Franz Lemmermeyer
It is well known that the Galois group of an extension puts constraints on the structure of the relative ideal class groups. Using only basic parts of the theory of group representations, we give a unified approach to such results.
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We construct an explicit example of a genus $2$ curve $C$ over a number field $K$ such that the adelic Galois representation arising from the action of $\operatorname{Gal}(\overline{K}/K)$ on the Jacobian of $C$ has image $\operatorname{GSp}_4(\widehat{\mathbb Z})$.
In this paper we investigate the connection between relations among various invariants of number field s $L^H$ coresponding to subgroups $H$ acting on $L$ and of linear relations among norm idempotents.
In this letter we show that for certain infinite families of modular forms of growing level it is possible to have a control result for the exceptional primes of the attached Galois representations. As an application, a uniform version of a result of Wiese on Galois realizations of linear groups over finite fields of arbitrarily large exponent is proved.
Given an elliptic curve E over a number field k, the Galois action on the torsion points of E induces a Galois representation, ρ_E : Gal(\bar{k}/k) \to GL_2(\hat{Z}). For a fixed number field k, we describe the image of ρ_E for a "random" elliptic curve E over k. In particular, if k\neq Q is linearly disjoint from the cyclotomic extension of Q, then ρ_E will be surjective for "most" elliptic curves over k.
An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a centr…
Using Shimura's reciprocity law, we calculate the Galois action on the special values of theta functions associated to primitive Dirichlet characters of odd conductor, normalised by the Dedekind eta function, at the point i. As a consequence, we prove some experimental results of Cohen and Zagier and we deduce a partial result on the non-vanishing of these special theta values with prime conductor.
To each Drinfeld module over a finitely generated field with generic characteristic, one can associate a Galois representation arising from the Galois action on its torsion points. Recent work of Pink and Rütsche has described the image of this representation up to commensurability. Their theorem is qualitative, and the objective of this paper is to complement this theory with a worked out example. In particular, we give examples of Drinfeld modules of rank 2 for which the Galois action on their torsion points is as large as possible. We will follow the approach that Serre used to give explici…
We gives an explicit genus 3 curve over Q such that the Galois action on the torsion points of its Jacobian is a large as possible. That such curves exist is a consequence of a theorem of D. Zureick-Brown and the author; however, those methods do not produce explicit examples. We shall apply the general strategies of Hall and Serre in their open image theorems. We also make use of Serre's conjecture to show that the modulo ell Galois actions are irreducible. While we computationally focus on a single curve, the methods of this paper can be applied to a large family of genus 3 curves.
Counting Polynomials via Galois Actions on Root Subsets
本文研究了高度不超过H的单项整系数多项式数量，其伽罗华群在根上的作用同构于给定的置换群(G,Ω)。针对多种群族获得了新的上界。
A simplified algorithmic realization of Galois actions on special values of modular functions
本研究基于扩展形式类群理论，提出一种显式实用的算法，用于计算虚二次序相关CM点处模函数特殊值的Galois共轭元与不可约多项式，改进了Stevenhagen和Cho的早期方法。
Given two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's u…
A del Pezzo surface of degree one defined over the rationals has 240 exceptional curves. These curves are permuted by the action of the absolute Galois group. We show how a solution to the classical inverse Galois problem for a subgroup of the Weyl group of type $D_8$ gives rise to a solution of the inverse Galois problem for the action of this subgroup on the 240 exceptional curves. A del Pezzo surface of degree one with such a Galois action contains a Galois invariant sublattice of type $D_8$ within its Picard lattice; this can be characterized in terms of a certain set of sixteen points in …
In this paper we present a description of the Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and $SL_2(\mathbb{F}_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over $\mathbb{Q}_2$ is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.
We study the Galois action attached to the Dwrok surfaces $X_λ:X_0^4+X_1^4+X_2^4+X_3^4-4λX_0X_1X_2X_3=0$ with parameter $λ$ in a number field $F$. We show that when $X_λ$ has geometric Picard number $19$, its Néron-Severi group $NS(\overline{X}_λ)\otimes \mathbb{Q}$ is a direct sum of quadratic characters. We provide two proofs to this conclusion in our article. In particular, the geometrically proof determines the conductor of each of quadratic characters. Our result matches the one in \cite{Voight2}. With this decomposition, we give another proof to a result of Wan.
