We extend earlier examples provided by Schoen, Nori and Bloch to show that when a surface has the property that the kernel of its Albanese map is non-zero over the field of complex numbers, this kernel is non-zero over a field of transcendence degree one. This says that the conjecture of Bloch and Beilinson that this kernel is zero for varieties over number fields is precise in the sense that it is not valid for fields of transcendence degree one.
We show how the notion of the transcendence degree of a zero-cycle on a smooth projective variety X is related to the structure of the motive M(X). This can be of particular interest in the context of Bloch's conjecture, especially for Godeaux surfaces, when the surface is given as a finite quotient of a suitable quintic in P^3.
We construct an example of a regular algebra over $\mathbb C$ of dimension $d$ and a rank $r$ projective module over it which is not generated by $d+r-1$ elements. This strengthens an example by Swan over the field of real numbers.
Throughout the paper, an analytic field means a non-archimedean complete real-valued one, and our main objective is to extend to these fields the basic theory of transcendental extensions. One easily introduces a topological analogue of the transcendence degree, but, surprisingly, it turns out that it may behave very badly. For example, a particular case of a theorem of Matignon-Reversat asserts that if $\mathrm{char}(k)>0$ then $\widehat{k((t))^a}$ possesses non-invertible continuous $k$-endomorphisms, and this implies that the topological transcendence degree is not additive in towers. Never…
For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending of the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of…
We present a conjecture in Diophantine geometry concerning the construction of line bundles over smooth projective varieties over $\bar{\mathbb Q}}$. This conjecture, closely related to the Grothendieck Period Conjecture for cycles of codimension 1, is also motivated by classical algebraization results in analytic and formal geometry and in transcendence theory. Its formulation involves the consideration of $D$-group schemes attached to abelian schemes over algebraic curves over $\bar{\mathbb Q}}$. We also derive the Grothendieck Period Conjecture for cycles of codimension 1 in abelian varieti…
It is a classical fact that the elliptic modular functions satisfies an algebraic differential equation of order 3, and none of lower order. We show how this generalizes to Siegel modular functions of arbitrary degree. The key idea is that the partial differential equations they satisfy are governed by Gauss--Manin connections, whose monodromy groups are well-known. Modular theta functions provide a concrete interpretation of our result, and we study their differential properties in detail in the case of degree 2.
The purpose of this paper is to combine classical methods from transcendental number theory with the technique of restriction to real scalars. We develop a conceptual approach relating transcendence properties of algebraic groups to results about the existence of homomorphisms to group varieties over real fields. Our approach gives a new perspective on Mazur's conjecture on the topology of rational points. We shall reformulate and generalize Mazur's problem in the light of transcendence theory and shall derive conclusions in the direction of the conjecture. Next to these new theoretical insigh…
These are notes on de Jong's proof of the period=index theorem over fields of transcendence degree two. They are actually about the simplified proof sketched by de Jong in the last section of his paper. These notes were meant as support for my lectures at the summer school "Central Simple Algebras over Function Fields" at the Universitat Konstanz between August, 26 and September, 1 2007 (other lectures on this subject were given by Philippe Gille, Andrew Kresch, Max Lieblich, Tamas Szamuely and Jan Van Geel). No originality is intended (except perhaps a little in the proof of the Artin splitti…
For $C$ a complete algebraically closed extension of $\mathbb{Q}_p$, we show that a one-dimensional $p$-divisible group $G/ \mathcal{O}_C$ can be defined over a complete discretely valued subfield $L \subset C$ with Hodge-Tate period ratios contained in $L$ if and only if $G$ has CM, if and only if the period ratios generate an extension of $\mathbb{Q}_p$ of degree equal to the height of the connected part of $G$. This is a $p$-adic analog of a classical transcendence result of Schneider which states that for $τ$ in the complex upper half plane, $τ$ and $j(τ)$ are simultaneously algebraic over…
Let $f : X \to S$ be a family of smooth projective algebraic varieties over a smooth connected base $S$, with everything defined over $\overline{\mathbb{Q}}$. Denote by $\mathbb{V} = R^{2i} f_{*} \mathbb{Z}(i)$ the associated integral variation of Hodge structure on the degree $2i$ cohomology. We consider the following question: when can a fibre $\mathbb{V}_{s}$ above an algebraic point $s \in S(\overline{\mathbb{Q}})$ be isomorphic to a transcendental fibre $\mathbb{V}_{s'}$ with $s' \in S(\mathbb{C}) \setminus S(\overline{\mathbb{Q}})$? When $\mathbb{V}$ induces a quasi-finite period map $φ:…
We give effective upper bounds for the number of purely inseparable points on non isotrivial curves over function fields of positive characteristic and of transcendence degree one. These bounds depend on the genus of the curve, the genus of the function field and the number of points of bad reduction of the curve.
We prove a few uniform versions of the Mordell-Lang Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points. The main focus is on function fields having high transcendence degree over the complex numbers.
