Applications of Computer Algebra 2014 Special Session on Arithmetic Geometry July 9  12, Fordham University New York, New York
OrganizersOverview
Arithmetic geometry is the study of the solutions in k^n of a system of polynomials in n variables with coefficients in a ring k such that k=Z, Q, Z/pZ, or a Dedekind domain. The subject is a combination of algebraic number theory, commutative algebra, and algebraic geometry. During the last 3040 years the subject has seen many developments both theoretical and computational. In this section we intend to bring together experts in the area provide survey talks on the subject and recent developments. Computational aspects of arithmetic geometry and applications in cryptography and coding theory will be encouraged.
Topics of the session include, but are not limited to: Integral extensions and integral closure
 Algebraic curves
 Jacobians of algebraic curves, rational torsion points in the Jacobian etc.
 Computational number theory, rational points on curves,
 Curves defined over Q
 Minimal discriminants and conductors
 Selmer groups in Jacobians
 Arithmetic invariant theory
 The arithmetic of hyperelliptic curves
 Pairings and Weil descent
 MordellWeil group
 Arithmetic Mirror Symmetry
 others
Talks:  L. Beshaj: Heights on algebraic curves (2 lectures)
 B. Trager: Good Reduction of Plane Curves
 J. Gutierrez, Recovering zeroes of hyperelliptic curves over finite fields
 A. Elezi: Quantum computing, error correcting quantum codes and algebraic curves.
 C. Shor: qWeierstrass point of hyperelliptic curves
 T. Shaska: Minimal models for superelliptic curves over their minimal field of definition
 F. Thompson: NeronTate heights for curves (cancelled)
 N. Pjero: Constructing minimal models for hyperelliptic curves. (cancelled)
Schedule
Friday afternoon:
 14:0015:00: L. Beshaj, Heights on algebraic curves
 15:0015:30: T. Shaska: Minimal models for superelliptic curves over their minimal field of definition
 15:3016:00: B. M. Trager, Good Reduction of Plane Curves
Saturday morning
 11:0011:30: A. Elezi: Quantum computing, error correcting quantum codes and algebraic curves.
 11:3012:00: C. Shor: qWeierstrass point of hyperelliptic curves
 12:0012:30: J. Gutierrez, Recovering zeroes of hyperelliptic curves over finite fields
If you are intending to give a talk, please send a title/abstract to T. Shaska (shaska@oakland.edu) and fill the following form.
Abstracts:
J. Gutierrez (University of Cantabria) Title: Recovering zeroes of hyperelliptic curves over finite fields
A. Elezi (American University) Title: Quantum computing, error correcting quantum codes and algebraic curves. Abstract: In this talk we give a mathematical introduction to quantum computing with emphasis on error correcting quantum codes. We will explore the construction of quantum stabilizer codes via self orthogonal classic codes. Self orthogonal Goppa codes arising from algebraic curves are particularly useful in this construction and will be treated in detail.
L. Beshaj (Oakland University)
Title: Heights on hyperelliptic and superelliptic curves
Abstract: We discuss heights of polynomials, heights of algebraic curves, and moduli height and determine relations among them. Furthermore, we use Segre embedding to determine relations among different heights for curves of small genii.
C. Shor (Western New England University)
Title: qWeierstrass point of hyperelliptic curves
F. Thompson (Oakland University) Title: NeronTate heights for curves
Abstract: We will briefly define the NeronTate heights on Abelian varieties and describe an algorithm of Holmes for computing such heights for hyperelliptic Jacobians. If time permits we will discuss some extension of this algorithm to nonhyperelliptic curves.
T. Shaska (Oakland University)
Title: Minimal models for superelliptic curves over their minimal field of definition
Abstract: This talk is about determining a minimal model of a given curve over its minimal field of definition. There are different ways of defining a minimal model due to the discriminant of the curve, the heights, etc. We will focus on finding equations of curves with minimal discriminant.
Special Session: Arithmetic GeometrySpecial Session: Arithmetic Geometry