On Kummer characters arising from Galois actions on the pro-$p$ fundamental groups of once-punctured CM elliptic curves
本文研究了来自CM椭圆曲线一次punctured时的pro-$p$基本群的Kummer字符，探讨了其非平凡性和满射性条件。
We make some observations concerning the Galois actions on Hurwitz curves and on the closely related but lesser-known Hurwitz origamis.
In this paper we introduce confluence relations for motivic Euler sums (also called alternating multiple zeta values) and show that all linear relations among motivic Euler sums are exhausted by the confluence relations. This determines all automorphisms of the de Rham fundamental groupoid of $\mathbb{P}^{1}\setminus\{0,\pm1,\infty\}$ coming from the action of the motivic Galois group of mixed Tate motives over $\mathbb{Z}[1/2]$. Moreover, we also discuss other applications of the confluence relations such as an explicit $\mathbb{Q}$-linear expansion of a given motivic Euler sum by their basis…
The fundamental group of Fermat and generalized Fermat curves is computed. These curves are Galois ramified covers of the projective line with abelian Galois groups $H$. We provide a unified study of the action of both cover Galois group $H$ and the absolute Galois group $\mathrm{Gal}(\bar{\Q}/\Q)$ on the pro-$\ell$ homology of the curves in study. Also the relation to the pro-$\ell$ Burau representation is investigated.
Information about the absolute Galois group $G_K$ of a number field $K$ is encoded in how it acts on the étale fundamental group $π$ of a curve $X$ defined over $K$. In the case that $K=\mathbb{Q}(ζ_n)$ is the cyclotomic field and $X$ is the Fermat curve of degree $n \geq 3$, Anderson determined the action of $G_K$ on the étale homology with coefficients in $\mathbb{Z}/n \mathbb{Z}$.The étale homology is the first quotient in the lower central series of the étale fundamental group.In this paper, we determine the structure of the graded Lie algebra for $π$. As a consequence, this determines the…
This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and Z…
Given a polynomial f of degree d defined over a complete local field, we construct a biholomorphic change of variables defined in a neighbourhood of infinity which transforms the action z->f(z) to the multiplicative action z->z^d. The relation between this construction and the Bottcher coordinate in complex polynomial dynamics is similar to the relation between the complex uniformization of elliptic curves, and Tate's p-adic uniformization. Specifically, this biholomorphism is Galois equivariant, reducing certain questions about the Galois theory of preimages by f to questions about multiplica…
Our aim of this and subsequent papers is to enlighten (a part of, presumably) arithmetic structures of knots. This paper introduces a notion of profinite knots which extends topological knots and shows its various basic properties. Particularly an action of the absolute Galois group of the rational number field on profinite knots is rigorously established, which is attained by our extending the action of Drinfeld's Grothendieck-Teichmüller group on profinite braid groups into on profinite knots.
At present, the inverse Galois problem over $\mathbb{Q}$ is unsolved for the Mathieu group $M_{23}$. Here an overview of the current state in realizing $M_{23}$ as Galois group using the rigidity method and the action of braids is given. Computing braid orbits for $M_{23}$ revealed new invariants of the action of braids in addition to Fried's lifting invariant. These invariants can be used to construct generic braid orbits and more Galois realizations over $\mathbb{Q}$ for the Mathieu group $M_{24}$, but until now did not lead to success for realising $M_{23}$ as Galois group over $\mathbb{Q}$…
We define and study a collection of special cycles on certain non-PEL Shimura varieties for $U(2,1) \times U(1,1)$ that appear naturally in the context of the recent conjectures of Gan, Gross and Prasad on restrictions of automorphic forms for unitary groups and conjectural generalizations of the Gross--Zagier formula. We prove a combinatorial formula expressing the Galois action in terms of the distance function on the Bruhat--Tits buildings for these groups. In addition, we calculate explicitly the Hecke polynomial appearing in the congruence relation conjectured by Blasius and Rogawski. Usi…
We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring $\mathcal{R}$ classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group $π_0 \mathcal{R}$ recovers Mazur's deformation ring. We give evidence that these rings $\mathcal{R}$ occur in the wild: For suitable Galois representations, the Langlands program predicts that $π_0 \mathcal{R}$ should act on the homology of an arithmetic group. We explain how the Taylor--Wiles method can be used to upgrade such an action to a graded action of $π_* \mathcal{R…
We examine which representations of the absolute Galois group of a field of finite characteristic with image over a finite field of the same characteristic may be constructed by the Galois group's action on the division points of an appropriate Drinfeld module.
