Alexander Schmidt
We extend the notion of a tame covering of a pair (X,D) where X is a regular scheme and D is a normal crossing divisor (cf. SGA1), to pairs (X,Y) where X is an arbitrary scheme and Y is a closed subset in X. We show that the abelianized tame fundamental group of a regular scheme which is flat and of finite type over Spec(Z) is finite and does not depend on the choice of a particular compactification.
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The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes.
An arithmetic valuative criterion for proper maps of tame algebraic stacks
针对代数栈真态射的通常赋值准则在算术应用中仅给出剩余域扩张上的特化点的问题，本文提出了一种适用于算术应用的驯顺栈真态射新赋值准则，并证明了 Lang-Nishimura 定理对驯顺栈成立。
Studying ceratin combinatorial properties of non-unique factorizations have been a subject of recent literatures. Little is known about two combinatorial invariants, namely the catenary degree and the tame degree, even in the case of numerical monoids. In this paper we compute these invariants for a certain class of numerical monoids generated by generalized arithmetic sequences. We also show that the difference between the tame degree and the catenary degree can be arbitrary large even if the number of minimal generators is fixed.
The arithmetic of tame quotient singularities in dimension $2$
本文完全分类了二维驯顺商奇点中类型 R 的奇点，证明所有非循环驯顺商奇点及大多数循环奇点均为类型 R，从而推广了 Lang–Nishimura 定理并应用于模域研究。
We extend the unramified class field theory for arithmetic schemes of K. Kato and S. Saito to the tame case. Let $X$ be a regular proper arithmetic scheme and let $D$ be a divisor on $X$ whose vertical irreducible components are normal schemes. Theorem: There exists a natural reciprocity isomorphism \[ \rec_{X,D}: \CH_0(X,D) \liso \tilde π_1^t(X,D)^\ab\. \] Both groups are finite. This paper corrects and generalizes my paper "Relative K-theory and class field theory for arithmetic surfaces" (math.NT/0204330)
Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields
本文对几何伽罗瓦闭包亏格1的可分不可约驯服例外射影直线映射分类，给出有限域上算术固定域形式的编码准则，获秩2支撑公式等结果，明确特征与障碍的相关性质。
Given a perfect field $k$ with algebraic closure $\overline{k}$ and a variety $X$ over $\overline{k}$, the field of moduli of $X$ is the subfield of $\overline{k}$ of elements fixed by field automorphisms $γ\in\operatorname{Gal}(\overline{k}/k)$ such that the twist $X_γ$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset\overline{k}$ such that $X$ descends to $k'$. In this paper we extend the formalism, and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is de…
We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular homology in degree zero and the abelianized modified tame fundamental group.
Parshin's higher Witt pairing on an arithmetic surface can be combined with the higher tame pairing to form a symbol taking values in the absolute abelian Galois group of the function field. We prove reciprocity laws for this symbol using techniques of Morrow for the Witt symbol and Romo for the higher tame symbol.
In this paper we use the theory of $ε$-constants associated to tame finite group actions on arithmetic surfaces to define a Brauer group invariant $μ(\X,G,V)$ associated to certain symplectic motives of weight one. We then discuss the relationship between this invariant and $w_2(π)$, the Galois theoretic invariant associated to tame covers of surfaces.
- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Γ$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $σ$ be a generator of Gal(K/k), we study the constraints posed on the …
We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras over such fields, which are given {\it a priori} as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or {\it period-index bound}) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.
On tame ramification and centers of $F$-purity
本文为一般有限覆盖引入 tame ramification 概念，推广经典情形，并应用于 Frobenius 映射得到 $F$-purity 中心，进而通过传递性质描述有限覆盖下中心的行为。
We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to cover the first n terms of A. A similar result is presented, with the role of arithmetic and geometric progressions reversed.
