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 note we prove the non existence of certain irreducible two dimensional representations of the the absolute Galois group. Such results are predicted by Serre's conjecture and we use Fontaine's methods to verify these predictions in a small number of cases. We also use a methods of Faltings to study certain finiteness conjectures for Galois representations. Key words: Serre's conjecture, Fontaine-Mazur conjectures, Galois representations.
We prove that a selfdual $GL_3$-Galois representation constructed by van Geemen and Top is isomorphic to a quadratic twist of the symmetric square of the Tate module of an elliptic curve. This is an application of our refinement of the Faltings-Serre method to $3$-dimensional Galois representations with the ground field not equal to $\mathbb{Q}$. The proof makes use of the Faltings-Serre method, $\ell$-adic Lie algebra, and Burnside groups.
Suppose O is a complete discrete valuation ring of positive characteristic with perfect residue field. The category of finite flat strict modules was introduced recently by Faltings and appears as an equal characteristic analogue of the classical category of finite flat group schemes. In this paper we obtain a classification of these modules and apply it to prove analogues of properties, which were known earlier for group schemes.
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.
Motivated by the intermediate Lang conjectures on hyperbolicity and rational points, we prove new finiteness results for non-constant morphisms from a fixed variety to a fixed variety defined over a number field by applying Faltings's finiteness results to moduli spaces of maps.
Given a number field $k$, we show that, for many finite groups $G$, all the Galois extensions of $k$ with Galois group $G$ cannot be obtained by specializing any given finitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions $E/k(T)$ with Galois group $G$ and $E/k$ regular of a certain type, under a conjectural "uniform Faltings" theorem".
A finiteness theorem for mod $p$ Galois representations over global function fields
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Research Institute of Intelligent Complex Systems, Fudan University（复旦大学智能复杂系统研究院）
In this paper, we introduce a condition $\mathrm{(F}_m'\mathrm{)}$ on a field $K$, for a positive integer $m$, that generalizes Serre's condition (F) and which still implies the finiteness of the Galois cohomology of finite Galois modules annihilated by $m$ and algebraic $K$-tori that split over an extension of degree dividing $m$, as well as certain groups of étale and unramified cohomology. Various examples of fields satisfying $\mathrm{(F}_m'\mathrm{)}$, including those that do not satisfy (F), are given.
For a fixed odd prime p and a representation ρof the absolute Galois group of Q into the projective group PGL(2,p), we provide the twisted modular curves whose rational points supply the quadratic Q-curves of degree N prime to p that realize ρthrough the Galois action on their p-torsion modules. The modular curve to twist is either the fiber product of the modular curves X_0(N) and X(p) or a certain quotient of Atkin-Lehner type, depending on the value of N mod p. For our purposes, a special care must be taken in fixing rational models for these modular curves and in studying their automorphis…
We prove the finiteness of Selmer groups attached to lifts of certain 2-dimensional mod p representations of the absolute Galois group of Q. The mod p representation can be either even or odd. The lifts considered are the ones that were proven to exist by the seond named author.
In this paper, we give a necessary and sufficient condition for the finiteness of Galois cohomology of unipotent groups over local fields of positive characteristic
In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and Tamagawa, related with the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion.
Revised: just some typos, reorganized a bit the article. It will be published in the VIASM Annual meeting, Hanoi. We give a detailed account of Deligne's letter to Drinfeld dated June 18, 2011, in which he shows that there are finitely many irreducible lisse $\bar \Q_\ell$-sheaves with bounded ramification, up to isomorphism and up to twist, on a smooth variety defined over a finite field. The proof relies on Lafforgue's Langlands correspondence over curves. In addition, Deligne shows the existence of affine moduli of finite type over $\mathbb{Q}$. A corollary of Deligne's finiteness theorem i…
We show that families of coverings of an algebraic curve where the associated Cayley-Schreier graphs form an expander family exhibit strong forms of geometric (genus and gonality) growth. Combining this general result with finiteness statements for rational points under such conditions, we derive results concerning the variation of Galois representations in one-parameter families of abelian varieties.
Let K/F be a finite Galois extension of global fields with Galois group G and let M be a 1-motive over F. We discuss the kernel and cokernel of the restriction map Sha^{i}(F,M) --> Sha^{i}(K,M)^{G} for i=1 and 2, independently of any finiteness hypotheses. We show that these groups are finite and obtain, in particular, formulas for the corresponding quotient of group orders.
