The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser–Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco–Shalen–Johannson decomposition of $M$.

We show that for the primes l = 2, 3, 5, 7 or 13, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ that have good reduction at every prime different from l and are semi-stable at l. We show that any semi-stable abelian variety over $\mathbb{Q}$ with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X0(11). In addition, we show that for l = 2, 3 and 5, there do not exist any non-zero abelian varieties over $\mathbb{Q}$ with good reduction outside l that acquire semi-stable reduction at l over a tamely ramified extension.

We construct general type surfaces in mixed characteristic whose geometric genera can be made to jump by an arbitrarily prescribed positive amount under specialization. We then show that this phenomenon of jumping geometric genus presents itself in some compact Shimura surfaces. Finally, we find a set of conditions, met by the latter Shimura surfaces, that forces the higher plurigenera to remain constant in reduction modulo p.

We prove that an automorphism of the group of birational transformations of the complex projective plane is the composition of an interior automorphism and an automorphism of the field $\mathbb{C}$. The proof is based on a study of maximal abelian subgroups of the Cremona group.

Let X be an algebraic variety and let f:X−−→X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

We show that the automorphic étale cohomology of a (possibly noncompact) PEL-type or Hodge-type Shimura variety in characteristic zero is canonically isomorphic to the cohomology of the associated nearby cycles over most of their mixed characteristics models constructed in the literature.

We explicitly solve a collection of binomial Thue equations with unknown degree and unknown $S$-unit coefficients, for a number of sets $S$ of small cardinality. Equivalently, we characterize integers $x$ such that the polynomial $x^2+x$ assumes perfect power values, modulo $S$-units. These results are proved through a combination of techniques, including Frey curves and associated modular forms, lower bounds for linear forms in logarithms, the hypergeometric method of Thue and Siegel, local methods, and computational approaches to Thue equations of low degree. Along the way, we derive some new results on Fermat-type ternary equations, combining classical cyclotomy with Frey curve techniques.

An abelian cover is a finite morphism X→Y of varieties which is the quotient map for a generically faithful action of a finite abelian group G. Abelian covers with Y smooth and X normal were studied in [R. Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math. 417 (1991), 191–213; MR 1103912(92g:14012)]. Here we study the non-normal case, assuming that X and Y are S2 varieties that have at worst normal crossings outside a subset of codimension greater than or equal to two. Special attention is paid to the case of ℤr2-covers of surfaces, which is used in [V. Alexeev and R. Pardini, Explicit compactifications of moduli spaces of Campedelli and Burniat surfaces, Preprint (2009), math.AG/arXiv:0901.4431] to construct explicitly compactifications of some components of the moduli space of surfaces of general type.

We study deformation rings of an n-dimensional representation $\overline{\rho}$, defined over a finite field of characteristic $\ell$, of the arithmetic fundamental group $\pi_1(X)$, where X is a geometrically irreducible, smooth curve over a finite field k of characteristic p ($ \neq \ell$). When $\overline{\rho}$ has large image, we are able to show that the resulting rings are finite flat over $\mathbf{Z}_\ell$. The proof principally uses a Galois-theoretic lifting result of the authors in Part I of this two-part work, a lifting result for cuspidal mod $\ell$ forms of Ogilvie, Taylor–Wiles systems and the result of Lafforgue. This implies a conjecture of de Jong for representations of $\pi_1(X)$ with coefficients in power series rings over finite fields of characteristic $\ell$, that have this mod $\ell$ representation $\overline{\rho}$ as their reduction. A proof of all cases of the conjecture for $\ell>2$ follows from a result announced by Gaitsgory. The methods are different.

On Shimura varieties of orthogonal type over totally real fields, we prove a product formula and the modularity of Kudla’s generating series of special cycles in Chow groups.

Let G be a connected, simply connected, simple complex algebraic group and let ϵ be a primitive ℓth root of one, ℓ odd and 3∤ℓ if G is of type G2. We determine all Hopf algebra quotients of the quantized coordinate algebra 𝒪ϵ(G).

Given two Lie 2-groups, we study the problem of integrating a weak morphism between the corresponding Lie 2-algebras to a weak morphism between the Lie 2-groups. To do so, we develop a theory of butterflies for 2-term L∞-algebras. In particular, we obtain a new description of the bicategory of 2-term L∞-algebras. An interesting observation here is that the role played by 1-connected Lie groups in Lie theory is now played by 2-connected Lie 2-groups. Using butterflies, we also give a functorial construction of 2-connected covers of Lie 2 -groups. Based on our results, we expect that a similar pattern generalizes to Lie n-groups and Lie n-algebras.

We prove that a refinement of Stark‘s Conjecture formulated by Rubin in Ann. Inst Fourier 4 (1996) is true up to primes dividing the order of the Galois group, for finite, Abelian extensions of function fields over finite fields. We also show that in the case of constant field extensions, a statement stronger than Rubin‘s holds true.

Conditionally on the generalized Lindelöf hypothesis, we obtain an asymptotic for the fourth moment of Hecke–Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the random wave conjecture.

Let k be a field and X = Spec (k[t,t−1]). Katz proved that a differential equations with coefficients in k((t−1)) is uniquely extended to a special algebraic differential equation on X when k is of characteristic 0. He also proved that a finite extension of k((t−1)) is uniquely extended to a special covering of X when k is of any characteristic. These theorems are called canonical extension or Katz correspondence. We shall prove a p-adic analogue of canonical extension for quasi-unipotent overconvergent isocrystals. As a consequence, we can show that the local index of a quasi-unipotent overconvergent is equal to its Swan conductor.

We classify log-canonical pairs $(X,{\rm\Delta})$ of dimension two such that $K_{X}+{\rm\Delta}$ is an ample Cartier divisor with $(K_{X}+{\rm\Delta})^{2}=1$, giving some applications to stable surfaces with $K^{2}=1$. A rough classification is also given in the case where ${\rm\Delta}=0$.

In this paper, we study the orthogonalities of Hecke eigenvalues of holomorphic cusp forms. An asymptotic large sieve with an unusually large main term for cusp forms is obtained. A family of special vectors formed by products of Kloosterman sums and Bessel functions is constructed for which the main term is exceptionally large. This surprising phenomenon reveals an interesting fact: that Fourier coefficients of cusp forms favor the direction of products of Kloosterman sums and Bessel functions of compatible type.

A notion of Hochschild cohomology HH*(𝒜) of an abelian category 𝒜 was defined by Lowen and Van den Bergh (Adv. Math. 198 (2005), 172–221). They also showed the existence of a characteristic morphism χ from the Hochschild cohomology of 𝒜 into the graded centre ℨ*(Db(𝒜)) of the bounded derived category of 𝒜. An element c∈HH2(𝒜) corresponds to a first-order deformation 𝒜c of 𝒜 (Lowen and Van den Bergh, Trans. Amer. Math. Soc. 358 (2006), 5441–5483). The problem of deforming an object M∈Db(𝒜) to Db(𝒜c) was treated by Lowen (Comm. Algebra 33 (2005), 3195–3223). In this paper we show that the element χ(c)M∈Ext𝒜2(M,M) is precisely the obstruction to deforming M to Db(𝒜c). Hence, this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a ‘derived deformation theory’.

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.