- 2002年12月12日 (木曜日) 16:00-17:00
- Henri Darmon 氏(McGill Univ. 数学)
- Elliptic curves and Hilbert's twelfth Problem II
Hilbert's twelfth problem is concerned with constructing
class fields of a number field K by transcendental means. A prototype of
what is sought
for is the theory of complex multiplication, allowing the construction of
abelian extensions of an imaginary quadratic field K by evaluating elliptic
modular functions at arguments in K.
In the 1960's Stark proposed a conjectural solution to Hilbert's twelfth
problem for a larger class of number fields (including totally real
fields), predicting the existence of special units in these extensions
constructed from derivatives of abelian L-series at s=0.
It has also been observed for some time that there is a special resonance
between the problem of constructing units in number fields and rational
points on elliptic curves. I will report on an emerging conjectural picture
which allows the construction of algebraic points on elliptic curves in
terms of periods of the associated modular forms. This picture is a
generalisation of the theory of complex multiplication, and any progress on
it would yield new insights into two important questions in number theory:
the construction of class fields, and of rational points on elliptic curves.
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