In this paper we characterise the action of the absolute Galois group on the geometric finite cyclic groups without étale factorization of stack inertia of the profinite geometric fundamental group of moduli spaces of marked curves. As a complementary result, we give the same action on prime order profinite elements in genus 2.
We show that the simple group PSL_2(F_p) occurs as the Galois group of an extension of the rationals for all primes p>3. We obtain our Galois extensions by studying the Galois action on the second etale cohomology groups of a specific elliptic surface.
This is a revision of the paper that was previously entitled "Weighted Completion of Galois Groups and Some Conjectures of Deligne". Fix a prime number $ł$. We prove a conjecture stated by Ihara, which he attributes to Deligne, about the action of the absolute Galois group on the pro-$ł$ completion of the fundamental group of the thrice punctured projective line. Similar techniques are also used to prove part of a conjecture of Goncharov, also about the action of the absolute Galois group on the fundamental group of the thrice punctured projective line. The main technical tool is the weighted …
We construct explicit examples of E_8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E_8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E_8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 over number fields. The geometric Picard group, considered as a lattice via the negative of the intersection pairing, contains a sublattice isomorphic to E_8. We construct examples of such surfaces for w…
This paper establishes restrictions on the possible Galois actions on the pro-l-unipotent fundamental group of a smooth variety X of good reduction over a local field K. In particular, if X is proper and l is not equal to the residue characteristic p of K, then this action is uniquely determined by the action on cohomology. A similar result for l=p is then proved via a comparison theorem between the crystalline and pro-p homotopy types.
We establish the faithfulness of a geometric action of the absolute Galois group of the rationals that can be defined on the discriminantal variety associated to a finite complex reflection group, and review some possible connections with the profinite Grothendieck-Teichmüller group.
We expand on Nekovář's construction of the plectic half transfer to define a plectic Galois action on Hilbert modular varieties. More precisely, we study in a unifying fashion Shimura varieties associated to groups that differ only in the centre from $R_{F/\mathbb{Q}}{\rm GL}_2$. We define plectic Galois actions on the CM points and on the set of connected components of these Shimura varieties, and show that these two actions are compatible. This extends the plectic conjecture of Nekovář--Scholl.
For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.
We are studing Galois actions on fundamental groups. Using towers of coverings we construct measures on the products of finite copies of Z_p. Using these measures we can calculate coefficients of Galois representations. In the simplest case of measures on Z_p, we get Kubota_Leopoldt p-adic L-functions and p-adic Hurwitz zeta functions.
The object of this paper is the study of a class of dessins d'enfants, the so-called diameter four trees. These objects, first introduced by G. Shabat, can be considered as the simplest non trivial example of etale covers of the projective line minus three points. Their arithmetic properties are still mysterious, and their study can inspire the understanding of more general situations. Here, the main interest is devoted to the action of the absolute Galois group on these (isomorphism classes of) coverings. In particular, in many cases, we are able to distinguish Galois orbits and to describe t…
This paper studies the Galois action on a special lattice of geometric origin, which is related to mod-$\ell$ abelian-by-central quotients of geometric fundamental groups of varieties. As a consequence, we formulate and prove the mod-$\ell$ abelian-by-central variant/strengthening of a conjecture due to Ihara/Oda-Matsumoto.
