The Tate conjecture for K3 surfaces over finite fields

By Franccois Charles. 2012

127 citations recorded by the source corpus.

Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/853f9a189e59072dd6bb09d75894cd32ba53bacc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. C. Liedtke
2. Max Lieblich
3. Davesh Maulik
4. D. Huybrechts
5. Keerthi Madapusi Pera
6. Wansu Kim

In Geometry and Topology

1. D. Huybrechts
2. C. Liedtke
3. Max Lieblich
4. Franccois Charles
5. Wansu Kim
6. Keerthi Madapusi Pera
7. Davesh Maulik
8. Yukinobu Toda

In Mathematical Physics

1. D. Huybrechts
2. C. Liedtke
3. Max Lieblich
4. Wansu Kim
5. Franccois Charles
6. Keerthi Madapusi Pera
7. Davesh Maulik
8. Emanuele Macrì

Citations in the corpus

• On the Picard number of K3 surfaces over number fields (2011, background): “By work of Nygaard [1983] and Nygaard and Ogus [1985] in the finite height case and the author in the supersingular case [Charles 2013], K3 surfaces over finite fields of characteristic at least 5 satisfy the Tate conjecture; namely, their geometric Picard number is equal to the dimension of the lar”
• Lectures on K3 Surfaces (2016, background): “In [108] Charles built upon Maulik’s approach and removed the dependence of the characteristic on the degree (and also avoiding the reduction of the supersingular case to the case of elliptic K3 surfaces dealt with in [20]).”
• 2-ADIC INTEGRAL CANONICAL MODELS (2016, result): “Of course, the case when the field has characteristic 6= 2, this is already known by the results of the second author in [17], and also by earlier work by Maulik [21] and Charles [7] (among many others).”
• A note on the cone conjecture for K3 surfaces in positive characteristic (2011, methodology): “At this stage, thanks to recent proofs of the Tate conjecture [4, 5, 6, 11, 12, 14], we could be done.”
• Supersingular K3 surfaces are unirational (2013, methodology): “It follows from recent progress in the Tate-conjecture for K3 surfaces due to Charles [7], Madapusi Pera [32], and Maulik [35] that a K3 surface in odd characteristic is Artin-supersingular if and only if it is Shiodasupersingular.”