Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

Abstract

Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline {K}}$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K.

Equidistribution of Hodge loci II

“…Several results analogous to Theorem 1.7 for algebraic families parameterized by Shimura varieties have been settled in arithmetic situations over rings of integers of number fields and over curves defined over finite fields: indeed Charles proved [Cha18] that there are infinitely many places where the reduction of two elliptic curves are isogenous; Shankar and Tang [ST20] proved that an abelian surface over a number field with real multiplication has infinitely many specializations which are isogenous to the self-product of an elliptic curve and in collaboration with Maulik in [MST22b] they derived similar results for ordinary abelian surfaces over the function field of a curve over a finite field. Finally the analogous statement of Theorem 1.7 for K3-type variations of Hodge structures over curves has been proved in the number field setting in [SSTT22,Tay22] and over curves over finite fields in [MST22a]. It is thus interesting to further explore other analogous statements of Theorems 1.1, 1.7, and 1.22 over number fields and function fields situations.…”

Intégration de Hecke de fonctions singulières

“…Par exemple, dans le cas modulaire, f peut être à support compact, continue sauf en un point où elle présente une singularité logarithmique ; cf. [3,10] ; f est alors intégrable, et on cherche à lui étendre la formule d'intégration (0.1), que nous appelons ici intégration de Hecke.…”

Rational curves on elliptic K3 surfaces

“…We interpret the locus S(X, α) as an intersection locus of S with a family of special divisors in a Shimura variety with level structure at r, the geometric torsion order of α. Following a method initiated by Charles in [Cha18] and generalized in [SSTT22], see also [MST22a,ST20,MST22b], we control the intersection numbers of S with a sequence of special divisors indexed by integers m at archimedean and non-archimedean places and compare the order of growths. If there were only finitely many primes where α vanishes, then this means that the intersection is supported at finitely many primes independent of m. As m grows, we get a contradiction by comparing the order of growths of the local and global estimates.…”

“…If there were only finitely many primes where α vanishes, then this means that the intersection is supported at finitely many primes independent of m. As m grows, we get a contradiction by comparing the order of growths of the local and global estimates. The assumption on the rank is used at this level, as the results of [SSTT22] are valid only when the rank of the transcendental lattice is at least 5, and the case of rank 3 follows from [FHVA22].…”