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1. Sato–Tate Groups of Abelian Threefolds

   https://doi.org/10.1090/memo/1582  

   Fité, Francesc; Kedlaya, Kiran; Sutherland, Andrew (August 2025, Memoirs of the American Mathematical Society)

   Given an abelian variety over a number field, its Sato–Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the $L$-function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato–Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato–Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato–Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura’s theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations inGAP,SageMath, andMagma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated $L$-functions. 

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2. Angle ranks of abelian varieties

   https://doi.org/10.1007/s00208-023-02633-7  

   Dupuy, Taylor; Kedlaya, Kiran S.; Zureick-Brown, David (June 2023, Mathematische Annalen)

   Abstract Using the formalism of Newton hyperplane arrangements, we resolve the open questions regarding angle rank left over from work of the first two authors with Roe and Vincent. As a consequence we end up generalizing theorems of Lenstra–Zarhin and Tankeev proving several new cases of the Tate conjecture for abelian varieties over finite fields. We also obtain an effective version of a recent theorem of Zarhin bounding the heights of coefficients in multiplicative relations among Frobenius eigenvalues. 

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3. On the Tate conjecture for divisors on varieties with ℎ ^2,0 = 1 in positive characteristics

   https://doi.org/10.1515/crelle-2025-0042  

   Hamacher, Paul; Yang, Ziquan; Zhao, Xiaolei (July 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove that the Tate conjecture for divisors is “generically true” for $\mathrm{mod}p$\operatorname{mod}preductions of complex projective varieties with $h^{2,0}=1$h^{2,0}=1, under a mild assumption on moduli.By refining this general result, we establish a new case of the BSD conjecture over global function fields, and the Tate conjecture for a class of general type surfaces of geometric genus 1. 

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4. ADJOINT FUNCTORS ON THE DERIVED CATEGORY OF MOTIVES

   https://doi.org/10.1017/S1474748016000104  

   Totaro, Burt (June 2018, Journal of the Institute of Mathematics of Jussieu)

   We show that the subcategory of mixed Tate motives in Voevodsky’s derived category of motives is not closed under infinite products. In fact, the infinite product $$\prod _{n=1}^{\infty }\mathbf{Q}(0)$$ is not mixed Tate. More generally, the inclusions of several subcategories of motives do not have left or right adjoints. The proofs use the failure of finite generation for Chow groups in various contexts. In the positive direction, we show that for any scheme of finite type over a field whose motive is mixed Tate, the Chow groups are finitely generated. 

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5. Potential automorphy for $$GL_n$$

   https://doi.org/10.1007/s00222-022-01161-6  

   Qian, Lie (September 2022, Inventiones mathematicae)

   We prove potential automorphy results for a single Galois representation 𝐺𝐹→𝐺𝐿𝑛(ℚ⎯⎯⎯⎯⎯𝑙) where F is a CM number field. The strategy is to use the p, q switch trick to go between the p-adic and q-adic realisation of a certain variant of the Dwork motive. We choose this variant to break self-duality shape of the motives, but not the Hodge-Tate weights. Another key result to prove is that certain p-adic representations we choose that come from the Dwork motives is ordinarily automorphic. One input is the automorphy lifting theorem in Allen et al.: (Potential automorphy over CM fields, Cornell University, New York 2018) . 

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