First of all - Emma was not the innocent she portrayed to the police when her flat was broken into that night that started it all. She lied to the police later and said that she was raped, it was proven to be a lie. In the end, it was her jealous boyfriend, Simon, that killed her.
Jane was interested in Emma and her death because she was starting to become interested in Edward. She reached out to Simon a few times because of that intrigue and realized - almost too late - that Simon was unbalanced and he killed Emma.
By that time Jane was pregnant with Edward's baby and desperate to keep her unborn baby safe. Simon dies and she eventually gives birth to a little boy. The baby is born with Down Syndrome and Edward proves to be the dick we all thought he was by insisting she give the baby up for adoption so they can try again for a "perfect" baby. It is then revealed that Jane actually sought Edward out and intentionally became pregnant with his child because she thought genetically he would be the perfect father. All this time the reader believed Edward to be a controlling ass (he is), but it was Jane manipulating him to get what she wanted. A baby.
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