The purpose of this paper is to construct non-trivial elements in the Abel-Jacobi kernels in any codimension by specializing correspondences with non-trivial Hodge-theoretical invariants at points with different transcendence degrees over a subfield in $\CC$.
Let $G$ be the automorphism group of an extension $F|k$ of algebraically closed fields of characteristic zero and of transcendence degree $n$, $1\le n\le\infty$. In this paper we (i) construct some maximal closed non-open subgroups $G_v$, and some (all, in the case of countable transcendence degree) maximal open proper subgroups of $G$; (ii) describe, in the case of countable transcendence degree, the automorphism subgroups over the intermediate subfields (a question of Krull, \cite[\S4, question 3b)]{krull}); (iii) construct, in the case $n=\infty$, a fully faithful subfunctor $(-)_v$ of the …
Let $k$ be an algebraically closed field of characteristic zero, $F$ be an algebraically closed extension of $k$ of transcendence degree one, and $G$ be the group of automorphisms over $k$ of the field $F$. The purpose of this note is to calculate the group of continuous automorphisms of $G$.
If there is a topologically locally constant family of smooth algebraic varieties together with an admissible normal function on the total space, then the latter is constant on any fiber if this holds on some fiber. Combined with spreading out, it implies for instance that an irreducible component of the zero locus of an admissible normal function is defined over k if it has a k-rational point where k is an algebraically closed subfield of the complex number field with finite transcendence degree. This generalizes a result of F. Charles that was shown in case the normal function is associated …
We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has maximal transcendence degree over the base fields. As an application, we give a criterion for when an analytic branch at infinity in the affine plane that is defined over a number field in a suitable sense is the branch of an algebraic curve.
We prove that every place $P$ of an algebraic function field $F|K$ of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field $FP$ over $K$ is equal to the transcendence degree of $F|K$, and the extension $FP|K$ is separable. We generalize this result to the case where $P$ dominates a regular local Nagata ring $R\subseteq K$ of Krull dimension $\dim R\leq 2$, assuming that the valued field $(K,v_P)$ is defectless, the factor group $v_P F/v_P K$ is torsion-free and the extension of resi…
We consider function fields of transcendence degree at least 2 over algebraic closures of finite fields, and describe a functorial way to recover such function fields form their pro-l Galois theory.
This is a common introduction to math.RT/0101170, math.RT/0306333, math.RT/0506043, math.RT/0601028. Compared to these references there are new results including (i) a description of a separable closure of an extension of transcendence degree one of an algebraically closed field of positive characteristic; (ii) a "Künneth formula" for the products with curves; (iii) the semi-simplicity of the module of regular forms of top degree $Ω^n_{F/k}$.
We show that Colliot-Thélène's conjecture on 0-cycles of degree 1 implies finiteness for the u-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers.
We show that there is no way to define degrees of 0-cycles on Artin stacks with proper good moduli spaces so that (i) the degree of an ordinary point is non-zero, and (ii) degrees are compatible with closed immersions.
Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the stalk cohomology of the Milnor fiber in lowest degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant.
The cohomological dimension of a field is the largest degree with non-vanishing Galois cohomology. Serre's "Conjecture II" predicts that for every perfect field of cohomological dimension $2$, every torsor over the field for a semisimple, simply connected algebraic group is trivial. A field is perfect and "pseudo algebraically closed" (PAC) if every geometrically irreducible curve over the field has a rational point. These have cohomological dimension $1$. Every transcendence degree $1$ extension of such a field has cohomological degree $2$. We prove Serre's "Conjecture II" for such fields of …
We show that given a smooth projective variety X over C with dim(X) > 2, an ample line bundle O(1) on X and an integer n > 1, any n distinct points on a generic hypersurface of degree d in X are linearly independent in CH_0(X) if d >> 0. This generalizes a result of C. Voisin.
For $θ$ a non-algebraic point on a quasi projective variety over a number field, I prove that $θ$ has an approximation by a series of algebraic points of bounded height and degree which is essentially best possible. Applications of this result will include a proof of a slightly strengthened version of the Philippon criterion, some new algebraic independence criteria, statements concerning metric transcendence theory on varieties of arbitrary dimension, and a rather accurate estimate for the number of algebraic points of bounded height and degree on quasi projective varieties over number fields…
Let $K$ be an algebraically closed field of arbitrary characteristic. Let $A$ be an affine domain over $K$ with transcendence degree 1 which is not isomorphic to $K[x]$, and let $B$ be a domain over $K$. We show that the AK invariant distributes over the tensor product of $A$ by $B$. As a consequence, we obtain a generalization of the cancellation theorem of S. Abhyankar, P. Eakin, and W. Heinzer.
Let G be an algebraic group defined over an algebraically closed field k of characteristic zero. We give a simple proof of the following result: if H^1(L, G) = {1} for some finitely generated field extension L/k of transcendence degree \ge 3 then H^1(K, G) = {1} for every field extension K/k.
