We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, expressing them in terms of the natural lambda-structure on the Grothendieck ring. More explicitly, we will see that if $L^*$ is the torus of invertible elements in the $n$-dimensional separable k-algebra $L$, then the class of $L^*$ can be expressed as an alternating sum of the images of the spectrum of $L$ under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To prove the formula will require some rather explicit calculations in the Grothendieck ring. To be able to perform these we introduce a homomorphism from the Burnside ring of the absolute Galois group of k, to the Grothendieck ring of varieties over~k. In the process we obtain some information about the structure of the subring generated by zero-dimensional varieties.

1 Using class field theory one associates to each curve C over a finite field, and each subgroup G of its divisor class group, unramified abelian covers of C whose genus is determined by the index of G. By listing class groups of curves of small genus one may get examples of curves with many points; we do this for all curves of genus 2 over the fields of cardinality 5,7,9,11,13 and 16, giving new entries for the tables of curves with many points [6].

1 We use class field theory to search for curves with many rational points over the finite fields of cardinality ≤ 5. By going through abelian covers of each curve of genus ≤ 2 over these fields we find a number of new curves. In particular, over F 2 we settle the question of how many points there can be on a curve of genus 17 by finding one with 18 points. The search is aided by computer; in some cases it is exhaustive for this type of curve of genus up to 50.

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