гипотеза
https://arxiv.org/abs/1809.10090 Convergence of measures on compactifications of locally symmetric spaces
Christopher Daw, Alexander Gorodnik, Emmanuel Ullmo
(Submitted on 26 Sep 2018)
We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space S=Γ∖G/K is compact. More precisely, given a sequence of homogeneous probability measures on S, we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including S itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when G=SL3(R) and Γ=SL3(Z).
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новое доказательство гипотезы Морделла
https://arxiv.org/abs/1807.02721 Diophantine problems and p-adic period mappings
Brian Lawrence, Akshay Venkatesh
(Submitted on 7 Jul 2018 (v1), last revised 30 Aug 2018 (this version, v2))
We give an alternative proof of Faltings's theorem (Mordell's conjecture): a curve of genus at least two over a number field has finitely many rational points. Our argument utilizes the set-up of Faltings's original proof, but is in spirit closer to the methods of Chabauty and Kim: we replace the use of abelian varieties by a more detailed analysis of the variation of p-adic Galois representations in a family of algebraic varieties. The key inputs into this analysis are the comparison theorems of p-adic Hodge theory, and explicit topological computations of monodromy.
By the same methods we show that, in sufficiently large dimension and degree, the set of hypersurfaces in projective space, with good reduction away from a fixed set of primes, is contained in a proper Zariski-closed subset of the moduli space of all hypersurfaces. This uses in an essential way the Ax--Schanuel property for period mappings, recently established by Bakker and Tsimerman.
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