We compare two approaches to the study of Galois module structures: on the one hand factor equivalence, a technique that has been used by Fröhlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We also use this comparison to derive a factorisability result on higher $K$-groups of rings of integers, which is a direct analogue of a theorem of de Smit on $S$-units.
We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $ρ$ is determined by an equality of an $m$-power character $g\mapsto Tr(ρ(g^m))$ for some natural number $m$. Using this, we extend Faltings' finiteness criteria to determine the equivalence of two $\ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.
In this paper, we define a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of $\mathbf{Q}_p$, over a Galois extension of $\mathbf{Q}_p$ whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localisation at the primes above p are extensions of the above type. In the example of the restriction to an imaginary quadratic field of the representation atta…
We develop a computational framework for the statistical characterization of Galois characters with finite image, with application to characterizing Galois groups and establishing equivalence of characters of finite images of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.
We study l-independence of monodromy groups G_l of any compatible system of l-adic representations (in the sense of Serre) of number field K assuming semisimplicity. We prove that the formal character of the derived group of the identity component of G_l is independent of l and the (complexified) Lie algebra g_l of G_l satisfies an equal-rank subalgebra equivalence for all l. This equivalence is equivalent to the l-independence of the number of A_n factors for all n belonging to {6,9,10,11,...} and the parity of A_4 factors in g_l.
I give another proof of the geometric Satake equivalence from Mirkovic-Vilonen over a separably closed field. Using Galois descent, I obtain a canonical construction of the Galois form of the full $L$-group.
In this expository paper we present a proof of the equivalence of the standard definition of descent data on schemes with another one mentioned in the literature that involves certain cartesian diagrams. Using this equivalence, we discuss the Galois descent of both schemes and morphisms of schemes.
The classification of the local Galois representations using $(φ,Γ)$-modules by Fontaine has been generalized by Kisin and Ren over the Lubin-Tate extensions of local fields using the theory of $(φ_q,Γ_{LT})$-modules. In this paper, we extend the work of (Fontaine) Herr by introducing a complex which allows us to compute cohomology over the Lubin-Tate extensions and compare it with the Galois cohomology groups. We further extend that complex to include certain non-abelian extensions. We then deduce some relations of this cohomology with those arising from $(ψ_q,Γ_{LT})$-modules. We also comput…
This is a revision of a McMaster University preprint, with extension. In this paper we prove that over local or global fields of characteristic 0, the Corestriction Principle holds for kernel and image of all maps which are connecting maps in group cohomology and the groups of $R$-equivalences. Some related questions over arbitrary fields of characteristic 0 are also discussed. AMS Mathematics Subject Classification (1991): Primary 11E72, Secondary 18G50, 20G10
Let p be an odd prime and let L/k be a Galois extension of number fields whose Galois group is isomorphic to the dihedral group of order 2p. Let S be a finite set of primes of L which is stable under the action of Gal(L/k). The Lichtenbaum conjecture on special values of the Dedekind zeta function at negative integers, together with Brauer formalism for Artin's L-functions, gives a (conjectural) formula relating orders of motivic cohomology groups of rings of S-integers and higher regulators of S-units of the subextensions of L/k. In analogy with the classical case of special values at 0, we g…
In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.
In this paper, we prove the "local epsilon-isomorphism conjecture" of Fukaya and Kato for a particular class of Galois modules obtained by tensoring a Zp-lattice in a crystalline representation of the Galois group of Qp with a representation of an abelian quotient of the Galois group with values in a suitable p-adic local ring. This can be regarded as a local analogue of the Iwasawa main conjecture for abelian p-adic Lie extensions of Qp, extending earlier work of Benois and Berger for the cyclotomic extension. We show that such an epsilon-isomorphism can be constructed using the Perrin-Riou r…
Given an object over the algebraic closure Qbar of Q, there is often no reason for invariants of the corresponding holomorphic object to be preserved by the absolute Galois group Gal(Qbar/Q), and in general this is not true, although it is sometimes surprising to observe in practice. The case of covers of the projective line branched only over the points 0, 1, and infinity, through Belyi's theorem, leads to Grothendieck's dessins d'enfants program for understanding the absolute Galois group through its faithful action on such covers. This note is motivated by Catanese's question about a higher…
This paper is a sequel to our earlier paper "Wach modules and Iwasawa theory for modular forms" (arXiv: 0912.1263), where we defined a family of Coleman maps for a crystalline representation of the Galois group of Qp with nonnegative Hodge-Tate weights. In this paper, we study these Coleman maps using Perrin-Riou's p-adic regulator L_V. Denote by H(Γ) the algebra of Qp-valued distributions on Γ= Gal(Qp(μ(p^\infty) / Qp). Our first result determines the H(Γ)-elementary divisors of the quotient of D_{cris}(V) \otimes H(Γ) by the H(Γ)-submodule generated by (ϕ* N(V))^{ψ= 0}, where N(V) is the Wac…
Let $K$ be a complete discretely valued field with mixed characteristic $(0, p)$ and imperfect residue field $k_K$. Let $Δ$ be a finite set. We construct an equivalence of categories between finite dimensional $\Bbb{F}_p$-representations of the product of $Δ$ copies of the absolute Galois group of $K$ and multivariable \' etale $(φ, Γ)$-modules over a multivariable Laurent series ring over $k_K$.
