Motivic Euler Product and its Applications

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 semi-stable 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 so-called motivic zetas for special linear groups. The joint work with Zagier on the Riemann Hypothesis for non-abelian 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 Harder-Narasimhan, Ramanathan, Atiyah–Bott and Behrend.