This paper is concerned with the conjectural correspondence between Galois representations and modular forms. Although such relation has been fully established in the case the Galois representation is realizable on the `-adic Tate module of an elliptic curve over Q (cf. [24],[20],[5]), very little is known in general. As a first step towards the understanding of more complicated cases, we consi...

4. Hermite Interpolants . . , . . . . . . . . . . . . . 104 4.1. The Co nine parameter interpolant ...... 104 4.2. C’ quintic interpolants .............. 104 4.3. The general case ................... 106 5. Split Triangle Interpolants . . . . . . . . . 107 5.1. The C’ Clough-Tocher interpolant . . _ . . 108 5.2. Limitations of the Clough-Tocher split . 110 5.3. The C’ Powell-Sabin interpolants ...

We describe here a remarkable relationship studied by Givental between hypergeometric series and the quantum cohomology of hypersurfaces in pro-jective space [G1]. As the quantum product involves genus 0 Gromov-Witten invariants, a connection between hypergeometric series and the geometry of rational curves on the hypersurfaces is made. While the most general context for such relationships has ...

We obtained center conditions for a O-symmetric system of degree 5 for which the origin is a uniformly isochronous singular point. Classification: Primary 34C05; Secondary 34C25

Let C be a nonsingular plane quintic curve over the complex number field C, and let πP : C → P be a projection from P ∈ C. Let LP be the Galois closure of the field extension C(C)/C(P) induced by πP , where C(C) and C(P) are the rational function fields of C and P, respectively. We call the point P a D4-point if the Galois group of LP /C(P) is isomorphic to the dihedral group D4 of order eight....

As a first step to a detailed study of orientifolds of Gepner models associated with CalabiYau manifolds, we construct crosscap states associated with anti-holomorphic involutions (with fixed points) of Calabi-Yau manifolds. We argue that these orientifolds are dual to M-theory compactifications on (singular) seven-manifolds with G2 holonomy. Using the spacetime picture as well as the M-theory ...

In this article we construct Lagrangian torus fibrations for general quintic Calabi-Yau hypersurfaces near the large complex limit and their mirror manifolds using gradient flowmethod. Then we prove the StromingerYau-Zaslow mirror conjecture for this class of Calabi-Yau manifolds in symplectic category.

Using a new regulator bound we determine unit groups and class groups of the 289040 quintic algebraic number fields with absolute discriminant less than 2 × 107 (totally real fields), respectively 5 × 106 (other signatures). We list significant data.

From the classical Voronoi algorithm, we derive an algorithm to classify quadratic positive definite forms by their minimal vectors; we define some new invariants for a class, for which several conjectures are proposed. Applying the algorithm to dimension 5 we obtain the table of the 136 classes in this dimension, we enumerate the 118 eutactic quintic forms, and we verify the Ash formula.

In this article we compute fundamental units for a family of number fields generated by a parametric polynomial of degree 5 with signature (1, 2) and Galois group D5.