Word representation that gives more weight to terms frequent in corpus?
I'm not aware of any standard representation which increases the importance of document-frequent words, but IDF can simply be reverted: instead of the usual
$$idf(w,D)=\log\left(\frac{N}{|d\in D\ |\ w \in d|}\right)$$
you could use the following:
$$revidf(w,D)=\log\left(\frac{N}{|d\in D\ |\ w \notin d|}\right)$$
However for the task you describe I would be tempted to try some more advanced feature engineering, typically by using features which represent how close the distribution of words in the current document is from the average distribution.