We give a classification of all possible 2-adic images of Galois representations associated to elliptic curves over Q. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.Date: May 19, 2015.

Dedicated to the memory of Robert Coleman.Abstract. Let X be a curve of genus g ≥ 2 over a number field F of degree d = [F : Q]. The conjectural existence of a uniform bound N (g, d) on the number #X(F ) of F -rational points of X is an outstanding open problem in arithmetic geometry, known by Caporaso, Harris, and Mazur to follow from the Bombieri-Lang conjecture. A related conjecture posits the existence of a uniform bound N tors, † (g, d) on the number of geometric torsion points of the Jacobian J of X which lie on the image of X under an Abel-Jacobi map. For fixed X this quantity was conjectured to be finite by the Manin-Mumford conjecture, and was proved to be so by Raynaud.We give an explicit uniform bound on #X(F ) when X has Mordell-Weil rank r ≤ g − 3. This generalizes recent work of Stoll on uniform bounds for hyperelliptic curves of small rank to arbitrary curves. Using the same techniques, we give an explicit, unconditional uniform bound on the number of F -rational torsion points of J lying on the image of X under an Abel-Jacobi map. We also give an explicit uniform bound on the number of geometric torsion points of J lying on X when the reduction type of X is highly degenerate.Our methods combine Chabauty-Coleman's p-adic integration, non-Archimedean potential theory on Berkovich curves, and the theory of linear systems and divisors on metric graphs.

We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack X , which specializes to the Baytev-Manin conjecture when X is a scheme and to Malle's conjecture when X is the classifying stack of a finite group.

We consider the distribution of the Galois groups Gal(K un /K) of maximal unramified extensions as K ranges over Γ-extensions of Q or F q (t). We prove two properties of Gal(K un /K) coming from number theory, which we use as motivation to build a probability distribution on profinite groups with these properties. In Part I, we build such a distribution as a limit of distributions on n-generated profinite groups. In Part II, we prove as q → ∞, agreement of Gal(K un /K) as K varies over totally real Γ-extensions of F q (t) with our distribution from Part I, in the moments that are relatively prime to q(q − 1)|Γ|. In particular, we prove for every finite group Γ, in the q → ∞ limit, the prime-to-q(q − 1)|Γ|-moments of the distribution of class groups of totally real Γ-extensions of F q (t) agree with the prediction of the Cohen-Lenstra-Martinet heuristics.

Generalizing the classical theorems of Max Noether and Petri, we describe generators and relations for the canonical ring of a stacky curve, including an explicit Gröbner basis. We work in a general algebro-geometric context and treat log canonical and spin canonical rings as well. As an application, we give an explicit presentation for graded rings of modular forms arising from finite-area quotients of the upper half-plane by Fuchsian groups.