

Seminar of the Department of Algebra and of the Department of Algebraic Geometry (Shafarevich Seminar)
May 27, 2014 15:00, Moscow, Steklov Mathematical Institute, room 540 (Gubkina 8)






Motivic Euler Product and its Applications
Lin Weng^{} ^{} Kyushu University, Japan

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Abstract:
Atiyah–Bott once asked whether
there is a uniform theory for common structures exposed by their
geometric approach and Harder–Narasimhan's arithmetic approach on
Poincare series for moduli spaces of bundles. In this talk, we offer
one. To be more precise, we begin with a construction of motivic
zeta functions for curves over any base field, using moduli stacks
of semistable bundles. Abelian one is due to Kapranov. Based on
it, we define motivic Euler products. As applications, we first
formulate the corresponding Tamagawa number conjecture; then we
explain the special uniformity of zeta functions, relating the above
motivic zetas and the socalled motivic zetas for special linear
groups. The joint work with Zagier on the Riemann Hypothesis for
nonabelian zeta functions of elliptic curves over finite fields
will be discussed. Finally we offer a pair of intrinsic relations
between total motivic mass of principal bundles and its semi stable
parts, using parabolic reduction of HarderNarasimhan, Ramanathan,
Atiyah–Bott and Behrend.