We show that the Galois cohomology groups of $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ can be computed via the generalization of Herr's complex to multivariable $(φ,Γ)$-modules. Using Tate duality and a pairing for multivariable $(φ,Γ)$-modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ are overconvergent and, moreover, passing to overconvergent multivariable $(φ,Γ)$-modules is an equivalence of cate…
Let $p$ be a prime number. In this article we present a theorem, suggested by Peter Scholze, which states that the absolute Galois group of $\mathbf{Q}_p$ is the étale fundamental group of a certain object $Z$ which is defined over an algebraically closed field. Thus, local Galois representations correspond to local systems on $Z$. In brief, $Z$ is a (non-representable) quotient of a perfectoid space. The construction combines two themes: the fundamental curve of $p$-adic Hodge theory (due to Fargues-Fontaine) and the tilting equivalence (due to Scholze).
A local analogue of the Grothendieck Conjecture is an equivalence of the category of complete discrete valuation fields $K$ with finite residue fields of characteristic $p\ne 0$ and the category of absolute Galois groups of fields $K$ together with their ramification filtrations. The case of characteristic 0 fields $K$ was considered by Mochizuki several years ago. Then the author proved it by different method if $p>2$ (but $\operatorname{char}K=0$ or $p$). This paper represents a modified approach: it covers the case $p=2$, contains considerable technical simplifications and replaces the Galo…
Suppose $K$ is a finite field extension of $\mathbb{Q} _p$ containing a primitive $p$-th root of unity. Let $Γ_{<p}$ be the Galois group of a maximal $p$-extension of $K$ with the Galois group of period $p$ and nilpotent class $<p$. In the paper we describe the ramification filtration $\{Γ_{<p}^{(v)}\}_{v\geqslant 0}$ and relate it to an explicit form of the Demushkin relation for $Γ_{<p}$. The results are given in terms of Lie algebras attached to involved groups by the classical equivalence of the categories of $p$-groups and Lie algebras of nilpotent class $<p$.
The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz $t$-motive, namely, the function field counterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out that these modules are in relation with fundamental objects of function field arithmetic: integral $t$-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denomina…
Let T be a Tannakian category over a field k of characteristic 0 and π(T) its fundamental group. In this paper we prove that there is a bijection between the otimes-equivalence classes of Tannakian subcategories of T and the normal affine group sub-T-schemes of π(T). We apply this result to the Tannakian category T_1(k) generated by motives of niveau \leq 1 defined over k, whose fundamental group is called the motivic Galois group G_mot(T_1(k)) of motives of niveau \leq 1. We find four short exact sequences of affine group sub-T_1(k)-schemes of G_mot(T_1(k)), correlated one to each other by in…
We introduce a notion of inertial equivalence for integral $\ell$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $\ell$-adic representations of the absolute Galois group of a global field lifting a fixed absolutely irreducible residual representation and with given inertial type outside a fixed finite set of places is uniformly bounded independent of the inertial type.
We show that correlators of the hermitian one-Matrix model with a general potential can be mapped to the counting of certain triples of permutations and hence to counting of holomorphic maps from world-sheet to sphere target with three branch points on the target. This allows the use of old matrix model results to derive new explicit formulae for a class of Hurwitz numbers. Holomorphic maps with three branch points are related, by Belyi's theorem, to curves and maps defined over algebraic numbers $\bmQ$. This shows that the string theory dual of the one-matrix model at generic couplings has wo…
We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that this absolute Galois group embeds into the Grothendieck-Teichmüller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_…
The paper deals with a particular type of a projective ring plane defined over the ring of double numbers over Galois fields, R\_{\otimes}(q) \equiv GF(q) \otimes GF(q) \cong GF(q)[x]/(x(x-1)). The plane is endowed with (q^2 + q + 1)^2 points/lines and there are (q + 1)^2 points/lines incident with any line/point. As R\_{\otimes}(q) features two maximal ideals, the neighbour relation is not an equivalence relation, i. e. the sets of neighbour points to two distant points overlap. Given a point of the plane, there are 2q(q+1) neighbour points to it. These form two disjoint, equally-populated fa…
In this paper we study an action of the absolute Galois group $Γ$ on bicolored plane trees induced by the action of $Γ$ on equivalence classes of conservative polynomials which are the simplest example of postcritically finite rational functions. We establish some basic properties of this action and compare it with the similar action provided by the Grothendieck theory of "Dessins d'enfants".
