## Code construction on modular curves

##### Author

Kara, Orhun

##### Advisor

Klyachko, Alexander

##### Date

2003##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

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Show full item record##### 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 (`).

##### Keywords

Modular curveelliptic curve

Goppa codes

isogeny

endomorphism ring

singularity

self intersection

supersingular elliptic curve

reduction

lifting

cusp

representations

characters