Coppersmith's method is an approach to finding small integral solutions to polynomial congruences. Given a monic polynomial $f(x)$ in $\textbf{Z}[x]$ of degree $d>1$ and a positive integer $N$, Coppersmith devised a polynomial time algorithm for finding all integers $r$ for which $f(r) = 0 \pmod{N}$ and $|r| < N^{1/d}$. In this talk we will show a connection between Coppersmith's method and adelic capacity theory, as developed by Cantor and Rumely. We will be able to use results from capacity theory to prove that the $N^{1/d}$ is sharp in Coppersmith's theorem. We will also explain how capacity theory proves that refinements to Coppersmith’s method cannot succeed unless $N$ has a “small” prime factor.

This talk will be more in depth and provide more details than my Oliver Club presentation. It should not, however, be necessary to have attended the Oliver Club talk. This talk is based on joint work with Ted Chinburg, Brett Hemenway and Nadia Heninger.