Timeline for Isomorphism of varieties in $\mathbb{C}$ implies isomorphism over finite fields

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May 30, 2023 at 17:20	comment	added	Will Sawin		@a_g It seems unlikely that you can deduce anything from $[X]$ and $[Y]$, rather than $X$ and $Y$ themselves, because saying that $[X]$ is equal to something is just another equality in the Grothendieck ring. Even with more normal geometric information (e.g. on the equations defining $X$ and $Y$), these results are typically very hard to make (usefully) effective.
May 30, 2023 at 16:58	comment	added	a_g		But it opens another question: If I have $[X] = [Y]$ in $K(Var_\mathbb{C})$, then your argument shows that $[X] = [Y]$ also holds in $K(Var_{F_q})$ for all $F_q$ extending a certain ring $R_0$. Is it possible to estimate the shape of $R_0$ from $[X]$ and $[Y]$? For instance, suppose that I know that both are polynomials in the Lefschetz "motive". Can I say something about $R_0$? Probably no apart from a coarse bound on its number of generators...
May 30, 2023 at 16:54	comment	added	a_g		Thanks a lot! I see your point, absolutely. Your last paragraph is actually exactly what I needed: To hold for all the finite fields extending a particular ring. I was absolutely too naïve to think it will hold for large characteristic.
May 30, 2023 at 11:53	history	edited	Alison Miller	CC BY-SA 4.0	fixed typo with missing exponent in Weierstrass equation
May 29, 2023 at 23:40	history	answered	Will Sawin	CC BY-SA 4.0