Here are some slides from recent talks and some relevant references.

An Exploration of Sato-Tate Groups of Abelian Varieties 

Main references:

• An Exploration of Degeneracy in Abelian Varieties of Fermat Type. Goodson. 2022. Preprint. 

• Sato-Tate Distributions of Catalan Curves. Goodson. 2021. Preprint. 

• Sato-Tate Distributions of y^2=x^p-1 and y^2=x^{2p}-1. Emory, Goodson. Journal of Algebra. Preprint. 

• Towards the Sato-Tate Groups of Trinomial Hyperelliptic Curves. Emory, Goodson, Peyrot. International Journal of Number Theory. Volume No. 17, Issue No. 10, pp. 2175 - 2206, Year 2021. Preprint.

Other references:

• (Start here! Great intro!) Sato-Tate distributions. Sutherland. Analytic Methods in Arithmetic Geometry, Contemporary Mathematics  740 (2019), 197-248.

• Sato-Tate groups of abelian threefolds. Fité, Kedlaya, Sutherland. arXiv (2021).

• Sato-Tate distributions of twists of the Fermat and the Klein quartics. Fité, Lorenzo Garcia, Sutherland. Research in the Mathematical Sciences 5 (2018), article 41.

• Sato-Tate groups of y^2=x^8+c and y^2=x^7-cx. Fité, Sutherland. Frobenius distributions: Lang-Trotter and Sato-Tate conjectures, Contemporary Mathematics  663 (2016), 103-126.

• Sato-Tate distributions and Galois endomorphism modules in genus 2. Fité, Kedlaya, Rotger, Sutherland. Compositio Mathematica 148 (2012), 1390-1442.

• The twisting Sato-Tate group of the curve y^2 = x^8 −14x^4 + 1. S. Arora, V. Cantoral-Farfan, A. Landesman, D. Lombardo, and J. S. Morrow. Math. Z., 290(3-4):991–1022, 2018.

• An algebraic Sato-Tate group and Sato-Tate conjecture. G. Banaszak and K. S. Kedlaya. Indiana Univ. Math. J., 64(1):245–274, 2015.

• The Sato-Tate conjecture for a Picard curve with complex multiplication (with an appendix by Francesc Fité). J.-C. Lario and A. Somoza.  In Number theory related to modular curves, volume 701 of Contemp. Math., pages 151–165. 2018

• Algebraic cycles on abelian varieties of Fermat type. Shioda. Math. Ann., 258(1):65–80 (1981/82).

• Non-isogenous abelian varieties sharing the same division fields. Lombardo 2021. Preprint.