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1. K-moduli of Fano threefolds and genus four curves

   https://doi.org/10.1515/crelle-2025-0016  

   Liu, Yuchen; Zhao, Junyan (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this article, we study the K-moduli space of Fano threefolds obtained by blowing up ${\mathbb{P}}^3$\mathbb{P}^{3}along $(2,3)$(2,3)-complete intersection curves.This K-moduli space is a two-step birational modification of the GIT moduli space of bidegree $(3,3)$(3,3)-curves on ${\mathbb{P}}^1\times {\mathbb{P}}^1$\mathbb{P}^{1}\times\mathbb{P}^{1}.As an application, we show that our K-moduli space appears as one model of the Hassett–Keel program for ${\bar{M}}_4$\overline{M}_{4}.In particular, we classify all K-(semi/poly)stable members in this deformation family of Fano varieties.We follow the moduli continuity method with moduli of lattice-polarized K3 surfaces, general elephants and Sarkisov links as new ingredients. 

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2. A new zero-free region for Rankin–Selberg 𝐿-functions

   https://doi.org/10.1515/crelle-2025-0009  

   Harcos, Gergely; Thorner, Jesse (March 2025, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝜋 and ${\pi }^{\prime }$\pi^{\prime}be cuspidal automorphic representations of $\mathrm{GL}\left(n\right)$\mathrm{GL}(n)and $\mathrm{GL}\left(n^{\prime }\right)$\mathrm{GL}(n^{\prime})with unitary central characters.We establish a new zero-free region for all $\mathrm{GL}\left(1\right)$\mathrm{GL}(1)-twists of the Rankin–Selberg 𝐿-function $L(s,\pi \times {\pi }^{\prime })$L(s,\pi\times\pi^{\prime}), generalizing Siegel’s celebrated work on Dirichlet 𝐿-functions.As an application, we prove the first unconditional Siegel–Walfisz theorem for the Dirichlet coefficients of $-L^{\prime }(s,\pi \times {\pi }^{\prime })/L(s,\pi \times {\pi }^{\prime })$-L^{\prime}(s,\pi\times\pi^{\prime})/L(s,\pi\times\pi^{\prime}).Also, for $n\le 8$n\leq 8, we extend the region of holomorphy and nonvanishing for the twisted symmetric power 𝐿-functions $L(s,\pi ,{\mathrm{Sym}}^n\otimes \chi )$L(s,\pi,\mathrm{Sym}^{n}\otimes\chi)of any cuspidal automorphic representation of $\mathrm{GL}\left(2\right)$\mathrm{GL}(2). 

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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. ADM mass for C 0 C^{0} metrics and distortion under Ricci–DeTurck flow

   https://doi.org/10.1515/crelle-2023-0085  

   Burkhardt-Guim, Paula (December 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We show that there exists a quantity, depending only on $C^0$C^{0}data of a Riemannian metric, that agrees with the usual ADM mass at infinity whenever the ADM mass exists, but has a well-defined limit at infinity for any continuous Riemannian metric that is asymptotically flat in the $C^0$C^{0}sense and has nonnegative scalar curvature in the sense of Ricci flow.Moreover, the $C^0$C^{0}mass at infinity is independent of choice of $C^0$C^{0}-asymptotically flat coordinate chart, and the $C^0$C^{0}local mass has controlled distortion under Ricci–DeTurck flow when coupled with a suitably evolving test function. 

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5. On Strichartz estimates for many-body Schrödinger equation in the periodic setting

   https://doi.org/10.1515/forum-2024-0105  

   Huang, Xiaoqi; Yu, Xueying; Zhao, Zehua; Zheng, Jiqiang (May 2024, Forum Mathematicum)

   Abstract In this paper, we prove Strichartz estimates for many body Schrödinger equations in the periodic setting, specifically on tori ${𝕋}^d${\mathbb{T}^{d}}, where $d\ge 3${d\geq 3}. The results hold for both rational and irrational tori, and for small interacting potentials in a certain sense. Our work is based on the standard Strichartz estimate for Schrödinger operators on periodic domains, as developed in [J. Bourgain and C. Demeter,The proof of the $l^2$l^{2}decoupling conjecture,Ann. of Math. (2) 182 2015, 1, 351–389]. As a comparison, this result can be regarded as a periodic analogue of [Y. Hong,Strichartz estimates forN-body Schrödinger operators with small potential interactions,Discrete Contin. Dyn. Syst. 37 2017, 10, 5355–5365] though we do not use the same perturbation method. We also note that the perturbation method fails due to the derivative loss property of the periodic Strichartz estimate. 

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