The arithmetic of Jacobian groups of superelliptic cubics

International audienceWe present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor's reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data

Quantum field theory on projective modules

We propose a general formulation of perturbative quantum field theory on (finitely generated) projective modules over noncommutative algebras. This is the analogue of scalar field theories with non-trivial topology in the noncommutative realm. We treat in detail the case of Heisenberg modules over noncommutative tori and show how these models can be understood as large rectangular pxq matrix models, in the limit p/q->theta, where theta is a possibly irrational number. We find out that the modele is highly sensitive to the number-theoretical aspect of theta and suffers from an UV/IR-mixing. We give a way to cure the entanglement and prove one-loop renormalizability.Comment: 52 pages, uses feynm

Real determinant line bundles

This article is an expanded version of the talk given by Ch. O. at the Second Latin Congress on "Symmetries in Geometry and Physics" in Curitiba, Brazil in December 2010. In this version we explain the topological and gauge-theoretical aspects of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces".Comment: LaTeX, 8 page

Improved Sobolev Embedding Theorems for Vector-valued Functions

The aim of this paper is to give an extension of the improved Sobolev embedding theorem for single-valued functions to the case of vector-valued functions which is involved with the three-dimensional massless Dirac operator together with the three- or two-dimensional Weyl--Dirac (or Pauli) operator, the Cauchy--Riemann operator and also the four-dimensional Euclidian Dirac operator.Comment: 40 pages. To appear in Funkcialaj Ekvacioj 57 (2014