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ACA 2014 Special Session: Arithmetic Geometry

Applications of Computer Algebra 2014 
Special Session on Arithmetic Geometry
July 9 - 12,
Fordham University
New York, New York






Organizers
Overview

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 30-40 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
  • Mordell-Weil 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:  q-Weierstrass point of hyperelliptic curves
  • T. Shaska: Minimal models for superelliptic curves over their minimal field of definition
  • F. Thompson: Neron-Tate heights for curves (cancelled)
  • N. Pjero: Constructing minimal models for hyperelliptic curves. (cancelled)
Schedule

Friday afternoon:  
  • 14:00-15:00:    L. Beshaj,  Heights on algebraic curves
  • 15:00-15:30:    T. Shaska: Minimal models for superelliptic curves over their minimal field of definition
  • 15:30-16:00:    B. M. Trager, Good Reduction of Plane Curves
Saturday morning
  • 11:00-11:30:     A. Elezi: Quantum computing, error correcting quantum codes and algebraic curves.
  • 11:30-12:00:     C. Shor:  q-Weierstrass point of hyperelliptic curves
  • 12:00-12:30:     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:  q-Weierstrass point of hyperelliptic curves

F. Thompson (Oakland University)
Title: Neron-Tate heights for curves

Abstract: We will briefly define the Neron-Tate 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 non-hyperelliptic 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 Geometry



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