On the distribution of the Picard ranks of the reductions of a K3 surface-Reference-Cited by-同舟云学术

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

Published:2020-06-23 Issue:3 Volume:6 Page:
ISSN:2522-0160
Container-title:Research in Number Theory
language:en
Short-container-title:Res. number theory

Author:

Costa Edgar,Elsenhans Andreas-Stephan,Jahnel Jörg

Abstract

AbstractWe 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 ) .

Funder

Simons Foundation

Publisher

Springer Science and Business Media LLC

Subject

Algebra and Number Theory

Link

https://link.springer.com/content/pdf/10.1007/s40993-020-00204-2.pdf

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