Research‎ > ‎

Algebra/Algebraic Geometry Group


Projects

  • Invariant theory 
  • Automorphisms of curves
  • Field of moduli versus field of definition
  • Theta functions and Jacobians 
  • Superelliptic curves

Theta Functions


Theta functions are some of the most amazing objects in mathematics.  They have been studied by Riemann, Picard, Kovalevski, Frobenious among many others.

We focus on theta functions of algebraic curves, and more explicitly on theta-nulls of superelliptic curves. 

For more information click below

Genus 3 curves


Genus 3 curves are ternary quartics.  If the curve is hyperelliptic then they are written as 
y^2=f(x)
where f(x) is a polynomial of degree 8. 

Genus 3 hyperelliptic curves are treated in the package of hyperelliptic curves.


Here we deal with non-hyperelliptic genus 3 curves. The most famous of them all is perhaps the Klein's quartic. 




Genus 2 Curves



A Maple package for genus 2 curves which computes the basic invariants of the curve: Igusa invariants, absolute invariants, field of moduli, field of definition, automorphism group, etc. 




Books for the Albanian audience


Algjebra Abstrakte (T. Shaska, L. Beshaj), AulonaPress, 2011, (Albanian).

  



Kurbat Algjebrike,  (work in progress)

Databases for curves

I have seen every curve at least once ...

Genus: 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Equation of curves over their field of definition

Let C be a curve defined over the complex numbers and F its field of definition.  Finding a curve $X$, isomorphic to C over the complex numbers, is an old problem in algebraic geometry.  

Algorithms exist for genus 2 curves due to work of Clebsch, Mestre, Shaska, Cardona, et al.  We focus on genus 3 curves and on superelliptic curves of any genus.  

More information can be found here. 

Conferences and Activities


  • Algebraic curves and their applications, December 2017 (TBA)
  • AJM 10-th anniversary conference, December 2016 (TBA)
    • A. Elezi, T. Shaska, ...
  • Special Session on Arithmetic of Algebraic Curves: AMS Sectional Meeting, University of North Carolina, Greensboro, NC. November 8-9, 2014. 
    • T. Shaska, J. D (TBA)
  • Moduli spaces and arithmetic dynamics, Special Session, ACA 2014, Fordham, New York, July 9-12, 2014
    • C. Shor, L. Beshaj
  • Michigan Computational Algebraic Geometry, MCAG III, Ann Arbor, MI, May 2014
  • Special Session on Arithmetic of Algebraic Curves: AMS Sectional Meeting, Tennessee, 21-23 March 2014. 
    • L. Beshaj, C. Shor, A. Malmendier
  • ACA 2013: Special Session on Arithmetic of Algebraic Curves
  • MCAG 2013 (Western Michigan University, Kalamazoo, May 3-4, 2013
    • Organizers: Gene Freuedenburg
  • AMS Special Session (Iowa, April 2013) 
    • E. Previato, L. Beshaj: 
  • Theta Functions, Special Session, SIAM Conference, Raleigh, Oct. 6-8. 2011.  Applied Algebraic Geometry, SIAM, Raleigh, Oct. 6-8. 2011.