International Conference on Jacobian varieties,  Abelian functions,  and Kummer surfaces

                      (held at the room K-21, University of Yamanashi; 2012, 30-31, March.   Organized by Yoshihiro Ônishi)

Speakers and Abstracts

• England, Matthew (Glasgow Univ.) : "Building Abelian functions with generalised Hirota operators"
  • Abstract : We consider symmetric generalisations of Hirota's bilinear operator and in particular how such operators can be used to build Abelian functions. The Abelian, or multiply periodic, functions associated to a curve have been the subject of increased study over recent years and have found a range of applications. A key problem for working with such functions is the identification of bases for the relevant vector spaces. We define new infinite classes of Abelian functions using generalised Hirota operators acting on the sigma function. These new functions have a prescribed poles structure and encompass both the Kleinian P-functions and their generalisation the Q-functions. We present some explicit examples of vector space bases built using the new functions, revealing some previously unseen similarities between bases associated to curves of the same genus.
  
  
• Nakayashiki, Atsushi (Tsuda College) : "The prime form as a derivative of sigma function"
  • Abstract : The prime function of an (n,s) curve is the prime form multiplied by certain half forms. We give an expression of the prime function in terms of a derivative of the sigma function. As an application we get an addition formula for sigma functions of (n,s) curves which generalizes that of Ônishi for hyperelliptic sigma function.
  
  
• Matsutani, Shigeki (Canon) : "Truncated Young diagrams and sigma functions"
  • Abstract : For a (r,s) plane curve, a symmetric Young diagram is induced. The sigma function of the curve has the Schur function as the leading term in its expansion around the origin. When we decompose the Young diagram into two parts, upper side truncated one and lower side one, there appear natural sigma functions related to these truncated Young diagrams. For the lower side truncated one, it corresponds to the sigma function of strata of the Jacobian whereas the other one does to sigma function for a space curve.
  
  
• Ayano, Takanori (Osaka Univ.) : "Sigma functions for telescopic curves"
  • Abstract : In this talk, we consider the sigma functions for algebraic curves expressed by a canonical form proposed by Miura. We construct the sigma functions for telescopic curves, i.e., the curves such that the number of defining equations is one less than that of variables in the Miura canonical form. In particular, our curves contain the (n,s)-curves.
  
  
• Koike, Kenji (Univ. of Yamanashi) : "Defining equations of Kummer surfaces of degree eight"
  • Abstract : It is classically known that (the minimal smooth model of) a Jacobian Kummer surfaces is a complete intersection of three quadrics in projective 5-space. I will give explicit equations of these quadrics by theta functions.
  
  
• Eilbeck, John Christopher (Heriot-Watt Univ.) : "Kummer varieties, Coble polynomials, heat equations, and sigma expansions".
  • Abstract: We examine generalized Weierstrass functions associated with algebraic curves of genus 2 and 3, and the various algebraic and differential equations they satisfy. We look at Kummer and other varieties associated with these functions, and of a remarkable connection with the unique Coble polynomials in these cases. In the trigonal genus 3 case we exhibit the associated set of heat equations associated with the curve, the corresponding operator algebra, and a recursion relation for the sigma function.
  
  
• Uchida, Yukihiro (Kyoto Univ.) : "The Tate-Lichtenbaum pairing on a hyperelliptic curve via hyperelliptic nets"
  • Abstract : In the area of cryptography, it is an important problem to find a fast algorithm to compute pairings on curves such as the Weil and Tate-Lichtenbaum pairings. Recently, Stange proposed a new algorithm to compute the Tate(-Lichtenbaum) pairing on an elliptic curve. This algorithm is based on maps called elliptic nets. In this talk, we define hyperelliptic nets as a generalization of elliptic nets to hyperelliptic curves by using the hyperelliptic sigma functions. We also give an algorithm to compute the Tate-Lichtenbaum pairing on a curve of genus 2 via hyperelliptic nets. This is joint work with Shigenori Uchiyama.

Schedule

  March 30
    11:00-12:00  England     : "Building Abelian functions with generalised Hirota operators"
      (lunch)
    13:30-14:30  Nakayashiki : "The prime form as a derivative of sigma function"
    15:00-16:00  Matsutani   : "Truncated Young diagrams and sigma functions"
    16:30-17:30  Ayano       : "Sigma functions for telescopic curves"
  
  March 31
     9:30-10:30  Koike       : "Defining equations of Kummer surfaces of degree eight"
    10:50-11:35  Eilbeck     : "Kummer varieties, Coble polynomials, heat equations, and sigma expansions (1)"
    11:45-12:30  Eilbeck     : "Kummer varieties, Coble polynomials, heat equations, and sigma expansions (2)"
      (lunch)
    13:30-14:30  Uchida      : "The Tate-Lichtenbaum pairing on a hyperelliptic curve via hyperelliptic nets"