TITLE: Estimating the size of reduction of rational points on $\mathbb{G}_m^2$
QUESTION [7 upvotes]: Hi,
Let $\Gamma$ be a free subgroup of rank 2 in $\mathbb{G}_m^2(\mathbb{Q})$. For all but finitely many primes p we can reduce $\Gamma$ modulo p. Let $S$ be the of primes for which $\Gamma$ does not reduce modulo p, and for any $p$ not in $S$, let $\gamma_p$ be the size of $\Gamma \mod p$. My question is what is known about the function

$f(x)= \sum_{p\not\in S,\ p\leq x}\frac{\log p }{\gamma_p}$

In particular what is the asymptotic behavior of $f$? Is the corresponding infinite series convergent whenever $\Gamma$ is not contained in an algebraic subgroup of $\mathbb{G}_m^2$? Do you know of any references that might be relevant to those questions?
Thanks in advance,

REPLY [3 votes]: I would just like to give a small update for the question. In my thesis https://epub.uni-bayreuth.de/1721/1/thesis.pdf I showed that the group $\Gamma\ \mod{p}$ has two generators for almost all primes p. So I would conjecture that on average $\gamma_p \sim p^2$ which would imply that the sum above indeed converges.