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Some of the latest projects which we have been involved are represented below.

Software for algebraic curves

We have started a project on computing with algebraic curves.  The project involves several people from mathematics and computer science.   If you are a graduate student in the Department of Mathematics or Department of Computer Science and would like to be involved in this project, you can contact me at shaska@oakland.edu.

Genus 2 curves

Genus two curves with split Jacobians have attracted the attention of many mathematicians recently, due to their applications in cryptography, factorization of large numbers, integrable systems, wave equation, ect.   

The Jacobian of a genus 2 curve is (n, n)-split if there is a degree n cover from C to an elliptic curve.  For a fixed n, the space of all genus 2 curves with (n, n)-split Jacobians, say L_n,  is an irreducible 2-dimensional variety if n is odd.  

L_3 was computed explicitly by this author in 2001 in terms of the absolute invariants of genus 2 curves. The space L_5 was computed by Magaard, Shaska, Voelklein in 2008.  

We know that M_g is not a fine moduli space.  It turns out that genus 2 curves for which the field of moduli is not a field of definition have automorphism group of order 2.  By work of Clebsch, Menstre, Shaska, Cardona we can explicitly give an equation of a genus 2 curve over its field of definition.  In recent work we are suggesting some improvement to these results such that the equation is minimal in some sense. 

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 has been implemented by this author. 

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.  

For superelliptic curves (i.e., curves with equation y^n = f(x) )which have extra automorphisms  we have algorithms which determine explicitly the field of moduli and an equation of the curve over its field of definition.  This algorithm makes use of dihedral invariants (some authors have called then Shaska invariants) of these curves.   Such invariants are very useful in genus g>3 since invariants of binary forms of degree > 8 are not known and have been used by many authors. 

The above algorithm will find some equation of the curve over its minimal field of definition.  However, this is not necessarily a minimal equation.  In recent work we develop some notion of minimality and suggest some techniques of how to find such equation.  

Theta functions and integrable systems

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.   Here are some suggested problems

  • For any g>2 determine the fundamental half-integer theta functions.  This is known for g=2,3 due to work of Shaska/Wijesiri.  
  • Prove a Picard like formula for hyperelliptic curves, only in terms of fundamental theta functions (see work of this author for such result in g=2, 3).
  • The moduli point  in H_3 is given in terms of ratios of half-integer fundamental theta functions.  Can you determine these ratios explicitly?  Can a this be generalized for any g>2. 
  • The above results are based on Thomae's formula for hyperelliptic curves.  Such formula has been generalized recently for Z_n curves (loosely speaking curves with equation y^n = f(x)) .  Can the results above be generalized to all curves with equation y^n=f(x) using this generalization of Thomae's formula?

Genus 3 curves

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

Genus 3 hyperelliptic curves have several open problems, namely
  • Determine a complete set of absolute invariants which determines the moduli point in the moduli space of hyperelliptic curves.
  • For any given curve, design an algorithm which determines the field of moduli, field of definition, and a minimal equation of the curve over its field of definition.
  • In the moduli space of genus 3 hyperelliptic curves, determine exactly the locus of curves for which the field of moduli is not a field of definition (This has to be a constructive algorithm which gives this locus in terms of invariants of the curves). 
Non hyperelliptic genus 3 curves are ternary quartics.  
The most famous of them all is perhaps the Klein's quartic which has 168 automorphisms meeting the Hurwitz bound of 84(g-1).  This group is the simple group of order 168.  On the left it is a sculpture of the Klein's quartic in the MSRI's backyard. 

Extending of the work of Diximier et al, we are working to determine invariants of ternary quartics and an explicit description of a moduli point in M_3. 

Subpages (1): Theta Functions