We give a full proof of the two dimensional Jacobian conjecture. We also give an algorithm to compute the inverse map of a polynomial map.

We give a proof of the two dimensional Jacobian conjecture. We also prove that if (F, G) is a Jacobian pair with deg y F ≥ 1, then F is a monic polynomial of y up to a scalar.

Using the p-adic uniformization of Shimura varieties we determine, for some of them, over which local fields they have rational points. Using this we show in some new curve cases that the jacobians are even in the sense of [PS].

We prove that there are exactly 149 genus two curves C defined over Q such that there exists a nonconstant morphism π : X1(N)→ C defined over Q and the jacobian of C is Q-isogenous to the abelian variety Af attached by Shimura to a newform f ∈ S2(Γ1(N)). We determine the corresponding newforms and present equations for all these curves.

Introduction and Review 2 1. Period domains and monodromy 4 2. Elliptic curves 8 3. Period mappings: an example 12 3.1. Asymptotics of the period map 14 4. Hodge structures of weight one 15 5. Hodge structures of weight two 18 6. Poincaré residues 20 7. Properties of the period mapping 24 8. The Jacobian ideal and the local Torelli theorem 25 9. The Horizontal Distribution Distance Decreasing P...

2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to `-adic representations 21 2.6 The isogen...

We confirm a conjecture of Merel describing a certain relation between the jacobians of various quotients of X(p) in terms of specific correspondences.

We prove non-vanishing results for the central value of high derivatives of the complete L-function Λ(f, s) attached to primitive forms of weight 2 and prime level q. For fixed k ≥ 0 the proportion of primitive forms f such that Λ(f, 1/2) 6= 0 is ≥ pk +o(1) with pk > 0 and pk = 1/2 + O(k−2), as the level q goes to infinity. This result is (asymptotically in k) optimal and analogous to a result ...