Motivated by Lang-Vojta's conjectures on hyperbolic varieties, we prove a new version of the Shafarevich conjecture in which we establish the finiteness of pointed families of polarized varieties. We then give an arithmetic application to the finiteness of integral points on moduli spaces of polarized varieties.
We prove a dynamical Shafarevich theorem on the finiteness of the set of isomorphism classes of rational maps with fixed degeneracies. More precisely, fix an integer d at least 2 and let K be either a number field or the function field of a curve X over a field k, where k is of characteristic zero or p>2d-2 that is either algebraically closed or finite. Let S be a finite set of places of K. We prove the finiteness of the set of isomorphism classes of rational maps over K with a natural kind of good reduction outside of S. We also prove auxiliary results on finiteness of reduced effective divis…
Finiteness of pointed families of symplectic varieties: a geometric Shafarevich conjecture
Université de Strasbourg（斯特拉斯堡大学）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；Saitama University（埼玉大学）；Universität Bielefeld（比勒费尔德大学）
We prove a finiteness theorem for the first flat cohomology group of finite flat group schemes over integral normal proper varieties over finite fields. As a consequence, we can prove the invariance of the finiteness of the Tate-Shafarevich group of Abelian schemes over higher dimensional bases under isogenies and alterations over/of such bases for the $p$-part. Along the way, we generalize previous results on the Tate-Shafarevich group in this situation.
We obtain a lower estimate for the Hilbert series of Jacobi algebras and their completions by providing analogue of the Golog-Shafarevich-Vinberg theorem for potential case. We especially treat non-homogeneous situation. This estimate allows to answer number of questions arising in the work of Wemyss-Donovan-Brown on noncommutative singularities and deformation theory. In particular, we prove that the only case when a potential algebra or its completion could be finite dimensional or of linear growth, is the case of two variables and potential having terms of degree three.
Kolyvagin proved that the Tate-Shafarevich group of an elliptic curve over Q of analytic rank 0 or 1 is finite, and that its algebraic rank is equal to its analytic rank. A program of generalisation of this result to the case of some motives which are quotients of cohomology motives of Shimura varieties is offered. We prove some steps of this program, mainly for quotients of $H^7$ of Siegel sixfolds. For example, we "almost" find analogs of Kolyvagin's trace and reduction relations. Some steps of the present paper are new contribution, because they have no analogs in the case of elliptic curve…
Shafarevich conjecture/problem is about the finiteness of isomorphism classes of a family of varieties defined over a number field with good reduction outside a finite collection of places. For K3 surfaces, such a finiteness result was proved by Y. She. For hyper-Kähler varieties, which are higher-dimensional analogs of K3 surfaces, Y. André has verified the Shafarevich conjecture for hyper-Kähler varieties of a given dimension and admitting a very ample polarization of bounded degree. In this paper, we provide a unification of both results by proving the (unpolarized) Shafarevich conjecture f…
Let $X$ be a smooth projective variety over a finitely generated field $K$ of characteristic $p>0$. We proved that the finiteness of the $\ell$-primary part of $\mathrm{Br}(X_{K^s})^{G_K}$ for a single prime $\ell\neq p$ will imply the finiteness of the prime-to-$p$ part of $\mathrm{Br}(X_{K^s})^{G_K}$, generalizing a theorem of Tate and Lichtenbaum for varieties over finite fields. For an abelian variety $A$ over $K$, we proved a similar result for the Tate-Shafarevich group of $A$, generalizing a theorem of Schneider for abelian varieties over global function fields.
We prove finiteness results for Tate--Shafarevich groups in degree 2 associated with 1--motives, rely them to Leopoldt's conjecture, and present an example of a semiabelian variety with an infinite Tate--Shafarevich group in degree 2. We also establish an arithmetic duality theorem for 1--motives over number fields which complements earlier results of Harari and Szamuely in this direction.
The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. André proved this conjecture for polarized K3 surfaces of fixed degree, and recently She proved it for polarized K3 surfaces of unspecified degree. In this paper, we prove a certain generalization of their results, which is stated by the unramifiedness of l-adic étale cohomology groups for K3 surfaces over finitely generated fields of characteristic 0. As a corollary, we get the original Shafare…
The paper's main result is an effective uniform bound for the finiteness statement of the Shafarevich Conjecture over function fields. Several results on the projective geometry of curves are established in the course of the proof. These results should be of independent interest. As a corollary, a uniform effective bound for the Mordell Conjecture over function fields is derived via Parshin's trick.
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 study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford--Tate conjecture. When applied to K3 surfaces, this affirms a conjecture of …
Let $K \subset \mathbb{C}$ be a number field, and let $\mathcal{O}_{K,N} = \mathcal{O}_{K}[N^{-1}]$ be its ring of $N$-integers. Recently, Lawrence and Venkatesh proposed a general strategy for proving the Shafarevich conjecture for the fibres of a smooth projective family $f : X \to S$ defined over $\mathcal{O}_{K,N}$. To carry out their strategy, one needs to be able to decide whether the algebraic monodromy group $\mathbf{H}_{Z}$ of any positive-dimensional geometrically irreducible subvariety $Z \subset S_{\mathbb{C}}$ is "large enough", in the sense that a certain orbit of $\mathbf{H}_{Z}…
We study arithmetic of the algebraic varieties defined over number fields by applying Lagrange interpolation to fibrations. Assuming the finiteness of the Tate-Shafarevich group of a certain elliptic curve, we show, for Châtelet surface bundles over curves, that the violation of Hasse principle being accounted for by the Brauer-Manin obstruction is not invariant under an arbitrary finite extension of the ground field.
We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties, and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general set-up for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley-Weil theorem for stacks.
We prove an effective version of the Shafarevich conjecture (as proven by Faltings) for smooth quartic curves. To do so, we establish an effective version of Scholl's finiteness result for smooth del Pezzo surfaces of degree at most four.
Zero-cycles on surfaces dominated by products of hyperelliptic curves
CNRS Laboratoire de Mathématiques d’Orsay Université Paris-Saclay（法国国家科学研究中心奥赛数学实验室巴黎萨克雷大学）；CNRS Institut Galilée Université Sorbonne Paris Nord（法国国家科学研究中心加利亚莱研究所索邦巴黎北大学）；Institute for the Information Transmission Problems Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
