On the distribution of the Picard ranks of the reductions of a K 3 surface
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RESEARCH
On the distribution of the Picard ranks of the reductions of a K 3 surface Edgar Costa1 , Andreas-Stephan Elsenhans2 and Jörg Jahnel3* * Correspondence:
[email protected]|-|-|http://www.unimath.gwdg.de/jahnel 3 Department Mathematik, University of Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany Full list of author information is available at the end of the article Edgar Costa was partially supported by the Simons Collaboration Grant #550033
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 rk Pic SK is even, we introduce a quadratic character, called the jump character, such that rk Pic SFp > rk Pic SK for all good primes at which the character evaluates to (−1). Keywords: Characteristic polynomial of the Frobenius, Functional equation, K 3 surface, Picard rank Mathematics Subject Classification: Primary 14J28, Secondary 14F20, 11G35, 14G25
1 Introduction 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. I.e., one always has rk Pic SFp rk Pic SK .
(1)
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. In an ideal case, one would be able to give a precise reason why the geometric Picard rank jumps at a given good prime. There are two well-known such reasons. (i) According to the Tate conjecture [26], the left hand side is always even. Thus, 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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E. Costa et al. Res. Number Theory (2020)6:27
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(ii) Generalising this, if S has real multiplication (RM) by an endomorphism field E and the integer (22 − rk Pic SK )/[E : Q] is odd, then again every good prime is a jump prime [11, Theorem 1(2)]. It is known due to F. Charles [11, Theorem 1] that these are the only cases in which every good prime is a jump prime. In this article, we describe a t
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