Let $k/\mathbb F_p$ denote a finite field. For any split connected reductive group $G/W(k)$ and certain CM number fields $F$, we deform certain Galois representations $\overlineρ:Gal(\overline F/F) \to G(k)$ to continuous families $X_{\overlineρ}$ of Galois representations $Gal(\overline F/F) \to G(\overline{\mathbb Q_p})$ lifting $\overlineρ$ such that the space of points of $X_{\overlineρ}$ which are geometric (in the sense of the Fontaine-Mazur conjecture) with parallel Hodge-Tate weights has positive codimension in $X_{\overlineρ}$. Thus the set of points in $X_{\overlineρ}$ which could (conjecturally) be associated to automorphic forms is sparse. This generalizes a result of Calegari and Mazur for $F/\mathbb Q$ quadratic imaginary and $G = GL_2$. The sparsity of automorphic points for $F$ a CM field contrasts with the situation when $F$ is a totally real field, where automorphic points are often provably dense.
We construct extensions of the field of rational numbers with the Galois group G_2(F_p) by reducing p-adic representations attached to automorphic representations.
A strategy to address the inverse Galois problem over Q consists of exploiting the knowledge of Galois representations attached to certain automorphic forms. More precisely, if such forms are carefully chosen, they provide compatible systems of Galois representations satisfying some desired properties, e.g. properties that reflect on the image of the members of the system. In this article we survey some results obtained using this strategy.
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $ρ$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $ρ$. This generalizes previous work of the author and James Newton.
In this paper we generalize the work of Harris-Soudry-Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on GL_2 over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semisimplification.
We show that a sufficient condition for an irreducible automorphic Galois representation $ρ: G_F\to\mathrm{GL}_2({\overline{\bf F}_p})$ of a totally real field $F$ to have an automorphic crystalline lift is that for each place $v$ of $F$ above $p$ the restriction $\mathrm{det}ρ|_{I_v}$ is a fixed power of the mod $p$ cyclotomic character. Moreover, we show that the only obstruction to controlling the level and character of such automorphic lifts arises for badly dihedral representations.
Many results are known regarding how much local information is required to determine a global object, such as a modular form, or a Galois or automorphic representation. We survey some things that are known and expected, and then explain recent joint work with Dinakar Ramakrishnan about comparing degree 2 Artin and automorphic representations which a priori may not correspond at certain infinite sets of places.
We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lo…
This article is the third and last part of a series of three articles about compatible systems of symplectic Galois representations and applications to the inverse Galois problem. This part proves the following new result for the inverse Galois problem for symplectic groups. For any even positive integer n and any positive integer d, PSp_n(F_{l^d}) or PGSp_n(F_{l^d}) occurs as a Galois group over the rational numbers for a positive density set of primes l. The result is obtained by showing the existence of a regular, algebraic, self-dual, cuspidal automorphic representation of GL_n(A_Q) with l…
By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for almost every prime. Moreover, by using the notion of (n,p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes.
In this paper we study the images of certain families $\{ρ_{π,\ell} \}_\ell$ of $G_2$-valued Galois representations of $\mbox{Gal}(\overline{F}/F)$ associated to $L$-algebraic regular, self-dual, cuspidal automorphic representations $π$ of $\mbox{GL}_7(\mathbb{A}_F)$, where $F$ is a totally real field. In particular, we prove that, under certain automorphic conditions, the images of the residual representations $\overlineρ_{π,\ell}$ are as large as possible for infinitely many primes $\ell$. Moreover, we apply our result to some examples constructed by Chenevier, Renard and Taïbi.
For a fixed mod $p$ automorphic Galois representation, $p$-adic automorphic Galois representations lifting it determine points in universal deformation space. In the case of modular forms and under some technical conditions, Böckle showed that every component of deformation space contains a smooth modular point, which then implies their Zariski density when coupled with the infinite fern of Gouvêa-Mazur. We generalize Böckle's result to the context of polarized Galois representations for CM fields, and to two dimensional Galois representations for totally real fields. More specifically, under …
The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic $L$-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois equivariant version of the formula. We also give some applications to special values of automorphic representations of $\GL_{n}\times\GL_{1}$. We show that our results are compatible with Deligne's conjecture.
