Consider the log-ratio between classes \(k\) and \(l\), \[ \log\frac{P(Y=k\,|\,\pmb{X} = \pmb{x})}{P(Y = l\,|\,\pmb{X} = \pmb{x})} = \log\frac{P(\pmb{X} = \pmb{x}\,|\,Y = k)}{P(\pmb{X} = \pmb{x}\,|\,Y = l)} + \log\frac{P(Y=k)}{P(Y=l)} \\ = \pmb{x}^T\pmb{\Sigma}^{-1}(\pmb{\mu_k} - \pmb{\mu_l}) - \frac{1}{2}(\pmb{\mu_k} - \pmb{\mu_l})^T\pmb{\Sigma}^{-1}(\pmb{\mu_k} - \pmb{\mu_l}) + \log\frac{P(Y=k)}{P(Y=l)}. \]
(Notice the normalizing constant \((2\pi)^{-p/2}|\pmb{\Sigma}|^{-1/2}\) and quadratic term \(-\frac{1}{2}\pmb{x}^T\pmb{\Sigma}^{-1}\pmb{x}\) cancel due to the common covariance assumption.)
If this ratio is positive, \(\pmb{X}\) is more likely to have arisen from class \(k\) than \(l\).