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Title: On the distribution of the Picard ranks of the reductions of a K3 surface
Abstract 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 $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S K ¯ is even, we introduce a quadratic character, called the jump character, such that $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{{\mathbb {F}}}}_{\!{{\mathfrak {p}}}}} > \mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S F ¯ p > rk Pic S K ¯ for all good primes at which the character evaluates to $$(-1)$$ ( - 1 ) .  more » « less
Award ID(s):
1439786
NSF-PAR ID:
10302684
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Research in Number Theory
Volume:
6
Issue:
3
ISSN:
2522-0160
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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