We develop the notion of a geometric covering of a rigid space X, which yields a much larger class of covering spaces than that studied previously by de Jong. Geometric coverings of X are closed under disjoint unions and are étale local on X. If X is connected, its geometric coverings form a tame infinite Galois category, and hence are classified by a topological group. The definition is based on the property of lifting of "geometric arcs," making it similar to geometric coverings of schemes studied by Bhatt and Scholze as well as semicoverings of topological spaces introduced by Brazas.
We introduce mod $l$ Milnor invariants of a Galois element associated to Ihara's Galois representation on the pro-$l$ fundamental group of a punctured projective line ($l$ being a prime number), as arithmetic analogues of Milnor invariants of a pure braid. We then show that triple quadratic (resp. cubic) residue symbols of primes in the rational (resp. Eisenstein) number field are expressed by mod $2$ (resp. mod $3$) triple Milnor invariants of Frobenius elements. For this, we introduce dilogarithmic mod $l$ Heisenberg ramified covering ${\cal D}^{(l)}$ of $\mathbb{P}^1$, which may be regarded…
Sparse $k$-AP Covering Sets and the Arithmetic Kakeya Conjecture
该研究否定了Kiss等人关于k-AP覆盖集的猜想，证明k≥6时存在满足特定密度界的k-AP覆盖集，并将该问题与算术Kakeya猜想建立关联。
In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$ of the first type, the systole of most of the principal congruence coverings $M_{I}$ satisfy $$sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$$ where $c$ is a constant independent of $I$. This generalizes previous work of Buser and Sarnak, and Katz, Schaps and Vishne in dimension 2 and 3. As applications, we obtain explicit estimates for systolic genus of hyperbolic manifolds studied by Belolipetsky and the distance of homological codes constructed by Guth and Lubotzky. In an appendix together with Cayo Dória w…
Covering shrinking polynomials by quasi progressions
本文将Erdős关于平方数无法被等差数列经济覆盖的猜想及Sárközy的下界证明，推广至收缩多项式与拟等差数列的情形。
On Hasse norm principle for 3-manifolds in arithmetic topology
本研究借助算术拓扑中纽结与素数、3-流形与数环的类比，推导得出3-流形有限循环覆盖下的Hasse范数原理拓扑对应形式，拓展了该原理的适用范畴。
Minimisation of 2-coverings of genus 2 Jacobians
本文针对计算算术几何中简化有理系数多项式方程组的问题，提出了一种在首个二次型固定约束下极小化特定二次型对的算法，并应用于亏格 2 曲线 Jacobian 的 2-descent。
We extend to convenient finite quotients of a noetherian Lambda-module the classical result of K. Iwasawa giving the asymptotic expression of the l-part of the number of ideal class groups in Zl-extensions of number fields. Then, in the arithmetic context, we compute the three characters associated by this way to the l-groups of T-infinitesimal S-classes in the cyclotomic tower and relate them to the classical invariants and the decomposition characters associated to the finite sets of places S and T. A main tool in this study is the so-called Spiegelungssatz of Georges Gras, which exchanges (…
Circle coverings driven by arithmetic sequences: a percolation approach to Diophantine approximation and fractal intersections
研究通过缩小区间覆盖[0,1)的问题，揭示了在特定序列下覆盖半径的最优性，并扩展了对分形交集的尺寸计算。
A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of covering systems with distinct moduli was initiated by Erdős in 1950, and over the following decades numerous problems were posed regarding their properties. One particularly notorious question, due to Erdős, asks whether there exist covering systems whose moduli are distinct and all odd. We show that if in addition one assumes the moduli are square-free, then there must be an even modulus.
Short arithmetic terms for the cardinality of elliptic curves over rings of remainder classes and over finite fields
该研究改进了椭圆曲线计数的算术项构造方法，降低了中间整数位数，对素模p≥17时大幅减少了运算次数，提升了有限域上魏尔斯特拉斯标准形式椭圆曲线基数的计算效率。
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