We study the p-adic valuation of the values of normalised Hecke eigenforms attached to non-isotrivial elliptic curves defined over function fields of transcendence degree one over finite fields of characteristic p. We derive upper bounds on the smallest attained valuation in terms of the minimal discriminant under a certain assumption on the function field and provide examples to show that our estimates are optimal. As an application of our results we also prove the analogue of the degree conjecture unconditionally for strong Weil curves with square-free conductor defined over function fields …
Let $G$ be a finite group and $K$ a number field. We construct a $G$-extension $E/F$, with $F$ of transcendence degree $2$ over $K$, that specializes to all $G$-extensions of $K_\mathfrak{p}$, where $\mathfrak{p}$ runs over all but finitely many primes of $K$. If furthermore $G$ has a generic extension over $K$, we show that the extension $E/F$ has the so-called Hilbert-Grunwald property. These results are compared to the notion of essential dimension of $G$ over $K$, and its arithmetic analogue.
In this paper, the homotopy groups of Chow variety $C_{p,d}(P^n)$ of effective $p$-cycles of degree $d$ is proved to be stable in the sense that $p$ or $n$ increases. We also obtain a negative answer to a question by Lawson and Michelsohn on homotopy groups for the space of degree two cycles.
The standard period-index conjecture for the Brauer group of a field of transcendence degree 2 over a $p$-adic field predicts that the index divides the cube of the period. Using Gabber's theory of prime-to-$\ell$ alterations and the deformation theory of twisted sheaves, we prove that the index divides the fourth power of the period for every Brauer class whose period is prime to $6p$, giving the first uniform period-index bounds over such fields.
We show that the characteristic homomorphism of Colliot-Th{é}l{è}ne and Sansuc from the Chow group of 0-cycles of degree~0 is compatible with restriction and corestriction maps.
In this paper, we develop the main step in the global theory for the mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we show how to reconstruct a function field $K$ of transcendence degree $\geq 5$ over an algebraically closed field, up-to inseparable extensions, from the mod-$\ell$ abelian-by-central Galois group of $K$ endowed with the collection of mod-$\ell$ rational quotients.
We discuss various results and questions around the Grothendieck period conjecture, which is a counterpart, concerning the de Rham-Betti realization of algebraic varieties over number fields, of the classical conjectures of Hodge and Tate. These results give new evidence towards the conjectures of Grothendieck and Kontsevich-Zagier concerning transcendence properties of the torsors of periods of varieties over number fields. We notably establish that the Grothendieck period conjecture holds in degree 1 for products of curves, of abelian varieties, and of K3 surfaces, and that it holds in degre…
Let X be a projective scheme over a finite field. In this paper, we consider the asymptotic behavior of the number of effective cycles on X with bounded degree as it goes to the infinity. By this estimate, we can define a certain kind of zeta functions associated with groups of cycles. We also consider an analogue in Arakelov geometry.
We study the class of 2-dimensional affine k-domains R satisfying ML(R) = k, where k is an arbitrary field of characteristic zero. In particular, we obtain the following result: Let R be a localization of a polynomial ring in finitely many variables over a field of characteristic zero. If ML(R) = K for some field K included in R and such that R has transcendence degree 2 over K, then R is K-isomorphic to K[X,Y,Z]/(XY-P(Z)) for some nonconstant polynomial P(Z) in K[Z].
We prove that the polar degree of an arbitrarily singular projective hypersurface can be decomposed as a sum of non-negative numbers which represent local vanishing cycles of two different types. This yields lower bounds for the polar degree of any singular projective hypersurface.
In this article we consider some questions raised by F. Benoist, E. Bouscaren and A. Pillay. We prove that infinitely $p$-divisible points on abelian varieties defined over function fields of transcendence degree one over a finite field are necessarily torsion points. We also prove that when the endomorphism ring of the abelian variety is $\mZ$ then there are no infinitely $p$-divisible points of order a power of $p$.
An abelian variety over a field K is said to have big monodromy, if the image of the Galois representation on l-torsion points, for almost all primes l contains the full symplectic group. We prove that all abelian varieties over a finitely generated field K with endomorphism ring Z and semistable reduction of toric dimension one at a place of the base field K have big monodromy. We make no assumption on the transcendence degree or on the characteristic of K. This generalizes a recent result of Chris Hall.
For any rank 2 Drinfeld module rho defined over an algebraic function field, we consider its period matrix P, which is analogous to the period matrix of an elliptic curve defined over a number field. Suppose that the characteristic of F_q is odd and rho is without complex multiplication. We show that the transcendence degree of the field generated by the entries of P over F_q(theta) is 4. As a consequence, we show also the algebraic independence of Drinfeld logarithms of algebraic functions which are linearly independent over F_q(theta).
Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which: - The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l di…
We present a dynamical proof of the well-known fact that the Neron-Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field k of transcendence degree 1 over an algebraically closed field K of characteristic 0. More generally, we investigate the mechanism for which the local canonical height for a rational function f defined over k can take irrational values (at points in a local completion of k), providing examples in all degrees greater than 1. Building on Kiwi's classification of non-archimedean Julia sets for quadratic maps…