We remove a parity condition from the construction of automorphic Galois representations carried out in the Paris Book Project. We subsequently generalize this construction to the case of `mixed-parity' (but still regular essentially self-dual) automorphic representations over totally real fields, finding associated geometric projective representations. Finally, we optimize some of the Galois lifting results of arxiv:1207.6724 and apply them to the previous mixed-parity setting.
We determine the sign of the polarization of any polarized irreducible factor of a Galois representation attached to a polarized cohomological cuspidal automorphic form of Gl_n of a CM field: it is always +1, as was conjectured by Gross.
We prove the vanishing of the geometric Bloch-Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic zero point induced from the automorphic representation is formally smooth of the correct dimension. We do this by employing the Taylor-Wiles-Kisin patching method together with Kisin's technique of analy…
Let $F$ be a CM field and let $(\overline{r}_{π,λ})_λ$ be the compatible system of residual $\mathcal{G}_n$-valued representations of $\operatorname{Gal}_{F}$ attached to a RACSDC automorphic representation $π$ of $\operatorname{GL}_n(\mathbb{A})$, as studied by Clozel, Harris and Taylor and others. Under mild assumptions, we prove that the fixed-determinant universal deformation rings attached to $\overline{r}_{π,λ}$ are unobstructed for all places $λ$ in a subset of Dirichlet density $1$, continuing the investigations of Mazur, Weston and Gamzon. During the proof, we develop a general framew…
In the 80's Aschbacher classified the maximal subgroups of almost all of the finite almost simple classical groups. Essentially, this classification divide these subgroups into two types. The first of these consist roughly of subgroups that preserve some kind of geometric structure, so they are commonly called subgroups of geometric type. In this paper we will prove the existence of infinitely many compatible systems $\{ ρ_\ell \}_\ell$ of $n$-dimensional Galois representations associated to regular algebraic, essentially self-dual, cuspidal automorphic representations of $\mbox{GL}_n(\mathbb{…
Let $F$ be a CM field with totally real subfield $F^+$ and let $π$ be a $C$-algebraic cuspidal automorphic automorphic representation of $\mathrm{U}(a,b)(\mathbf{A}_{F^+})$ whose archimedean components lie in the (non-degenerate limit of) discrete series. We attach to $π$ a Galois representation $R_π:\mathrm{Gal}(\overline F/ F^+)\to{}^C\mathrm{U}(a,b)(\overline{\mathbf Q}_\ell)$ such that, for any complex conjugation element $c$, $R_π(c)$ is as predicted by the Buzzard--Gee conjecture. As a corollary, we deduce that the Galois representations attached to certain irregular, $C$-algebraic (esse…
Let $m$ be an integer greater than three and $\ell$ be an odd prime. In this paper, we prove that at least one of the following groups: $\mbox{P}Ω^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PSO}^\pm_{2m}(\mathbb{F}_{\ell^s})$, $\mbox{PO}_{2m}^\pm(\mathbb{F}_{\ell^s})$ or $\mbox{PGO}^\pm_{2m}(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for infinitely many integers $s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorp…
Let pi be a regular, algebraic, essentially self-dual cuspidal automorphic representation of GL_n(A_F), where F is a totally real field and n is at most 5. We show that for all primes l, the l-adic Galois representations associated to pi are irreducible, and for all but finitely many primes l, the mod l Galois representations associated to pi are also irreducible. We also show that the Lie algebras of the Zariski closures of the l-adic representations are independent of l.
