Open questions and recent developments in Iwasawa theory

Open questions and recent developments in Iwasawa theory
in honor of Ralph Greenberg's 60th birthday
June 2005

Prime divisors of special values of theta functions in the ray class fields of a certain quartic field modulo 2n


Takashi Fukuda (speaker) and Keiichi Komatsu

Nihon and Waseda Universities

Abstract

For K=Q(ζ+ζ3+ζ9), where ζ=e2πi/13, which corresponds to the Jacobian variety of

y2 = x5 − 156x4 + 10816x3 − 421824x2 + 8998912x − 80427776,

we try to construct certain type of units using special values of 2-dimensional theta functions hoping for a connection with the work of Coates-Wiles about the Birch and Swinnerton-Dyer conjecture for CM elliptic curves.