Amy Ksir

Automorphism groups of some AG codes

In this correspondence, we show that in many cases, the automorphism group of a curve and the permutation automorphism group of a corresponding AG code are the same. This generalizes a result of Wesemeyer beyond the case of planar curves.

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

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.

Codes from Riemann-Roch spaces for y 2 = x p - x over GF(p)

Let Χ denote the hyperelliptic curve y 2 = x p - x over a field F of characteristic p. The automorphism group of Χ is G = PSL(2, p). Let D be a G-invariant divisor on Χ(F). We compute explicit F-bases for the Riemann-Roch space of D in many cases as well as G-module decompositions. AG codes with good parameters and large automorphism group are constructed as a result. Numerical examples using GAP and SAGE are also given.

Enumerative Algebraic Geometry of Conics

In this expository paper, we describe the solutions to several enumerative problems involving conies, including Steiners problem. The results and techniques presented here are not new; rather, we use these problems to introduce and demonstrate several of the key ideas and tools of algebraic geometry. The problems we discuss are the following: Given p points, / lines, and c conies in the plane, how many conies are there that contain the given points, are tangent to the given lines, and are tangent to the given conies? It is not even clear a priori that these questions are well-posed. The answers may depend on which points, lines, and conies we are given. Nineteenth and twentieth century geometers struggled to make sense of these questions, to show that with the proper interpretation they admit clean answers, and to put the subject of enumerative algebraic geometry on a firm mathematical foundation. Indeed, Hilbert made this endeavor the subject of his fifteenth challenge problem.

Suzuki-invariant codes from the Suzuki curve

In this paper we consider the Suzuki curve

Automorphism groups on tropical curves: some cohomology calculations

y^q + y = x^{q_0}(x^q + x)

Automorphisms of tropical curves and Berkovich analytic curves and their skeletons

yq+y=xq0(xq+x) over the field with