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1. Eisenstein cocycles in motivic cohomology

   https://doi.org/10.1112/S0010437X24007322  

   Sharifi, Romyar; Venkatesh, Akshay (October 2024, Compositio Mathematica)

   Several authors have studied homomorphisms from first homology groups of modular curves to$$K_2(X)$$, with$$X$$either a cyclotomic ring or a modular curve. These maps send Manin symbols in the homology groups to Steinberg symbols of cyclotomic or Siegel units. We give a new construction of these maps and a direct proof of their Hecke equivariance, analogous to the construction of Siegel units using the universal elliptic curve. Our main tool is a$$1$$-cocycle from$$\mathrm {GL}_2(\mathbb {Z})$$to the second$$K$$-group of the function field of a suitable group scheme over$$X$$, from which the maps of interest arise by specialization. 

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2. Hodge and Teichmüller

   https://doi.org/10.3934/jmd.2022007  

   Kahn, Jeremy; Wright, Alex (January 2022, Journal of Modern Dynamics)

   We consider the derivative \begin{document}$$ D\pi $$\end{document} of the projection \begin{document}$$ \pi $$\end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form \begin{document}$$ \eta $$\end{document} determines a relative cohomology class \begin{document}$$ [\eta]_\Sigma $$\end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of \begin{document}$$ D\pi([\eta]_\Sigma) $$\end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory. 

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3. On the ergodic theory of the real Rel foliation

   https://doi.org/10.1017/fmp.2024.6  

   Chaika, Jon; Weiss, Barak (January 2024, Forum of Mathematics, Pi)

   Abstract Let$${{\mathcal {H}}}$$be a stratum of translation surfaces with at least two singularities, let$$m_{{{\mathcal {H}}}}$$denote the Masur-Veech measure on$${{\mathcal {H}}}$$, and let$$Z_0$$be a flow on$$({{\mathcal {H}}}, m_{{{\mathcal {H}}}})$$obtained by integrating a Rel vector field. We prove that$$Z_0$$is mixing of all orders, and in particular is ergodic. We also characterize the ergodicity of flows defined by Rel vector fields, for more general spaces$$({\mathcal L}, m_{{\mathcal L}})$$, where$${\mathcal L} \subset {{\mathcal {H}}}$$is an orbit-closure for the action of$$G = \operatorname {SL}_2({\mathbb {R}})$$(i.e., an affine invariant subvariety) and$$m_{{\mathcal L}}$$is the natural measure. These results are conditional on a forthcoming measure classification result of Brown, Eskin, Filip and Rodriguez-Hertz. We also prove that the entropy of$$Z_0$$with respect to any of the measures$$m_{{{\mathcal L}}}$$is zero. 

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4. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan; Zhang, Yanzhi (January 2022, Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$$ (- \Delta)^\frac{{ \alpha}}{{2}} $$\end{document} for \begin{document}$$ \alpha \in (0, 2) $$\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^2) $$\end{document}, while \begin{document}$$ {\mathcal O}(h^4) $$\end{document} for quadratic basis functions with \begin{document}$ h $$\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$$ \alpha \in (0, 2) $$\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$$ u \in C^{m, l}(\bar{ \Omega}) $$\end{document} for \begin{document}$$ m \in {\mathbb N} $$\end{document} and \begin{document}$$ 0 < l < 1 $$\end{document}, our method has an accuracy of \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $$\end{document} for constant and linear basis functions, while \begin{document}$$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $$\end{document}$ for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency. 

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5. Quadratic Chabauty for modular curves: algorithms and examples

   https://doi.org/10.1112/S0010437X23007170  

   Balakrishnan, Jennifer S.; Dogra, Netan; Müller, J. Steffen; Tuitman, Jan; Vonk, Jan (June 2023, Compositio Mathematica)

   We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus$$g>1$$whose Jacobians have Mordell–Weil rank$$g$$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients$$X_0^+(N)$$of prime level$$N$$, the curve$$X_{S_4}(13)$$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve$$X_{\scriptstyle \mathrm { ns}} ^+ (17)$$. 

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