Modular representations on some Riemann-Roch spaces of modular curves X(N)

TL;DR

This paper analyzes the modular representation structure of Riemann-Roch spaces on modular curves X(N), providing explicit computations and applications to algebraic geometry codes, with examples for N=7 and 11.

Contribution

It explicitly determines the PSL(2,N)-module structure of Riemann-Roch spaces on X(N) and explores applications to algebraic geometry codes.

Findings

Explicit module structure for N=7, 11 computed

Ramification module explicitly described

Applications to AG codes demonstrated

Abstract

We compute the PSL(2,N)-module structure of the Riemann-Roch space L(D), where D is an invariant non-special divisor on the modular curve X(N), with N > 5 prime. This depends on a computation of the ramification module, which we give explicitly. These results hold for characteristic p if X(N) has good reduction mod p and p does not divide the order of PSL(2,N). We give as examples the cases N=7, 11, which were also computed using GAP. Applications to AG codes associated to this curve are considered, and specific examples are computed using GAP and MAGMA.