Plectic Galois action on CM points and connected components of Hilbert modular varieties
Marius Leonhardt
备注：
This is the version to appear in Bull. Lond. Math. Soc., https://doi.org/10.1112/blms.12692. The start of Section 2.3 has been rewritten to expand on the moduli space interpretation of Hilbert modular varieties. Comments welcome!
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.
We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovar and Scholl. This is achieved with the help of Morel's work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.
Looking for a geometric framework to study plectic Heegner points, we define a collection of abelian varieties - called plectic Jacobians - using the middle degree cohomology of quaternionic Shimura varieties (QSVs). The construction is inspired by the definition of Griffiths' intermediate Jacobians and rests on Nekovar-Scholl's notion of plectic Hodge structures. Moreover, we construct exotic Abel-Jacobi maps sending certain zero-cycles on QSVs to plectic Jacobians.
We describe extension classes arising in the $\ell$-adic and Hodge cohomology of Hilbert modular varieties, generalising results of Caspar to arbitrary dimensions. We show that this description is consistent with the "plectic conjectures" of Nekovář and the second author.
We study Galois representations attached to nonsimple abelian varieties over finitely generated fields of arbitrary characteristic. We give sufficient conditions for such representations to decompose as a product, and apply them to prove arithmetical analogues of results shown by Moonen and Zarhin in the context of complex abelian varieties (of dimension at most 5).
We prove in this paper the Ax-Lindemann-Weierstrass theorem for all mixed Shimura varieties and discuss the lower bounds for Galois orbits of special points of mixed Shimura varieties. In particular we reprove a result of Silverberg in a different approach. Then combining these results we prove the André-Oort conjecture for any mixed Shimura variety whose pure part is a subvariety of A_6^n.
We study toric varieties over a field k that split in a Galois extension K/k using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k-forms of projective spaces when K/k is cyclic, and we also study k-forms of surfaces.
Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order.
We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.
Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any discrete place of K. The Mumford-Tate group is an object of analytical nature whereas having good reduction is an arithmetical notion, linked to the ramification of Galois representations. This conjecture has been proved by Morita for particular abelian varieties with many endomorphisms (called of PEL type). Noot obtained results for abelian varieties without non…
Let $S$ be a Shimura variety with reflex field $E$. We prove that the action of $\operatorname{Gal}(\overline{\mathbb{Q}}/E)$ on $S$ maps special points to special points and special subvarieties to special subvarieties. Furthermore, the Galois conjugates of a special point all have the same complexity (as defined in the theory of unlikely intersections). These results follow from Milne and Shih's construction of canonical models of Shimura varieties, based on a conjecture of Langlands which was proved by Borovoi and Milne.
In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.
We consider the Kisin variety associated to a $n$-dimensional absolutely irreducible mod $p$ Galois representation $\barρ$ of a $p$-adic field $K$ and a cocharacter $μ$. Kisin conjectured that the Kisin variety is connected in this case. We show that Kisin's conjecture holds if $K$ is totally ramfied with $n=3$ or $μ$ is of a very particular form. As an application, we also get a connectedness result for the deformation ring associated to $\barρ$ of given Hodge-Tate weights. We also give counterexamples to show Kisin's conjecture does not hold in general.
Let $k$ be a totally real field, and let $A/k$ be an absolutely irreducible, polarized Abelian variety of odd, prime dimension whose endomorphisms are all defined over $k$. Then the only strictly compatible families of abstract, absolutely irreducible representations of $\gal(\ov{k}/k)$ coming from $A$ are tensor products of Tate twists of symmetric powers of two-dimensional $λ$-adic representations plus field automorphisms. The main ingredients of the proofs are the work of Borel and Tits on the `abstract' homomorphisms of almost simple algebraic groups, plus the work of Shimura on the fields…
Galois groups of low dimensional abelian varieties over finite fields
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Inria and Laboratoire d’Informatique de l’École polytechnique, CNRS, Institut Polytechnique de Paris（法国国家信息与自动化研究所、巴黎综合理工学院计算机实验室、法国国家科学研究中心、巴黎理工学院）；Harvard University（哈佛大学）
Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…
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Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell$, the $\ell^n$-torsion points of $A$ give rise to a representation $ρ_{A, \ell^n} : \gal(\bar{K} / K) \to \gl_{2g}(\zz/\ell^n\zz)$. In particular, we get a mod-$\ell$ representation $ρ_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\ff_\ell)$and an $\ell$-adic representation $ρ_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\zz_\ell)$. In this paper, we describe the possible determinants of subrepresentations (or more generally, subquotients) of these two representation for $\ell$ a prime number, as $A$ varies ove…
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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…
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…
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In a recent paper, Moshe Jarden proposed a conjecture, later named the Kuykian conjecture, which states that if A is an abelian variety defined over a Hilbertian field K, then every intermediate field of K(A_{tor})/K is Hilbertian. We prove that the conjecture holds for Galois extensions of K in K(A_{tor}).
We discuss Galois properties of points of prime order on an abelian variety that imply the simplicity of its endomorphism algebra. Applications to hyperelliptic jacobians are given. In particular, we improve some of our earlier results.
Let $\{ρ_\ell\}_\ell$ be the system of $\ell$-adic representations arising from the $i$th $\ell$-adic cohomology of a complete smooth variety $X$ defined over a number field $K$. Let $Γ_\ell$ and $\mathbf{G}_\ell$ be respectively the image and the algebraic monodromy group of $ρ_\ell$. We prove that the reductive quotient of $\mathbf{G}_\ell^\circ$ is unramified over every degree 12 totally ramified extension of $\mathbb{Q}_\ell$ for all sufficiently large $\ell$. We give a necessary and sufficient condition $(\ast)$ on $\{ρ_\ell\}_\ell$ such that for all sufficiently large $\ell$, the subgrou…
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Given a natural number n and a number field K, we show the existence of an integer \ell_0 such that for any prime number \ell\geq \ell_0, there exists a finite extension F/K, unramified in all places above \ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting \ell-torsion representation ρ_{A,\ell} from G_F to GSp(A[\ell](\bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(\mathbb{F}_\ell) as the Galois group of a finite tame extension of number fields F'/F such that F is unramified above \ell.
The paper is concerned with the following version of Hilbert's irreducibility theorem: if $π: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and sufficiently divisible $n$ there exist a degree $n$ closed point $y \in |Y|$ and $x \in π^{-1}(y)$ for which $k(x)/k(y)$ is a Galois $H$-extension, and $k(y)/k$ is an $S_n$-extension. The result has interesting corollaries when applied to moduli spaces of various kinds. For instance, for every finite group $G$ there is a constant $N$ such that for all $n>N$ there …
We construct group-theoretical generalizations of the Hasse invariant on strata closures of the stacks $G$-Zip$^μ$. Restricting to zip data of Hodge type, we obtain a group-theoretical Hasse invariant on every Ekedahl-Oort stratum closure of a general Hodge-type Shimura variety. A key tool is the construction of a stack of zip flags $G$-ZipFlag$^μ$, fibered in flag varieties over $G$-Zip$^μ$. It provides a simultaneous generalization of the "classical case" homogeneous complex manifolds studied by Griffiths-Schmid and the "flag space" for Siegel varieties studied by Ekedahl-van der Geer. Four …
