Code construction on modular curves

Abstract

In this thesis, we have introduced two approaches on code construction on modular curves and stated the problems step by step. Moreover, we have given solutions of some problems in road map of code construction. One of the approaches uses mostly geometric and algebraic tools. This approach studies local invariants of the plane model Z0(`) of the modular curve Y0(`) given by the modular equation Φ` in affine coordinates. The approach is based on describing the hyperplane of regular differentials of Z0(`) vanishing at a given Fp 2 rational point. As constructing a basis for the regular differentials of Z0(`), we need to investigate its singularities. We have described the singularities of Z0(`) for prime ` in both characteristic 0 and positive characteristic. We have shown that all singularities of of the affine part, Z0(`), are self intersections. These self intersections are all simple nodes in characteristic 0 whereas the order of contact of any two smooth branches passing though a singular point may be arbitrarily large in characteristic p > 3 where p 6= `. Moreover the self intersections in characteristic zero are double. Indeed, structure of singularities of the affine curve Z0(`) essentially depends on two types of elliptic curves: The singularities corresponding to ordinary elliptic curves and the singularities corresponding to supersingular elliptic curves. The singularities corresponding to ordinary elliptic curves are all double points even though they are not necessarily simple nodes as in the case of characteristic 0. The singularities corresponding to supersingular elliptic curves are the most complicated ones and it may happen that there are more then two smooth branches passing though such kind of a singular point. We have computed the order of contact of any two smooth branches passing though a singular point both for ordinary case and for supersingular case.We have also proved that two points of Z0(`) at ∞ are cusps for odd prime ` which are analytically equivalent to the cusp of 0, given by the equation x ` = y `−1 . These two cusps are permuted by Atkin-Lehner involution. The multiplicity of singularity of each cusp is (`−1)(`−2) 2 . This result is valid in any characteristic p 6= 2, 3. The second approach is based on describing the Goppa codes on modular curve Y (`) as P SL2(F`) module. The main problem in this approach is investigating the structure of a group code as P SL2(F`) module. We propose a way of computing the characters of representations of a group code by using the localization formula. Moreover, we give an example of computing the characters of the code which associated to a canonical divisor on Y (`).