Counting Points

Grothendieck group[edit | edit source]

A useful way to study the Grothendieck ring of varieties is via its realisation maps into a group , also known as motivic measures or additive invariants. These are maps from isomorphism classes of varieties that behave additive under disjoint unions, i.e. descend to . One such measure is counting points:

For example:
From the Weil Conjectures, it will follow that if a projective variety of dimension is the union of smooth, geometrically irreducible (i.e. after base change to an algebraic closure) components of maximal dimension, then

Exercise[edit | edit source]

Complete this argument to show the claim about .

Square polynomials[edit | edit source]

We can also tackle the first problem now and prove that assuming square values for all integers is itself the square of a polynomial in (although showing is also possible). For this, observe that is a square if and only if the hyperelliptic curve given by

has two irreducible components. This however can be established over finite fields and then be transferred. Reducing the curve modulo (not a power of ), the number of -rational points is governed by : For each (except for the roots of whose number is bounded by ), we get two solutions for .

Alternatively (or equivalently), one can use the Weil estimate for smooth, projective, geom. irreducible curves of genus :

Exercise[edit | edit source]

Formulate and prove the most general version of this type of claim you can imagine. Possible answer: Let be a multivariate polynomial that assumes perfect power values at all integral points (possibly with different exponents). Then is a perfect power...