We give a classification theorem for certain four-dimensional families of geometric $λ$-adic Galois representations attached to a pure motive. More precisely, we consider families attached to the cohomology of a smooth projective variety defined over $\Q$ with coefficients in a quadratic imaginary field, non-selfdual and with four different Hodge-Tate weights. We prove that the image is as large as possible for almost every $λ$ provided that the family is irreducible and not induced from a family of smaller dimension. If we restrict to semistable families an even simpler classification is given. A version of the main result is given for the case where the family is attached to an automorphic form.
We introduce Galois families of modular forms. They are a new kind of family coming from Galois representations of the absolute Galois groups of rational function fields over the rational field. We exhibit some examples and provide an infinite Galois family of non-liftable weight one Katz modular eigenforms over an algebraic closure of F_p for p in {3,5,7,11}.
We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular Galois extension can be prescribed has been investigated by Dèbes and Ghazi in the unramified case, and by Legrand, Neftin and the author in general. Here, we generalize these results and give a partial solution to Grunwald problems using Galois extensions arising as specializations of a family of regular Galois extensions.
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.
In 2002, M. A. Tsfasman and S. G. Vlăduţ formulated the generalized Brauer-Siegel conjecture for asymptotically exact families of number fields. In this article, we establish this conjecture for asymptotically good towers and asymptotically bad families of number fields with solvable normal closure.
We propose an approach for the computation of multi-parameter families of Galois extensions with prescribed ramification type. More precisely, we combine existing deformation and interpolation techniques with recently developed strong tools for the computation of $3$-point covers. To demonstrate the applicability of our method in relatively large degrees, we compute several families of polynomials with symplectic Galois groups, in particular obtaining the first totally real polynomials with Galois group PSp(6,2).
We study compatible families of four-dimensional Galois representations constructed in the étale cohomology of a smooth projective variety. We prove a theorem asserting that the images will be generically large if certain conditions are satisfied. We only consider representations with coefficients in an imaginary quadratic field. We apply our result to an example constructed by Jasper Scholten, obtaining a family of linear groups and one of unitary groups as Galois groups over $\mathbb{Q}$.
We formulate a notion of purity for $p$-adic big Galois representations and pseudorepresentations of Weil groups of $\ell$-adic number fields for $\ell\neq p$. This is obtained by showing that all powers of the monodromy of any big Galois representation stay "as large as possible" under pure specializations. The role of purity for families in the study of the variation of local Euler factors, local automorphic types along irreducible components, the intersection points of irreducible components of $p$-adic families of automorphic Galois representations is illustrated using the examples of Hida…
Let $p$ be a prime, and let $K$ be a finite extension of $\mathbf{Q}_p$, with absolute Galois group $\cal{G}_K$. Let $π$ be a uniformizer of $K$ and let $K_\infty$ be the Kummer extension obtained by adjoining to $K$ a system of compatible $p^n$-th roots of $π$, for all $n$, and let $L$ be the Galois closure of $K_\infty$. Using these extensions, Caruso has constructed étale $(φ,τ)$-modules, which classify $p$-adic Galois representations of $K$. In this paper, we use locally analytic vectors and theories of families of $φ$-modules over Robba rings to prove the overconvergence of $(φ,τ)$-module…
The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that $\barρ:\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow \operatorname{GL}_2(\bar{\mathbb{F}}_p)$ lifts to a Hida line for which the weights range over a congruence class modulo-$p^2$. The advantage of the purely Galois theoretic approach is that it allows us to construct $p$-adic families of Galois representations lifting the actual representation $\barρ$, and not just the semisimplification.
In a previous article, we have proved a result asserting the existence of a compatible family of Galois representations containing a given crystalline irreducible odd two-dimensional representation. We apply this result to establish new cases of the Fontaine-Mazur conjecture, namely, an irreducible Barsotti-Tate $λ$-adic 2-dimensional Galois representation unramified at 3 and such that the traces $a_p$ of the images of Frobenii verify $\Q(\{a_p^2 \}) = \Q $ always comes from an abelian variety. We also show the non-existence of irreducible Barsotti-Tate 2-dimensional Galois representations of …
We prove the following uniformity principle: if one of the Galois representations in the family attached to a genus two Siegel cusp form of weight $k>3$, "semistable" and with multiplicity one, is reducible (for an odd prime $p$),then all the representations in the family are reducible. This, combined with Serre's conjecture (which is now a theorem) gives a proof of the Endoscopy Conjecture.