In this paper we study the compatible family of degree-4 Scholl representations $ρ_{\ell}$ associated with a space $S$ of weight $κ> 2$ noncongruence cusp forms satisfying Quaternion Multiplications over a biquadratic field $K$. It is shown that when either $K$ is totally real or $κ$ is odd, $ρ_\ell$ is automorphic, that is, its associated L-function has the same Euler factors as the L-function of an automorphic form for $GL_4(\mathbb Q)$. Further, it yields a relation between the Fourier coefficients of noncongruence cusp forms in $S$ and those of certain automorphic forms via the three-term …
It is known due to Baker and Montgomery that almost all Fekete polynomials under certain ordering have at least one zero on the interval (0, 1). In terms of the positive-definiteness, Fekete polynomial has no zero on the interval (0, 1) if and only if the corresponding automorphic form is positive-definite. On generalizing their result, we formulate an axiomatic result about sets of automorphic forms satisfying certain averages when suitably ordered, which ensures that almost all forms are not positive-definite within such sets. We then apply the result to various families, including the famil…
In 1994, van Geemen and Top constructed a non-selfdual motive of rank three over $\mathbb{Q}$ conjecturally associated with a cuspidal non-selfdual automorphic representation of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$ of level $Γ_0(128)$. They experimentally confirmed the coincidence of the local $L$-factors at finitely many primes using computer. In this paper, we shall prove the coincidence of the local $L$-factors at every prime. To show this, we use the recent results of Harris-Lan-Taylor-Thorne and Scholze on the construction of Galois representations, and Grenié's results to compare thr…
Let $m$ be an integer such that $m \geq 7$ and $m \equiv 0,1,7 \mod 8$. We construct strictly compatible systems of representations of $Γ_{\mathbb Q} \to \mathrm{Spin}_m(\overline{\mathbb Q}_l) \xrightarrow{\mathrm{spin}} \mathrm{GL}_N(\overline{\mathbb Q}_l)$ that is potentially automorphic and motivic. As an application, we prove instances of the inverse Galois problem for the $\mathbb F_p$--points of the spin groups. For odd $m$, we compare our examples with the work of A. Kret and S. W. Shin, which studies automorphic Galois representations valued in $\mathrm{Spin}_m$.
Let $π$ be a cuspidal automorphic representation of $\mathrm{GSp}_4(\mathbf{A_Q})$, whose archimedean component is a holomorphic discrete series or limit of discrete series representation. If $π$ is not CAP or endoscopic, then we show that its associated $\ell$-adic Galois representations are irreducible and crystalline for $100\%$ of primes $\ell$. If, moreover, $π$ is neither an automorphic induction nor a symmetric cube lift, then we show that, for $100\%$ of primes $\ell$, the image of its mod $\ell$ Galois representation contains $\mathrm{Sp}_4(\mathbf{F}_{\ell})$.
Clausen has constructed a homotopical enrichment of the Artin reciprocity symbol in class field theory. On the Galois side, Selmer K-homology replaces the abelianized Galois group, while on the automorphic side the K-theory of locally compact vector spaces replaces classical idelic objects. We supply proofs for some predictions of Clausen regarding the automorphic side.
We prove a companion forms theorem for ordinary n-dimensional automorphic Galois representations, by use of automorphy lifting theorems developed by the second author, and a technique for deducing companion forms theorems due to the first author. We deduce results about the possible Serre weights of mod l Galois representations corresponding to automorphic representations on unitary groups. We then use functoriality to prove similar results for automorphic representations of GSp4 over totally real fields.
For many finite groups, the Inverse Galois Problem can be approached through modular/automorphic Galois representations. This is a report explaining the basic strategy, ideas and methods behind some recent results. It focusses mostly on the 2-dimensional case and underlines in particular the importance of understanding coefficient fields.
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…
We prove the compatibility of local and global Langlands correspondences for $GL_n$ up to semisimplification for the Galois representations constructed by Harris-Lan-Taylor-Thorne and Scholze. More precisely, let $r_p(π)$ denote an $n$-dimensional $p$-adic representation of the Galois group of a CM field $F$ attached to a regular algebraic cuspidal automorphic representation $π$ of $GL_n(\mathbb{A}_F)$. We show that the restriction of $r_p(π)$ to the decomposition group of a place $v\nmid p$ of $F$ corresponds up to semisimplification to $rec(π_v)$, the image of $π_v$ under the local Langlands…
Vincent Lafforgue has recently made a spectacular breakthrough in the setting of the global Langlands correspondence for global fields of positive characteristic, by constructing the `automorphic--to--Galois' direction of the correspondence for an arbitrary reductive group $G$. We discuss a result that starts with Lafforgue's work and proceeds in the opposite (`Galois--to--automorphic') direction.
We show that an irreducible cuspidal automorphic representation of the group GSp(4,A), which is not CAP and whose infinite component belongs to the discrete series, is weakly equivalent to an irreducible generic automorphic cuspidal representation, whose infinite component belongs to the same archimedean local L-packet.
