Sharif is accused of conspiracy to kill Americans and supporting a terrorist group that took part in a 2009 suicide bombing in Iraq that killed five U.S. soldiers.
His lawyer Bob Aloneissi argued in final submissions Tuesday that the Crown has provided no clear evidence that Sharif helped support a terrorist group or that he agreed to help kill anyone.
He said the Crown's case is based on police interpretations of vague statements made by Sharif that have been translated into English from Arabic.
"What did he do? Where did he sign on to killing American soldiers in Iraq?" Aloneissi asked during the extradition hearing. "Where is that?"
Aloneissi also said Sharif's right to legal advice was violated when RCMP, FBI and U.S. Justice Department officials interrogated him immediately following his Jan. 19, 2011, arrest at an Edmonton apartment.
The Crown contends that evidence from intercepted Internet and phone conversations shows that Sharif was directly involved in supporting Tunisian terrorists who conducted the suicide bombing by helping them make contact with other supporters as they made their way across the Middle East to Iraq.
The terror group network is also accused of blowing up an Iraqi police station, killing seven Iraqi officers.
Federal prosecutor Moiz Rahman said Tuesday the Crown isn't required to enter evidence such as verbatim quotes from Sharif to support its case for extradition or prove that he was a member of a specific terror group.
"The conspiracy here is not confined to two attacks, it is an overarching conspiracy to kill Americans," he said. "The evidence is that he was interested in engaging in jihad attacks."
Rahman also challenged the defence's submission that Sharif was simply in contact with people who turned out to be terrorists and really didn't know about the suicide bombing or help support the attack.
"There is no allegation that he was in charge of this. He was a member of a network," Rahman said. "He was helping the jihadists."
Sharif, 40, was born in Iraq but moved to Toronto as a refugee in 1993 and became a Canadian citizen.
Sharif has acknowledged that his real name is Faruq Muhammad'Isa, but that he changed it to escape a Turkish refugee camp when he was young man. Canadian and U.S. police have referred to him by Sharif and Isa.
He said he feared that using his real name again would have made it more difficult for him to immigrate to Canada.
Justice Adam Germain of Edmonton Court of Queen's Bench reserved his ruling until Friday.
If he rules there is enough evidence to extradite Sharif, the case would be sent to federal Justice Minister Rob Nicholson, who would make the final decision on whether to hand him over to U.S. authorities to stand trial in New York.
Both Germain's ruling and the minister's final decision could be challenged in the Alberta Court of Appeal.
Germain pointed out the standard of evidence required to extradite someone is much lower than is required for use in a criminal trial.
On Monday, Germain ruled that videotaped statements that Sharif made to police during his interrogation would be admitted as evidence.
If convicted of terrorism charges in the United States, Sharif could face a maximum sentence of life imprisonment.
Outside court, Aloneissi raised questions about Canada's extradition laws, and said his team will exhaust every legal opportunity his client has in the case.
"Any Canadian who values their citizenship ought to be concerned with a law that allows them to be sent to a foreign jurisdiction on the strength of a letter from a lawyer," he said.
"The problem is you cannot cross-examine witnesses, you cannot test the evidence. There is not much to work with and there is not much judges can do either."
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