Connection between integer lattice and quadratic number field

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I read from http://www.mathunion.org/fileadmin/IMU/Prizes/2014/news_release_bhargava.pdf

"If one has a quadratic polynomial, counting the number of integer lattice points in a certain region of 3-dimensional space provides information about the associated quadratic number field."

My questions are

  1. Suppose polynomial is $ax^2+bx+c$, what is the corresponding lattice?
  2. Which region one should try to get information?

    3.What kind of information one will get?