The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\GL_{2n}/F$ whose multiplicative character is "odd". We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are "many" points correspon…
Let $F$ be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations $Gal(\bar{F}/F) \to PGL_n(C)$ lift to $GL_n(C)$. We take special interest in the interaction of this result with algebraicity (on the automorphic side) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, we study refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison …
We prove the existence of GSpin-valued Galois representations corresponding to cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. This confirms the Buzzard-Gee conjecture on the global Langlands correspondence in new cases. As an application we complete the argument by Gross and Savin to construct a rank seven motive whose Galois group is of type G_2 in the cohomology of Siegel modular varieties of genus three. Under some additional local hypotheses we also show automorp…
We introduce a method for producing congruences between Hecke eigenclasses, possibly torsion, in the coherent cohomology of automorphic vector bundles on certain good reduction Shimura varieties. The congruences are produced using some "generalized Hasse invariants" adapted to the Ekedahl-Oort stratification of the special fiber.
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…
Given a pair of n-dimensional complex Galois representations over Q, we define their matching density to be the density, if it exists, of the set of places at which the traces of Frobenius of the two Galois representations are equal. We will show that the set of matching densities of such pairs of irreducible Galois representations (for all n) is dense in the interval [0, 1]. Under the strong Artin conjecture, this also implies the corresponding statement for cuspidal automorphic representations.
We prove the existence of $\mathrm{GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of $\mathrm{GSO}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb{H}}$, arising from forms of $\mathrm{GSO}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (…
We investigate local-global compatibility for cuspidal automorphic representations $π$ for GL(2) over CM fields that are regular algebraic of weight $0$. We prove that for a Dirichlet density one set of primes $l$ and any $ι: \overline{\mathbf{Q}}_l \cong \mathbf{C}$, the $l$-adic Galois representation attached to $π$ and $ι$ has nontrivial monodromy at any $v \nmid l$ in $F$ at which $π$ is special.
We extend the strong multiplicity one theorem of Jacquet, Piatetski-Shapiro and Shalika. Let $π$ be a unitary, cuspidal, automorphic representation of $GL_n(\A_K)$. Let $S$ be a set of finite places of $K$, such that the sum $\sum_{v\in S}Nv^{-2/(n^2+1)}$ is convergent. Then $π$ is uniquely determined by the collection of the local components $\{π_v\mid v\not\in S, ~v \~\text{finite}\}$ of $π$. Combining this theorem with base change, it is possible to consider sets $S$ of positive density, having appropriate splitting behavior with respect to solvable extensions of $K$, and where $π$ is deter…
We show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the +/--Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this we relate the oddness of the associated polarized Galois representations (in the sense of the Bellaïche-Chenevier sign being +1) to the parity condition for criticality. Under an assumption similar to Vandiver's conjecture this also provides evidence for the Fontaine-Mazur conjecture for polarized Galois representations of any even dimension.
Let $X$ be a smooth, projective, geometrically connected curve over a finite field $\mathbb{F}_q$, and let $G$ be a split semisimple algebraic group over $\mathbb{F}_q$. Its dual group $\hat{G}$ is a split reductive group over $\mathbb{Z}$. Conjecturally, any $l$-adic $\hat{G}$-local system on $X$ (equivalently, any conjugacy class of continuous homomorphisms $π_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$) should be associated to an everywhere unramified automorphic representation of the group $G$. We show that for any homomorphism $π_1(X) \to \hat{G}(\bar{\mathbb{Q}}_l)$ of Zariski dense image, the…
For every finite field F and every positive integer r, there exists a finite extension F' of F such that either SO(2r+1,F') or its simple derived group can be realized as a Galois group over Q. If the characteristic of F is 3 or 5 (mod 8), then we can guarantee that the derived group of SO(2r+1,F') can be realized. Likewise, for every finite field F, there exists a finite extension F' of F such that the finite simple group G_2(F') can be realized a Galois group over Q. The proof uses automorphic forms to construct Galois representations which cut out Galois extensions of the desired type.
We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n = 3 some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n = 4 using automorphic inductions of Hecke characters.
