Title: On the Grothendieck ring of varieties in positive characteristic Show Chinese title

Title: 关于正特征中的代数簇的格罗滕迪克环

Abstract: This paper proves two theorems (1) Let $k$ be an algebraically closed field of characteristic $p>0$. I prove (Theorem 2.1.1) that if, $p > 13$ or $p = 11$, then the isomorphism class of any supersingular elliptic curve is a zero divisor in the ring of smooth, complete $k$-varieties and Bittner relations. In particular, this ring contains zero divisors. The proof proceeds via establishing (in Theorem 2.2.1) that the Albanese variety functor is a motivic measure. (2) I prove (Theorem 3.1) that the etale fundamental group of a smooth, proper variety over any alg. clsoed field k (in any characteristic) also provides a motivic measure on this ring. In particular, the etale fundamental group is a motivic measure on the Grothendieck ring of varieties over complex numbers. Show Chinese abstract

Abstract: 本文证明了两个定理 (1) 设 $k$ 是特征为 $p>0$ 的代数闭域。 我证明（定理 2.1.1）如果， $p > 13$ 或 $p = 11$，那么任何超奇异椭圆曲线的同构类在光滑、完备的 $k$-簇和 Bittner 关系的环中是一个零因子。 特别地，这个环包含零因子。 证明通过建立（定理 2.2.1）阿贝尔雅可比簇函子是一个动机测度来完成。 (2) 我证明（定理 3.1）在任何代数闭域 k 上的光滑、紧致簇的平展基本群（在任何特征下）也在该环上提供一个动机测度。 特别地，平展基本群是在复数上代数簇的格罗滕迪克环上的一个动机测度。