Abstract
For an arithmetic model of a Fermat surface or a hyperkahler variety with Betti number over a purely imaginary number field , we prove the finiteness of the -components of for all primes . This yields a variant of a conjecture of M. Artin.If is a smooth projective irregular surface over a number field and , then the -primary component of is an infinite group for every prime . Let be the universal family of elliptic curves with a Jacobian structure of level over a number field . Assume that . If is a smooth projective compactification of the surface , then the -primary component of is a finite group for each sufficiently large prime .
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