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Register a new userDiophantine Approximation of non-algebraic points on varieties II: Explicit estimates for arithmetic Hilbert Functions
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Authors
Heinrich Massold
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Because of its ineffectiveness, the usual arithmetic Hilbert-Samuel formula is not applicable in the context of Diophantine Approximation. In order to overcome this difficulty, the present paper presents explicit estimates for arithmetic Hilbert Functions of closed subvarieties in projective space.
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Let $mathcal{X}$ be a regular projective arithmetic variety equipped with an ample hermitian line bundle $overline{mathcal{L}}$. We prove that the proportion of global sections $sigma$ with $leftlVert sigma rightrVert_{infty}<1$ of $overline{mathcal{L}}^{otimes d}$ whose divisor does not have a singular point on the fiber $mathcal{X}_p$ over any prime $p<e^{varepsilon d}$ tends to $zeta_{mathcal{X}}(1+dim mathcal{X})^{-1}$ as $drightarrow infty$.
On any smooth algebraic variety over a $p$-adic local field, we construct a tensor functor from the category of de Rham $p$-adic etale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a $p$-adic analogue of Delignes classical Riemann--Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this $p$-adic Riemann--Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.
We survey some recent work on the geometric Satake of p-adic groups and its applications to some arithmetic problems of Shimura varieties. We reformulate a few constructions appeared in the previous works more conceptually.
We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than $d$, and for the size of the sum of $mod 2$ Betti numbers for the real form of complex manifolds of complex degree less than $d$.
Consider a finite polysquare or square tiled region, a connected, but not necessarily simply-connected, polygonal region tiled with aligned unit squares. Using ideas from diophantine approximation, we prove that a half-infinite billiard orbit in such a region is superdense, a best possible form of time-quantitative density, if and only if the initial slope of the orbit is a badly approximable number. As the traditional approach to questions of density and uniformity via ergodic theory depends on results such as Birkhoffs ergodic theorem which are essentially time-qualitative in nature and do not appear to lead naturally to time-quantitative statements, we appeal to a non-ergodic approach that is based largely on number theory and combinatorics. In particular, we use the famous 3-distance theorem in diophantine approximation combined with an iterative process. This paper improves on an earlier result of the authors and Yang where it is shown that badly approximable numbers that satisfy a quite severe technical restriction on the digits of their continued fractions lead to superdensity. Here we overcome this technical impediment.
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