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1. Sato-Tate distributions of y^2 = x^p − 1 and y^2 = x^{2p} − 1

   https://doi.org/10.1016/j.jalgebra.2022.01.002  

   Emory, Melissa; Goodson, Heidi (May 2022, Journal of Algebra)

   We determine the Sato-Tate groups and prove the generalized Sato-Tate conjecture for the Jacobians of curves of the form y^2 = x^p−1 and y2 = x^{2p}−1, where p is an odd prime. Our results rely on the fact the Jacobians of these curves are nondegenerate, a fact that we prove in the paper. Furthermore, we compute moment statistics associated to the Sato-Tate groups. These moment statistics can be used to verify the equidistribution statement of the generalized Sato-Tate conjecture by comparing them to moment statistics obtained for the traces in the normalized L-polynomials of the curves. 

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2. SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

   https://doi.org/10.1017/fmp.2020.1  

   LE, DANIEL; LE HUNG, BAO V.; LEVIN, BRANDON; MORRA, STEFANO (January 2020, Forum of Mathematics, Pi)

   We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $$p$$ . This is a generalization to $$\text{GL}_{3}$$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $$n$$ -dimensional Galois representations’, Duke Math. J.   149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.   212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $$\text{GL}_{3}(\mathbb{F}_{q})$$ . 

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3. Dimension drop for diagonalizable flows on homogeneous spaces

   https://doi.org/10.3934/jmd.2024012  

   Kleinbock, Dmitry; Mirzadeh, Shahriar (January 2024, Journal of Modern Dynamics)

   Let X =G/Γ, where G is a connected Lie group and Γ is a lattice in G. Let O be an open subset of X, and let F = {g_t : t ≥ 0} be a one-parameter subsemigroup of G. Consider the set of points in X whose F-orbit misses O; it has measure zero if the flow is ergodic. It has been conjectured that, assuming ergodicity, this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture has been proved when X is compact or when G is a simple Lie group of real rank 1, or, most recently, for certain flows on the space of lattices. In this paper we prove this conjecture for arbitrary Addiagonalizable flows on irreducible quotients of semisimple Lie groups. The proof uses exponential mixing of the flow together with the method of integral inequalities for height functions on G/Γ. We also derive an application to jointly Dirichlet-Improvable systems of linear forms. 

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4. 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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5. A periodic isoperimetric problem related to the unique games conjecture

   https://doi.org/10.1002/rsa.20877  

   Heilman, Steven (July 2019, Random Structures & Algorithms)

   We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Letn ≥ 2. Suppose a subset Ω ofn‐dimensional Euclidean spacesatisfies −Ω = Ω^cand Ω + v = Ω^c(up to measure zero sets) for every standard basis vector. For anyand for anyq ≥ 1, letand let. For anyx ∈ ∂Ω, letN(x) denote the exterior normal vector atxsuch that ‖N(x)‖₂ = 1. Let. Our main result shows thatBhas the smallest Gaussian surface area among all such subsets Ω, less a small error:In particular,Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10⁻⁹would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture. 

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