On the distribution of the Picard ranks of the reductions of a K3 surface

We report on our results concerning the distribution of the geometric Picard ranks of K 3 surfaces under reduction modulo various primes. In the situation that rk Pic S K ¯ is even, we introduce a quadratic character, called the jump character, such that rk Pic S F ¯ p > rk Pic S K ¯ for all good primes at which the character evaluates to ( - 1 ) .