Version 2.0 March 16, 2008These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form.

A 2-group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2-categories of weak and coherent 2-groups and an ‘improvement ’ 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the ‘fundamental 2-group ’ of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2-groups are built using Chern–Simons theory, and are closely related to the Lie 2-algebras g � ( � ∈ R) described in a companion paper. 1 1

Would a reader be able to predict the branch of mathematics that is the subject of this article if its title had not included the phrase “in Number Theory”? The distinction “local ” versus “global”, with various connotations, has found a home in almost every part of mathematics, local problems being often a stepping-stone to

Electromagnetism can be generalized to Yang–Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of ‘higher-dimensional Yang–Mills theory’. It turns out that to do this, one should replace the Lie group by a ‘Lie 2-group’, which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a ‘Lie crossed module’: a pair of Lie groups G, H with a homomorphism t: H → G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing’s ideas on the geometry of nonabelian gerbes, one can define ‘principal 2-bundles ’ for any Lie 2-group C and do gauge theory in this new context. Here we only consider trivial 2-bundles, where a connection consists of a g-valued 1-form together with an h-valued 2-form, and its curvature consists of a g-valued 2-form together with a h-valued 3-form. We generalize the Yang–Mills action for this sort of connection, and use this to derive ‘higher Yang– Mills equations’. Finally, we show that in certain cases these equations admit self-dual solutions in five dimensions. 1

Abstract We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program. For a smooth projective curve X we introduce an algebraic stack ˜ BunG of metaplectic bundles on X. It also has a local version ˜ GrG, which is a gerbe over the affine grassmanian of G. We define a categorical version of the (nonramified) Hecke algebra of the metaplectic group. This is a category Sph ( ˜ GrG) of certain perverse sheaves on ˜ GrG, which act on ˜ BunG by Hecke operators. A version of the Satake equivalence is proved describing Sph ( ˜ GrG) as a tensor category. Further, we construct a perverse sheaf on ˜ BunG corresponding to the Weil representation and show that it is a Hecke eigen-sheaf with respect to Sph ( ˜ GrG). 1.

this article, but beware of misprints. A more thorough treatment is given in May, [18]. We will use (standard) notation from [18] wherever possible. The way found initially around the restriction that K had to be reduced in the above loop construction was to take a maximal tree in K and to contract it to a point. In 1984, a groupoid version of the loop group construction was given by Dwyer and Kan, [12]. (Unfortunately the published paper has many misprints and the cleaned-up version that we will use was prepared by my student Phil Ehlers as part of his master's dissertation, [13]. Alternatives have been proposed by Joyal and Tierney, and by Moerdijk and Svensson. They end up with simplicial objects in the category of groupoids, whilst the Dwyer - Kan version gives a simplicially enriched groupoid, i.e. a groupoid all of whose Hom-objects are simplicial sets. A simplicially enriched groupoid is also a simplicial groupoid (simplicial object in the category of groupoids), but is one whose object of objects is a constant simplicial set.) Let SS denote the category of simplicial sets and SGpds that of simplicially enriched groupoids or as we will often call them, simply, simplicial groupoids. The loop groupoid functor is a functor

I categorify the definition of fibre bundle, replacing smooth manifolds with differentiable categories, Lie groups with coherent Lie 2-groups, and bundles with a suitable notion of 2-bundle. To link this with previous work, I show that certain 2-categories of principal 2-bundles are equivalent to certain 2-categories of (nonabelian) gerbes. This relationship can be (and has been) extended to connections on 2-bundles and gerbes. The main theorem, from a perspective internal to this paper, is that the 2-category of 2-bundles over a given 2-space under a given 2-group is (up to equivalence) independent of the fibre and can be expressed in terms of cohomological data (called 2-transitions). From the perspective of linking to previous work on gerbes, the main theorem is that when the 2-space is the 2-space corresponding to a given space and the 2-group is the automorphism 2-group of a given group, then this 2-category is equivalent to the 2-category of gerbes over that space under that group (being described by the same cohomological data).

Abstract. Let X be a smooth projective algebraic curve of genus> 1 over and algebraically closed field k of characteristic p> 0. Denote by Bunn (resp. Locn) the moduli stack of vector bundles of rank n on X (resp. the moduli stack of vector bundles of rank n endowed with a connection). Let also DBunn denote the sheaf of crystalline differential operators on Bunn (cf. e.g. [3]). In this paper we construct an equivalence Φn between the

this technical result to determine the exact structure of Pic(MG ) where G = SL r = s (theorem 5.6). I would like to thank L. Breen to have taught me both the notion of torsor and linearization of a vector bundle in the set-up of group-stack action and for his comments on a preliminary version of this paper.

The object of this paper is the notion of r-spin structure: a line bundle whose rth power is isomorphic to the canonical bundle. Over the moduli functor Mg of smooth genus-g curves, r-spin structures form a finite torsor under the group of r-torsion line bundles. Over the moduli functor Mg of stable curves, r-spin structures form an étale stack, but the finiteness and the torsor structure are lost. In the present work, we show how this bad picture can be definitely improved simply by placing the problem in the category of Abramovich and Vistoli’s twisted curves. First, we find that within such category there exist several different compactifications of Mg; each one corresponds to a different multiindex ⃗ l = (l0, l1,...) identifying a notion of stability: ⃗ l-stability. Then, we determine the suitable choices of ⃗ l for which r-spin structures form a finite torsor over the moduli of ⃗ l-stable curves. 1