Seminar on Arithmetic Algebraic Geometry: Real regulator of K_1 of elliptic surfaces

2012-11-15 10:00 - 2012-11-15 12:00
Faculty of Science Building #3 Room 413
Masanori Asakura
By the theory of higher Chern class, there is the canonical map from higher K-theory to the
Deligne-Beilinson cohomolgy with coefficients in real numbers, which we call the real
regulator map. As is well-known, it can be described by extensions of mixed Hodge
structures. Beilinson conjectured the mysterious relationship between the real regulator
and special values of L-functions. However, in spite of much efforts by many people,
it is still a widely open problem, together with the fact that the explicit computations of
regulator are very hard in many (most?) cases. In this talk I explain a certain method for
computation of real regulator of K_1 of elliptic surface, where the key ingredient is the
Picard-Fuchs operator. I have worked out it only to a certain example though it seems to
work as well in more general situation. Required preliminary knowledge is only standard
knowledge of mixed Hodge theory.