Estimating the size of reduction of rational points on $\mathbb{G}_m^2$ - MathOverflow

Estimating the size of reduction of rational points on $\mathbb{G}_m^2$

2011-05-30T15:19:32Z

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,

Answer by Joe Silverman for Estimating the size of reduction of rational points on $\mathbb{G}_m^2$

2011-05-30T21:14:42Z

Presumably "exceptional" means primes where either one of the generators of $\Gamma$ is 0 or $\infty$ mod p, or where $\Gamma$ mod $p$ has rank smaller than $2$. The following reference is possibly relevant to your question, although we consider a somewhat different sum. We give an upper bound (that should be fairly sharp) for the sum $$\sum_{p} \frac{\log p}{p\cdot\gamma_p^\epsilon}.$$ In particular, we prove that $$\limsup_{\epsilon\to0} ~~\epsilon \cdot \sum_{p} \frac{\log p}{p\cdot\gamma_p^\epsilon} \le 1+\frac{1}{\text{rank}~\Gamma}.$$ The article is

Murty, M. Ram and Rosen, Michael and Silverman, Joseph H., Variations on a theme of Romanoff, Internat. J. Math. 7 (1996), 373-391 (MR1395936).