This talk will be about connections between number theory, arithmetic geometry and cryptography. The security of the RSA method for sending encrypted messages depends on the difficulty of factoring a large integer N into primes. Coppersmith proved in the 1990's that if one knows a prime factor p of N to within the 4th root of N, one can factor N quickly. I will explain how this result can be generalized using arithmetic capacity theory. The idea is to view the roots of polynomials as repelling electric charges.

One can then use potential theory to determine whether or not polynomials with various special properties exist. This work is joint with Brett Hemenway, Nadia Heninger and Zach Scherr.