"We took the view we should prosecute, because we have to ensure there is a level playing field for all parents.
"The difficulty is that there is no clear law of what sanction applies if parents puts false information on their application form."
The BBC's education correspondent, James Westhead, said the council had been concerned the Fraud Act might only apply to property, rather than school places, and that its prosecution might fail.
He added anecdotal evidence suggested that the problem of parents lying about their home addresses in order to get into good schools had trebled over the last couple of years.
Pinner Park First School received 411 applications for its 90 available places in September 2008.
The council says it allocated places to children living closest to the school, up to a maximum distance of 0.685 miles.
When she applied for a school place, Mrs Patel gave the address of her mother, which was within the school's catchment area, rather than the family address.
Mrs Patel said that she was separated from her husband at this time and had "no intention" of moving back to the matrimonial home but that four weeks later she changed her mind.
Harrow Council said it had brought the case "with the greatest of reluctance" after Mrs Patel failed to explain irregularities regarding her application.
It said it had invited Mrs Patel to a series of meetings to explain her case but Mrs Patel had failed to attend them.
Mrs Patel said her solicitor had advised her she did not need to attend the meetings.
Mr Ashton said: "While we stand by the substance of our case, subsequent legal advice is that technical legal arguments over the interpretation of the Act could pose a risk to the success of the action.
"We have therefore decided that, rather than incur potentially costly legal fees over a Crown Court case that might be undermined by legal argument, the interests of residents are better served by Harrow Council withdrawing this action.
"This case was never about persecuting mothers who wish to do the best for their children; it was about defending the integrity of the school system against those who might seek to flout it."
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