congruences.

We give an algebraic description of the structure of the analytic universal cover of a complex abelian variety which suffices to determine the structure up to isomorphism. More generally, we classify the models of theories of ‘universal covers’ of rigid divisible commutative finite Morley rank groups.

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over $\overline{\mathbb{F}}_{p}$ of parallel weight 1 and level prime to $p$ is unramified above $p$. This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic $p$ embed into the ordinary part of parallel weight $p$ forms in two different ways per prime dividing $p$, namely via ‘partial’ Frobenius operators.

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

In this paper, we consider the Stokes equations in a perforated domain. When the number of holes increases while their radius tends to 0, it is proven in Desvillettes et al. [J. Stat. Phys. 131 (2008) 941–967], under suitable dilution assumptions, that the solution is well approximated asymptotically by solving a Stokes–Brinkman equation. We provide here quantitative estimates in $L^{p}$-norms of this convergence.

We prove a compactness theorem for metrics with bounded integral curvature on a fixed closed surface $\unicode[STIX]{x1D6F4}$. As a corollary we obtain a new convergence result for sequences of metrics with conical singularities, where an accumulation of singularities is allowed.

Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of $k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on $\text{GL}(k)$ associated with the maximal parabolic of type $(k-1,1)$, and the exact sup-norm exponent is determined to be $(k-2)/8$ for $k\geqslant 4$. In particular, if $k$ is odd, this exponent is not in $\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

Let $G$ be a semisimple Lie group with associated symmetric space $D$, and let $\unicode[STIX]{x1D6E4}\subset G$ be a cocompact arithmetic group. Let $\mathscr{L}$ be a lattice inside a $\mathbb{Z}\unicode[STIX]{x1D6E4}$-module arising from a rational finite-dimensional complex representation of $G$. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup $H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$ as $\unicode[STIX]{x1D6E4}_{k}$ ranges over a tower of congruence subgroups of $\unicode[STIX]{x1D6E4}$. In particular, they conjectured that the ratio $\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{k}]$ should tend to a nonzero limit if and only if $i=(\dim (D)-1)/2$ and $G$ is a group of deficiency $1$. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including $\operatorname{GL}_{n}(\mathbb{Z})$ for $n=3,4,5$ and $\operatorname{GL}_{2}(\mathscr{O})$ for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture.

Colmez [Périodes des variétés abéliennes a multiplication complexe, Ann. of Math. (2)138(3) (1993), 625–683; available at http://www.math.jussieu.fr/∼colmez] conjectured a product formula for periods of abelian varieties over number fields with complex multiplication and proved it in some cases. His conjecture is equivalent to a formula for the Faltings height of CM abelian varieties in terms of the logarithmic derivatives at $s=0$ of certain Artin $L$-functions. In a series of articles we investigate the analog of Colmez’s theory in the arithmetic of function fields. There abelian varieties are replaced by Drinfeld modules and their higher-dimensional generalizations, so-called $A$-motives. In the present article we prove the product formula for the Carlitz module and we compute the valuations of the periods of a CM $A$-motive at all finite places in terms of Artin $L$-series. The latter is achieved by investigating the local shtukas associated with the $A$-motive.