  We initiate an approach to simultaneously treat numerators and denominators
of Green's functions arising from quasi-periodic Schr\"odinger operators, which
in particular allows us to study completely resonant phases of the almost
Mathieu operator. Let $ (H_{\lambda,\alpha,\theta}u) (n)=u(n+1)+u(n-1)+
2\lambda \cos2\pi(\theta+n\alpha)u(n)$ be the almost Mathieu operator on
$\ell^2(\mathbb{Z})$, where $\lambda, \alpha, \theta\in \mathbb{R}$.
  Let $$ \beta(\alpha)=\limsup_{k\rightarrow \infty}-\frac{\ln
||k\alpha||_{\mathbb{R}/\mathbb{Z}}}{|k|}.$$ We prove that for any $\theta$
with $2\theta\in \alpha \mathbb{Z}+\mathbb{Z}$, $H_{\lambda,\alpha,\theta}$
satisfies Anderson localization if $|\lambda|>e^{2\beta(\alpha)}$.
  This confirms a conjecture of Avila and Jitomirskaya [The Ten Martini
Problem. Ann. of Math. (2) 170 (2009), no. 1, 303--342] and a particular case
of a conjecture of Jitomirskaya [Almost everything about the almost Mathieu
operator. II. XIth International Congress of Mathematical Physics (Paris,
1994), 373--382, Int. Press, Cambridge, MA, 1995].
