The Vermont Supreme Court has rejected a claim by a former Republican US Senate candidate from Vermont that President Obama’s name should not have been placed on last year’s Vermont ballot because he is not a ‘‘natural-born citizen’’ of the United States. H. Brooke Paige had made the claim based on what he said were legal treatises and other writings from around the time of the US Constitution. Obama is a US citizen by virtue of his birth in Hawaii. His father, Barack Obama Sr., was born in Kenya. Paige, who lost in last year’s Republican US Senate primary, sought a court order barring Secretary of State Jim Condos from putting Obama’s name on the ballot.