Let X be a nonsingular complex variety and D a reduced effective divisor in X. In this paper we study the conditions under which the formula cSM (1U ) = c(DerX(− logD)) ∩ [X] is true. We prove that this formula is equivalent to a RiemannRoch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singul...

We summarise recent advances in techniques for solving Diophantine problems on hyperelliptic curves; in particular, those for finding the rank of the Jacobian, and the set of rational points on the curve.

Representations of the wavefunction of the multiparticle Schrödinger equation using an unconstrained sum of Slater determinants have the potential to be very efficient, except that they cannot capture the interelectron cusp. To remedy this shortcoming, we extend these representations to include interelectron geminals and analyze the algorithmic implications of doing so.

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...

Reflection algebras is a class of algebras associated with integrable models with boundaries. The coefficients of Sklyanin determinant generate the center of the reflection algebra. We give a combinatorial description of Sklyanin determinant suitable for explicit computations.

The structures and distributions of light nuclei are investigated within a microscopic correlation model. Two particle correlations are responsible for the scattering of model particles either to low momentum-or to high momentum-states. The low momentum states form the model space while the high momentum states are used to calculate the G-matrix. The three and higher order particle correlations...

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...