In his paper titled "Torsion points on Fermat Jacobians, roots of circular units and relative singular homology", Anderson determines the homology of the degree $n$ Fermat curve as a Galois module for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_n)}$. In particular, when $n$ is an odd prime $p$, he shows that the action of $G_{\mathbb{Q}(ζ_p)}$ on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial $1-(1-x^p)^p$. If $p$ satisfies Vandiver's conjecture, we prove that the Galois group of this splitting field over $\mathbb{…
Universality of the Galois action on the fundamental group of $\mathbb{P}^1\setminus\{0,1,\infty\}$
本文证明数域 Galois 群来自几何的任意半单表示都能实现为三穿孔射影直线基本群 pro-代数完备的正则函数环的子商，揭示了该基本群上 Galois 作用的普适性。
The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure it …
In this note, we consider function fields of higher-dimensional algebraic varieties defined over non-local fields, and show how the Galois action on the cohomology such function fields can be used to parameterize their divisorial valuations. By applying a recent theorem of Pop, we observe that, in dimension $\geq 3$, this information is enough to completely determine the function field and the base-field in question.
We study a symplectic variant of algebraic $K$-theory of the integers, which comes equipped with a canonical action of the absolute Galois group of $\mathbf{Q}$. We compute this action explicitly. The representations we see are extensions of Tate twists $\mathbf{Z}_p(2k-1)$ by a trivial representation, and we characterize them by a universal property among such extensions. The key tool in the proof is the theory of complex multiplication for abelian varieties.
We study the absolute Galois group by looking for invariants and orbits of its faithful action on Grothendieck's dessins d'enfants. We define a class of functions called Belyi-extending maps, which we use to construct new Galois invariants of dessins from previously known invariants. Belyi-extending maps are the source of the ``new-type'' relations on the injection of the absolute Galois group into the Grothendieck-Teichmuller group. We make explicit how to get from a general Belyi-extending map to formula for its associated invariant which can be implemented in a computer algebra package. We …
Galois Action on Homology of the Heisenberg Curve
本文研究海森堡曲线的基本群与同调结构，构造了阿廷辫子群在其上的作用，并用特征0域上海森堡群的不可约表示描述其同调。
We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.
In this paper, we study the actions of profinite groups on Cantor sets which arise from representations of Galois groups of certain fields of rational functions. Such representations are associated to polynomials, and they are called profinite iterated monodromy groups. We are interested in a topological invariant of such actions called the asymptotic discriminant. In particular, we give a complete classification by whether the asymptotic discriminant is stable or wild in the case when the polynomial generating the representation is quadratic. We also study different ways in which a wild asymp…
The Galois Action on Consistent Maps
本文研究了 consistent maps 上的自然 Galois 作用，确立了其不变性条件，推广了作者先前关于非 Archimedes 位上有理取值 consistent maps 的工作。
Let $X$ be a fine and saturated log scheme, and let $G$ be a commutative finite flat group scheme over the underlying scheme of $X$. If $G$-torsors for the fppf topology can be thought of as being unramified objects by nature, then $G$-torsors for the log flat topology allow us to consider tame ramification. Using the results of Kato, we define a concept of Galois structure for these torsors, then we generalize the author's previous constructions (class-invariant homomorphism for semi-stable abelian varieties) in this new setting, thus dropping some restrictions.
An important piece of information in the theory of the arithmetic Galois action on the geometric fundamental groups of schemes is that divisorial inertia is acted on cyclotomically. We detail in this note the content of this fact in the case of the profinite braid groups arising from complex reflection groups, naturally viewing them as the geometric fundamental groups of the attending classifying spaces. We also include the case of the full (non colored) braid groups, whose completed classifying spaces are Deligne-Mumford stacks rather than schemes.