We study the Galois groups of polynomials arising from a compatible family of representations with big orthogonal monodromy. We show that the Galois groups are usually as large as possible given the constraints imposed on them by a functional equation and discriminant considerations. As an application, we consider the Frobenius polynomials arising from the middle étale cohomology of hypersurfaces in $\mathbb{P}_{\mathbb{F}_q}^{2n+1}$ of degree at least $3$. We also consider the $L$-functions of quadratic twists of fixed degree of an elliptic curve over a function field $\mathbb{F}_q(t)$. To de…
Let K_{f} be the finite unramified extension of Q_{p} of degree f and E any finite large enough coefficient field containing K_{f}. We construct analytic families of étale (Phi,Gamma)-modules which give rise to families of crystalline E-representations of the absolute Galois group G_{K_{f}} of K_{f}. For any irreducible effective two-dimensional crystalline E-representation of G_{K_{f}} with labeled Hodge-Tate weights {0,-k_{i}}_{τ_{i}} induced from a crystalline character of G_{K_{2f}}, we construct an infinite family of crystalline E-representations of G_{K_{f}} of the same Hodge-Tate type w…
We investigate the relation between p-adic Galois representations and overconvergent (phi,Gamma)-modules in families. Especially we construct a natural open subspace of a family of (phi,Gamma)-modules, over which it is induced by a family of Galois-representations.
Let $p$ be a prime. We produce two new families of pro-$p$ groups which are not realizable as absolute Galois groups of fields. To prove this we use the 1-smoothness property of absolute Galois pro-$p$ groups. Moreover, we show in these families one has one-relator pro-$p$ groups which may not be ruled out as absolute Galois groups employing the quadraticity of Galois cohomology (a consequence of Rost-Voevodsky Theorem), or the vanishing of Massey products in Galois cohomology.
We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with $p$-adic CM forms.
Associated to an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $ρ_A\colon Gal(\bar{K}/K)\to GL_{2g}(\hat{\mathbb{Z}})$. The representation $ρ_A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a nonempty open subvariety $U\subseteq \mathbb{P}^n_K$. We show that away from a set of density $0$, the image of $ρ_A$ will be very large; more precisely, it will have uniformly bounded in…
We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…
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 $p$ be an odd prime number, $K_{f}$ the finite unramified extension of $\mathbb{Q}_{p}$ of degree $f$, and $G_{K_{f}}$ its absolute Galois group. We construct analytic families of étale $(φ,Γ)$-modules which give rise to some families of 2-dimensional crystalline representations of $G_{K_{f}}$ with length of filtration $\geq p$. As an application, we prove that the modulo $p$ reductions of the members of each such family (with respect to appropriately chosen Galois-stable lattices) are constant.
We introduce the notion of finite slope families to encode the local properties of the p-adic families of Galois representations appearing in the work of Harris, Lan, Taylor and Thorne on the construction of Galois representations for (non-self dual) regular algebraic cuspidal automorphic representations of GL(n) over CM fields. Our main result is to prove the analytic continuation of semi-stable (and crystalline) periods for such families.
Let F be a totally real field and G=GSp(4)_{/F}. In this paper, we show under a weak assumption that, given a Hecke eigensystem lambda which is (p,P)-ordinary for a fixed parabolic P in G, there exists a several variable p-adic family underline{lambda} of Hecke eigensystems (all of them (p,P)-nearly ordinary) which contains lambda. The assumption is that lambda is cohomological for a regular coefficient system. If F=Q, the number of variables is three. Moreover, in this case, we construct the three variable p-adic family rho_{underline{lambda}} of Galois representations associated to underline…
We address the problem of the determination of the images of three-dimensional geometric and modular Galois representations. In the modular case the existence of these representations is only conjectural. We give conditions to ensure that the images are as large as possible for almost every prime. We compute some examples and stress the applications to inverse Galois theory.
