There is some interesting, recent work on the PORC conjecture (which is about Group Theory, specifically, finite $p$-groups). In some sense this is treading close to number theory even in its statement, I guess:
PORC Conjecture (Higman) Let $n$ be a fixed positive integer. The number $f(p,n)$ of nonisomorphic $p$-groups of order $p^n$ is given by a polynomial in $p$ whose coefficients depend only on the residue class of $p$ modulo some fixed $N$.
(PORC="Polynomial On Residue Classes")
The statement itself makes reference to number theory, of course; but there cent the recent work, by du Satoy Sautoy and Vaughan-Lee (Non-PORC behaviour of a class of descendant $p$-groups, in the arXiv) delves deep into number theory and arithmetic geometry (as does a previous paper by du Satoy Sautoy associating the problem of counting nilpotent groups with elliptic curves).