In the recent paper arXiv:1807.02721, B. Lawrence and A. Venkatesh develop a method of proving finiteness theorems in arithmetic geometry by studying the geometry of families over a base variety. Their results include a new proof of both the $S$-unit theorem and Faltings' theorem, obtained by constructing and studying suitable abelian-by-finite families over $\mathbb{P}^1\setminus\{0,1,\infty\}$ and over an arbitrary curve of genus $\geq 2$ respectively. In this paper, we apply this strategy to reprove Siegel's theorem: we construct an abelian-by-finite family on a punctured elliptic curve to …
Let $t$ and $x$ be indeterminates, let $ϕ(x)=x^2+t\in\mathbb Q(t)[x]$, and for every positive integer $n$ let $Φ_n(t,x)$ denote the $n^{\text{th}}$ dynatomic polynomial of $ϕ$. Let $G_n$ be the Galois group of $Φ_n$ over the function field $\mathbb Q(t)$, and for $c\in\mathbb Q$ let $G_{n,c}$ be the Galois group of the specialized polynomial $Φ_n(c,x)$. It follows from Hilbert's irreducibility theorem that for fixed $n$ we have $G_n\cong G_{n,c}$ for every $c$ outside a thin set $E_n\subset\mathbb Q$. By earlier work of Morton (for $n=3$) and the present author (for $n=4$), it is known that $E…
We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.
We prove (under certain assumptions) the irreducibility of the limit $σ_2$ of a sequence of irreducible essentially self-dual Galois representations $σ_k: G_{\mathbf{Q}} \to \mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$ approaches 2 in a $p$-adic sense) which mod $p$ reduce (after semi-simplifying) to $1 \oplus ρ\oplus χ$ with $ρ$ irreducible, two-dimensional of determinant $χ$, where $χ$ is the mod $p$ cyclotomic character. More precisely, we assume that $σ_k$ are crystalline (with a particular choice of weights) and Siegel-ordinary at $p$. Such representations arise in the study of $p$-adi…
A number field $K$ is Hilbert-Speiser if all of its tame abelian extensions $L/K$ admit NIB (normal integral basis). It is known that $\mathbb{Q}$ is the only such field, but when we restrict $\text{Gal}(L/K)$ to be a given group $G$, the classification of $G$-Hilbert-Speiser fields is far from complete. In this paper, we present new results on so-called $G$-Leopoldt fields. In their definition, NIB is replaced by ``weak NIB'' (defined below). Most of our results are negative, in the sense that they strongly limit the class of $G$-Leopoldt fields for some particular groups $G$, sometimes even …
We establish the finiteness of the kernel and cokernel of the restriction map III^{i}(F,M) ---> III^{i}(K,M)^{G} for i=1 and 2, where M is a (Deligne) 1-motive over a global field F and K/F is a finite Galois extension of global fields with Galois group G.
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 w…
Let $X$ be a smooth curve over a finitely generated field $k$, and let $\ell$ be a prime different from the characteristic of $k$. We analyze the dynamics of the Galois action on the deformation rings of mod $\ell$ representations of the geometric fundamental group of $X$. Using this analysis, we prove analogues of the Shafarevich and Fontaine-Mazur finiteness conjectures for function fields over algebraically closed fields in arbitrary characteristic, and a weak variant of the Frey-Mazur conjecture for function fields in characteristic zero. For example, we show that if $X$ is a normal, conne…
This paper is a continuation to \cite{Gazaki2017}. For every integer $n\geq 1$, we consider the generalized Galois symbol $K(k;G_1,G_2)/n\xrightarrow{s_n} H^2(k,G_1[n]\otimes G_2[n])$, where $k$ is a finite extension of $\mathbb{Q}_p$, $G_1,G_2$ are semi-abelian varieties over $k$ and $K(k;G_1,G_2)$ is the Somekawa K-group attached to $G_1, G_2$. Under some mild assumptions, we describe the exact annihilator of the image of $s_n$ under the Tate duality perfect pairing, $H^2(k,G_1[n]\otimes G_2[n])\times H^0(k,Hom(G_1[n]\otimes G_2[n],μ_n))\rightarrow\mathbb{Z}/n$. An important special case is …
Let K be a complete discrete valuation field of mixed characteristic (0,p) and G_K the absolute Galois group of K. In this paper, we will prove the p-adic monodromy theorem for p-adic representations of G_K without any assumption on the residue field of K, for example the finiteness of a p-basis of the residue field of K. The main point of the proof is a construction of (phi,G_K)-module Nrig^+(V) for a de Rham representation V, which is a generalization of Pierre Colmez' Nrig^+(V). In particular, our proof is essentially different from Kazuma Morita's proof in the case when the residue field a…
We prove a finiteness result for the p-adic cohomology of the Lubin-Tate tower. For any n>=1 and p-adic field F, this provides a canonical functor from admissible p-adic representations of GL_n(F) towards admissible p-adic representations of Gal_F x D^*, where Gal_F is the absolute Galois group of F, and D/F is the central division algebra of invariant 1/n. Moreover, we verify a local-global-compatibility statement for this correspondence, and compatibility with the patching construction of Caraiani-Emerton-Gee-Geraghty-Paskunas-Shin.
Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same covolume and pro-congruence completion. We also make effective a f…