Motives and automorphic forms of arithmetic type give rise to Galois representations that occur in {\it compatible families}. These compatible families are of p-adic representations with p varying. By reducing such a family mod p one obtains compatible families of mod p representations. While the representations that occur in such a p-adic or mod p family are strongly correlated, in a sense each member of the family reveals a new face of the motive. In recent celebrated work of Wiles playing off a pair of Galois representations in different characteristics has been crucial. In this paper we in…
We construct and study the moduli of continuous representations of a profinite group with integral $p$-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a $p$-adic local field, we show that these moduli spaces admit Zariski-closed loci cutting out Galois representations that are potentially semi-stable with bounded Hodge-Tate weights and a given Hodge and Galois type. As a consequence, we show that these loci descend to the universal de…
We show that the Vojta (or Hall-Lang) conjecture implies that the arboreal Galois representations in a 1-parameter family of quadratic polynomials are surjective if and only if they surject onto some finite and uniform quotient. As an application, we use the Vojta conjecture, our uniformity theorem over $\mathbb{Q}(t)$, and Hilbert's irreducibility theorem to prove that the prime divisors of many quadratic orbits have density zero.
We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral $D_{2n}$ or (generalized) quaternion $\mathbb H_{2n}$ of two-power order. In the spirit of recent work of Fiorilli and Jouve arXiv:2001.05428, we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical aspect, where the degree goes to infinity. Our main contribution uncovers in families of extensions a phenomenon, for which Ng gave numerical evidence : real zeros…
We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the $V_4$ case, each family is a two-dimensional space of orthorhombic lattices and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity (with respect to natural measures). We also show that the shape is a complete invariant in some natural families of $V_4$-quartic fields. For $C_4$-quartic fields, each family is a one-dimensional spa…
For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we study the existence of a lift to characteristic zero that is geometric in the sense of the Fontaine-Mazur conjecture. For two-dimensional representations, Ramakrishna proved that under technical assumptions odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. A key step is generalizing and studying a local deformation condition at $p$ arising from Fontaine-Laffaille theory.
Moments in the Chebotarev density theorem: non-Gaussian families
Université Paris Cité, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche（巴黎西岱大学，索邦大学，法国国家科学研究中心，朱西厄-巴黎左岸数学研究所）；Univ. Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay（巴黎萨克雷大学，法国国家科学研究中心，奥赛数学实验室）；Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251（波尔多大学，法国国家科学研究中心，波尔多国立高等电子与计算机学院，IMB）
We establish the automorphy of some families of 2-dimensional representations of the absolute Galois group of a totally real field, which do not satisfy the so-called `Taylor--Wiles hypothesis'. We apply this to the problem of the modularity of elliptic curves over totally real fields
In this article, we show that for any non-isotrivial family of abelian varieties over a rational base with big monodromy, those members that have adelic Galois representation with image as large as possible form a density-$1$ subset. Our results can be applied to a number of interesting families of abelian varieties, such as rational families dominating the moduli of Jacobians of hyperelliptic curves, trigonal curves, or plane curves. As a consequence, we prove that for any dimension $g \geq 3$, there are infinitely many abelian varieties over $\mathbb Q$ with adelic Galois representation havi…
We study $p$-adic Hodge theory for families of Galois representations over pseudorigid spaces. Such spaces are non-archimedean analytic spaces which may be of mixed characteristic, and which arise naturally in the study of eigenvarieties at the boundary of weight space. We introduce perfect and imperfect overconvergent period rings, and we use the Tate--Sen method to construct overconvergent $(φ, Γ)$-modules for Galois representations over pseudorigid spaces.
