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 K3 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 ) .
2018 ◽
Vol 2020
(20)
◽
pp. 7306-7346
Keyword(s):
Keyword(s):
2021 ◽
Vol 67
(4)
◽
pp. 2236-2244
Keyword(s):
1967 ◽
Vol s1-42
(1)
◽
pp. 73-80
◽
Keyword(s):
2015 ◽
Vol 45
(5)
◽
pp. 1481-1509
2011 ◽
Vol 63
(3)
◽
pp. 481-499
◽
Keyword(s):
1981 ◽
Vol 10
(2)
◽
pp. 209-238
◽