In a nutshell, the major arguements are:
1) He was born in Kenya, (some also claim Canada), and his mother was too young to pass on her US citizenship to him. This is based on various internet claims, particularly Corsi and anti-semite Andy Martin. His Hawaiian birth certificate is therefore a forgery.
2) He was born in Hawaii, but both him and his mother lost their US citizenship when she married Lolo Soetoro and moved to Indonesia. This is based on a faulty reading of the US Nationality Act of 1940 and 1952. While it is true that a woman marrying a non-citizenship husband would have lost her citizenship (if the marriage had happened before 1922 that is), nothing the parents did would have affected the minor's citizenship. The parents cannot renounce the child's US citizenship, and the child cannot renounce their US citizenship until the child is 18. The basis of the argument comes from a Neo-Confederate named "Judah Benjamin" on the PUMA blog "TexasDarlin".
3) He was born in Hawaii and has US citizenship though his mother. However, due to the fact his father was not a US citizen, and Obama inherited UK/Kenyan citizenship from him, he is not a "Natural Born citizen" but rather has divided loyalties like a naturalized citizen and can be considered at the same level of citizenship. "Judah Benjamin" and various right-wing sources are responsible for this.
Berg uses 1 and 2 in his lawsuit, Donofrio, (which will be heard by SCOTUS tomorrow) uses 3. Most of the other lawsuits use various combos, many are simple copycats of Berg or Donofrio.
As for the Birth Certificate circus, the touchstone these cultists use for their belief is a nameless internet "expert" found at Townhall.com and Free Republic called "Polarik". He did the initial flawed analysis of the COLB images and immediately claimed them a forgery. Since then, any attempt to prove them as legit will result in Polarik, (Or as he's now styling himself "Dr. Ron Polarik, Ph.D." I suspect the Ph.D. is in bullshitology) within 24-48 hours claiming that THAT attempt is also a forgery. Several real computer forensic experts have checked his work and claimed it to be bogus, including Dr. Neal Krawetz at hackerfactor.com. Note, Polarik or Ron Polarik is not his real name, he claims he's keeping that a secret due to threats against his family.
Another player was "TechDude" originall from Pam Geller's "Atlas Shrugs" blog. "TechDude" originally claimed to have confirmed "Polarik's" findings, but then started to make more and more wild eyed claims, ending when he claimed that Maya Soetoro's name was on the Daily Kos and Fight The Smears copies of the COLB. When nobody else could prove this, he claimed a threat against his family and disappeared. Most sites expect "Atlas Shrugs" eventually rejected his claim, but his claims are still in Berg's lawsuit and Berg has mentioned them on the air once or twice - even though officially they don't use him as an example.
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http://whatsyourevidence.com/- Patrick McKinnion