We compute the arithmetic L-invariants (of Greenberg-Benois) of twists of symmetric powers of p-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on p-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of GL(2), we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the L-invari…
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
In this article new cases of the Inverse Galois Problem are established. The main result is that for a fixed integer n, there is a positive density set of primes p such that PSL_2(F_{p^n}) occurs as the Galois group of some finite extension of the rational numbers. These groups are obtained as projective images of residual modular Galois representations. Moreover, families of modular forms are constructed such that the images of all their residual Galois representations are as large as a priori possible. Both results essentially use Khare's and Wintenberger's notion of good-dihedral primes. Pa…
Let k be an algebraically closed field of arbitrary characteristic,let K/k be a finitely generated field extension and let X be a separated scheme of finite type over K. For each prime ell, the absolute Galois group of K acts on the ell-adic etale cohomology modules of X. We prove that this family of representations varying over ell is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure of K of the kernels of the representations for all ell become linearly disjoint over a finite extension of K. In doing this, we also prove a number of interesting f…
In this paper we construct families of rotated $D_n$-lattices which may be suitable for signal transmission over both Gaussian and Rayleigh fading channels via subfields of cyclotomic fields. These constructions exhibit full diversity and good minimum product distance which are important parameters related to the signal transmission error probability. It is also shown that for some Galois extensions $\matK|\matQ$ it is impossible to construct rotated $D_n$-lattices via fractional ideals of $\mathcal O_{\mattK}$.
We construct two-parameter analytic families of Galois cohomology classes interpolating the etale Abel--Jacobi images of generalised Heegner cycles, with both the modular form and Grossencharacter varying in p-adic families.
We prove modularity of some two dimensional, 2-adic Galois representations over totally real fields that are nearly ordinary and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the 2-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above 2.
Let $p$ be an odd prime, and let $K/K_0$ be a quadratic extension of number fields. Denote by $K_\pm$ the maximal $\mathbb{Z}_p$-power extensions of $K$ that are Galois over $K_0$, with $K_+$ abelian over $K_0$ and $K_-$ dihedral over $K_0$. In this paper we show that for a Galois representation over $K_0$ satisfying certain hypotheses, if it has odd Selmer rank over $K$ then for one of $K_\pm$ its Selmer rank over $L$ is bounded below by $[L:K]$ for $L$ ranging over the finite subextensions of $K$ in $K_\pm$. Our method or proof generalizes a method of Mazur--Rubin, building upon results of N…
We study the problem of existence of one-parameter, linear families of polynomials of degree n all of whose polynomials have Galois group A_n. The methods we use have a strong geometric flavour.
Let $p\ge 5$ be a prime, and let $f$ be a cuspidal eigenform of weight at least $2$ and level coprime to $p$ of finite slope $α$. Let $\barρ_f$ denote the mod $p$ Galois representation associated with $f$ and $ω$ the mod $p$ cyclotomic character. Under an assumption on the weight of $f$, we prove that there exists a cuspidal eigenform $g$ of weight at least $2$ and level coprime to $p$ of slope $α+1$ such that $$\barρ_f \otimes ω\simeq \barρ_g,$$ up to semisimplification. The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions…
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.
We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N \geq 5$ divisible by a prime number $p \neq 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained as a quotient of its étale cohomology. For any compact $\mathbb{Z}_p[[1 + N \mathbb{Z}_p]]$-algebra $Λ_1$ satisfying certain suitable conditions, we construct a representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over $Λ_1$ associated to a $Λ_1$-adic cuspidal eigenform of finite slope as a scalar extension of a quotient of the étale cohomology.
In previous work we described when a single geometric representation, valued in a linear algebraic group, of the Galois group of a number field lifts through a central torus quotient to a geometric representation. In this paper we prove a much sharper result for systems of l-adic representations, such as the l-adic realizations of a motive, having common "good reduction" properties. Namely, such systems admit geometric lifts with good reduction outside a common finite set of primes. The method yields new proofs of theorems of Tate (the original result on lifting projective representations over…
We study irreducible odd mod $p$ Galois representations $\barρ \colon \mathrm{Gal}(\overline{F}/F) \to G(\overline{\mathbb{F}}_p)$, for $F$ a totally real number field and $G$ a general reductive group. For $p \gg_{G, F} 0$, we show that any $\barρ$ that lifts locally, and at places above $p$ to de Rham and Hodge-Tate regular representations, has a geometric $p$-adic lift. We also prove non-geometric lifting results without any oddness assumption.