We study the Galois actions on the l-adic schematic and Artin-Mazur homotopy groups of algebraic varieties. For proper varieties of good reduction over a local field K, we show that the l-adic schematic homotopy groups are mixed representations explicitly determined by the Galois action on cohomology of Weil sheaves, whenever l is not equal to the residue characteristic p of K. For quasi-projective varieties of good reduction, there is a similar characterisation involving the Gysin spectral sequence. When l=p, a slightly weaker result is proved by comparing the crystalline and p-adic schematic…
Let $K$ be a field, let $G$ be a finite group, and let $\bar X\rightarrow \bar Y$ be a $G$-Galois branched cover of varieties over $K^{sep}$. Given a mere cover model $X\rightarrow Y$ of this cover over $K$, in Part I of this paper I observe that there is a unique minimal field $E$ over which $X\rightarrow Y$ becomes Galois, and I prove that $E/K$ is Galois with group a subgroup of $Aut(G)$. In Part II of this paper, by making the additional assumption that $K$ is a field of definition (i.e., that there exists \it some \rm Galois model over $K$), I am able to give an explicit description of th…
We prove two "large images" results for the Galois representations attached to a degree $d$ Q-curve $E$ over a quadratic field $K$: if $K$ is arbitrary, we prove maximality of the image for every prime $p >13$ not dividing $d$, provided that $d$ is divisible by $q$ (but $d \neq q$) with $q=2$ or 3 or 5 or 7 or 13. If $K$ is real we prove maximality of the image for every odd prime $p$ not dividing $d D$, where $D = \disc(K)$, provided that $E$ is a semistable Q-curve. In both cases we make the (standard) assumptions that $E$ does not have potentially good reduction at all primes $p \nmid 6$ an…
We discuss an action of the Grothendieck-Teichmüller proalgebraic group on the linear span of proalgebraic tangles, oriented tangles completed by a filtration of Vassiliev. The action yields a motivic structure on tangles. We derive distinguished properties of the action particularly on proalgebraic string links and on proalgebraic knots which can not be observed in the action on proalgebraic braids. By exploiting the properties, we explicitly calculate the inverse image of the trivial (the chordless) chord diagram under the Kontsevich isomorphism.
Let $ L/K $ be a finite separable extension of local or global fields in any characteristic, let $ H_{1}, H_{2} $ be two Hopf algebras giving Hopf-Galois structures on the extension, and suppose that the actions of $ H_{1}, H_{2} $ on $ L $ commute. We show that a fractional ideal $ {\mathfrak B} $ of $ L $ is free over its associated order in $ H_{1} $ if and only if it is free over its associated order in $ H_{2} $. We also study which properties these associated orders share.
Galois Action and Localization in Number Fields
本文利用伽罗瓦群在类群上的自然作用，通过研究整数环局部化的类群，从新视角阐明并扩展了已知结果，最后讨论了与作用相关的范数集算术与复杂性。
Inspired by recent work of Farb, Kisin and Wolfson, we develop a method for using actions of finite group schemes over a mixed characteristic dvr R to get lower bounds for the essential dimension of a cover of a variety over K = Frac(R). We then apply this to prove p-incompressibility for congruence covers of a class of unitary Shimura varieties for primes p at which the reduction of the Shimura variety (at any prime of the reflex field over p) does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the p-incompressibility of the multiplication by p map…
Suppose that C is a smooth, projective, geometrically connected curve of genus g > 2 defined over a number field K. Suppose that x is a K-rational point of C. Denote the Lie algebra of the unipotent completion (over Q_ell) of the fundamental group of the corresponding complex analytic curve by p(C,x). This is acted on by both G_K (the Galois group of K) and Gamma, the mapping class group of the pointed complex curve. In this paper we show that the algebraic cycle C_x-C_x^- in the jacobian of C controls the size of the image of G_K in Aut p(C,x). More precisely, we give necessary and sufficient…
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
In this article, we prove that if $H$ is a skew field of center $k$ and $σ$ an automorphism of finite order of $H$ such that the fixed subfield $k^{\langle σ\rangle}$ of $k$ under the action of $σ$ contains an ample field, then the inverse Galois problem has a positive answer over the skew field $H(t,σ)$ of twisted rational fractions. Moreover, if $k^{\langle σ\rangle}$ contains either a real closed field, or an Henselian field of residue characteristic $0$ and containing all roots of unity, then the profree group of countable rank $\widehat{F}_ω$ is a Galois group over $H(t,σ)$.
Uniform irreducibility of Galois action on the $\ell$-primary part of Abelian $3$-folds of Picard type
本文研究了皮卡型阿贝尔3折的ℓ-幂部分上格罗森德克尔作用的均匀不可约性，并将其推广到具有虚二次域乘法的二维阿贝尔3折族。
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