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e-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.   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?

L. Beshaj, A. Elezi, T. Shaska, Theta functions of superelliptic curves

This paper was written as part of the lecture notes for the NATO Advanced Study Institute held in Ohrid, Macedonia in 2014.  We give a brief review of the Abelian integrals, Abel's theorem, Jacobi's inversion problem, and Riemann's theta function.  Then, we focus on finding algebraic dependencies for theta-nulls for superelliptic curves.


T. Shaska; G. Wijesiri; Theta functions and algebraic curves with automorphisms,  New Challenges in digital communications,  NATO Advanced Study Institute, 2009, pg. 193-237.



E. Previato, T. Shaska, G. Wijesiri; Thetanulls of cyclic curves of small genus, Albanian J. Math., vol. 1, Nr. 4, 2007, 265-282.



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