ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 14, 30 Horiuchi, Ryutaro, Doshisha University, Japan On Weierstrass points of the Fermat Curve Weierstrass points of the Fermat curve z_{0}^{n} + z_{1}^{n} + z_{2}^{n} = 0 have been studied by Hasse, Leopoldt, Matzat, Rohrlich, Towse, Watanabe. Set x = z_{1}/z_{0}, y = z_{2}/z_{0}. We put the Wronskians W_{n}(x, y) in the form W_{n}(x, y) = cx^{ℓ(n)}f_{n}(x)/y^{m(n)}. The polynomial f_{n}(x) can be put in the form

\begin{matrix}\{(X − 1)(X + 2)(X + 1/2)\}^{k(n)} ∏_{i=1}^{N(n)} (X − a_{i}(n))(X \frac{1}{a_{i}(n)})· \\ (X + \frac{a_{i}(n)}{1 + a_{i}(n)})(X + 1 + a_{i}(n))(X + \frac{1}{1 + a_{i}(n)})(X + \frac{1 + a_{i}(n)}{a_{i}(n)}) \end{matrix} ,

where X = x^{n}, a_{i}(n) ≠ 0, 1, −2, −1/2. We obtain ℓ(n), m(n), k(n), N(n), a_{i}(n) for n = 7, 8, 9, 10. Reference: K. Watanabe, Weierstrass points of the Fermat curve, Proceedinds of the Congress ISAAC’97, Univ. of Delaware, U. S. A., 3–7 June, 1997, Kluwer Academic, to appear.