Determinant Expressions for Abelian Functions in Genus Two

Determinant Expression for Abelian Functions in Genus Two

X iv :m at h/ 01 05 18 8v 1 [ m at h. N T ] 2 3 M ay 2 00 1 DETERMINANT EXPRESSION FOR ABELIAN FUNCTIONS IN GENUS TWO YOSHIHIRO ÔNISHI Faculty of Humanities and Social Sciences, Iwate University, Morioka, 020-8550, Japan e-mail:[email protected] Abstract. In this paper we generalize the formula of Frobenius-Stickelberger (see (0.1) below) and the formula of Kiepert (see (0.2) below) to genus...

Tau-functions on spaces of Abelian differentials and higher genus generalizations of Ray-Singer formula

Let w be an Abelian differential on compact Riemann surface of genus g ≥ 1. We obtain an explicit holomorphic factorization formula for ζ-regularized determinant of the Laplacian in flat conical metrics with trivial holonomy |w|2, generalizing the classical Ray-Singer result in g = 1.

Hyper-kähler Fourfolds Fibered by Elliptic Products

Every fibration of a projective hyper-Kähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We classify those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypotheses. Unlike the genus two case, we prove there are infinitely many deformat...

Applications des fonctions thêta à la cryptographie sur courbes hyperelliptiques. (Applications of theta functions for hyperelliptic curve cryptography)

Since the mid 1980’s, abelian varieties have been widely used in cryptography: the discrete logarithm problem and the protocols that rely on it allow asymmetric encryption, signatures, authentification... For cryptographic applications, one of the most interesting examples of principally polarized abelian varieties is given by the Jacobians of hyperelliptic curves. The theory of theta functions...

Determinant Expressions for Hyperelliptic Functions in Genus Three