Abstract
We report on our results concerning the distribution of the geometric Picard ranks of K3 surfaces under reduction modulo various primes. In the situation that {mathop {{mathrm{rk}}}nolimits mathop {{mathrm{Pic}}}nolimits S_{{overline{K}}}} is even, we introduce a quadratic character, called the jump character, such that {mathop {{mathrm{rk}}}nolimits mathop {{mathrm{Pic}}}nolimits S_{{overline{{mathbb {F}}}}_{!{{mathfrak {p}}}}} > mathop {{mathrm{rk}}}nolimits mathop {{mathrm{Pic}}}nolimits S_{{overline{K}}}} for all good primes at which the character evaluates to (-1).
Highlights
Let S be a K 3 surface over a number field K
We report on our results concerning the distribution of the geometric Picard ranks of K 3 surfaces under reduction modulo various primes
One would be able to give a precise reason why the geometric Picard rank jumps at a given good prime
Summary
Let S be a K 3 surface over a number field K. It is a well-known fact that the geometric Picard rank of S may not decrease under reduction modulo a good prime p of S. It would certainly be interesting to understand the sequence (rk Pic SFp )p, or at least the set of jump primes jump(S) := { p prime of K | p good for S, rk Pic SFp > rk Pic SK } , for a given surface. One would be able to give a precise reason why the geometric Picard rank jumps at a given good prime. In the case that rk Pic SK is odd, inequality (1) is always strict and every good prime is a jump prime
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