Lieblich Project

The aim of this project will be to understand recent progress on the Tate conjectures. More details on the project will be provided here as they become available.

The organizers of this working group are Harrison Chen and Ravi Fernando. To participate in this working group, send an email to one of these organizers.

Both students and faculty are encouraged to participate in the working groups.

References:

• F. Charles, Birational boundedness for holomorphic symplectic varieties. Zarhin's trick for K3 surfaces and the Tate conjecture. Annals of Math 184 (2016) available here

The original Tate work for abelian varieties:

• J. Tate, Endomorphisms of abelian varieties over finite fields, Invent. Math. 2 (1966), 134–144. MR 0206004. Zbl 0147.20303. http://dx.doi.org/10.1007/BF01404549.

The Artin--Swinnerton-Dyer proof for elliptic K3s:

• M. Artin and H. P. F. Swinnerton-Dyer, The Shafarevich-Tate conjecture for pencils of elliptic curves on K3 surfaces, Invent. Math. 20 (1973), 249–266. MR 0417182. Zbl 0289.14003. http://dx.doi.org/10.1007/BF01394097.

The Tate conjecture for K3 surfaces of finite height, as proven by Nygaard and Ogus:

• N. Nygaard and A. Ogus, Tate’s conjecture for K3 surfaces of finite height, Ann. of Math. (2) 122 (1985), 461–507. MR 0819555. DOI 10.2307/1971327. https://www.jstor.org/stable/1971327

Reference on supersingular K3 surfaces:

• M. Artin, Supersingular K3 surfaces, Ann. Sci. École Norm. Sup. 7 (1974), 543–567 (1975). MR 0371899. Zbl 0322.14014. Available at http://www.numdam.org/item?id=ASENS_1974_4_7_4_543_0.

An earlier approach to finishing the Tate conjecture for K3s based on the Kuga-Satake construction:

• D. Maulik, Supersingular K3 surfaces for large primes, Duke Math. J. 163 (2014), 2357–2425. MR 3265555. Zbl 1308.14043. http://dx.doi.org/10.1215/00127094-2804783.
• F. Charles, The Tate conjecture for K3 surfaces over finite fields, Invent. Math. 194 (2013), 119–145. MR 3103257. Zbl 1282.14014. http://dx.doi.org/10.1007/s00222-012-0443-y.
• K. Madapusi Pera, The Tate conjecture for K3 surfaces in odd characteristic, Invent. Math. 201 (2015), 625–668. MR 3370622. Zbl 1329.14079. http://dx.doi.org/10.1007/s00222-014-0557-5.

From Zarhin’s trick to the Tate conjecture:

• M. Lieblich, D. Maulik, and A. Snowden, Finiteness of K3 surfaces and the Tate conjecture, Ann. Sci. Éc. Norm. Supér. 47 (2014), 285–308. MR 3215924. Zbl 1329.14078. https://arxiv.org/abs/1107.1221