Analytic Curves in Algebraic Varieties over Number Fields

@article{Bost2009AnalyticCI,
  title={Analytic Curves in Algebraic Varieties over Number Fields},
  author={Jean-Beno{\^i}t Bost and Antoine Chambert-Loir},
  journal={arXiv: Number Theory},
  year={2009},
  pages={69-124}
}
We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions on algebraic curves that extends the classical rationality theorems of Borel–Dwork and Polya–Bertrandias, valid over the projective line, to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and… 
S ep 2 00 7 Analytic curves in algebraic varieties over number fields
We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions on algebraic
Algebraic points on meromorphic curves
The classic Schneider-Lang theorem in transcendence theory asserts that there are only finitely many points at which algebraically independent complex meromorphic functions of finite order of growth
Algebraization, Transcendence, and D-Group Schemes
  • J. Bost
  • Computer Science, Mathematics
    Notre Dame J. Formal Log.
  • 2013
TLDR
The Grothendieck Period Conjecture for cycles of codimension 1 in abelian varieties over $\bar{\mathbb Q}}$ is derived from a classical transcendence theorem \`a la Schneider-Lang.
The Unbounded Denominators Conjecture
We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL2(Z). Our result includes also Mason’s generalization of the original
Algebraicity of formal varieties and positivity of vector bundles
We propose a positivity condition for vector bundles on a projective variety and prove an algebraicity criterion for formal schemes. Then we apply the algebraicity criterion to the study of formal
Differential forms on arithmetic jet spaces
We study derivations and differential forms on the arithmetic jet spaces of smooth schemes, relative to several primes. As applications, we give a new interpretation of arithmetic Laplacians, and we
Simply connected varieties in characteristic $p>0$
We show that there are no non-trivial stratified bundles over a smooth simply connected quasi-projective variety over an algebraic closure of a finite field if the variety admits a normal projective
Algebraic solutions of differential equations over ℙ1 −{0,1,∞}
  • Yunqing Tang
  • Mathematics
    International Journal of Number Theory
  • 2018
The Grothendieck–Katz [Formula: see text]-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo [Formula: see text] has vanishing [Formula: see
Numerical equivalence of $\mathbb R$-divisors and Shioda-Tate formula for arithmetic varieties
Let $X$ be an arithmetic variety over the ring of integers of a number field $K$, with smooth generic fiber $X_K$. We give a formula that relates the dimension of the first Arakelov-Chow vector space

References

SHOWING 1-10 OF 69 REFERENCES
Evaluation maps, slopes, and algebraicity criteria
We discuss criteria for the algebraicity of a formal subscheme V� in the completionX�P at some rational point P of an algebraic variety X over some field K. In particular we consider the case where K
Algebraic leaves of algebraic foliations over number fields
Summary — We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C, a smooth algebraic variety X over K,
Heights of projective varieties and positive Green forms
Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil,
Formal subgroups of abelian varieties
Abstract.In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Néron model of A over the ring of integers Ok of k.
Formal groups and the isogeny theorem
In this paper, we prove an isogeny criterion for abelian varieties that involves conditions on the formal groups of the varieties (see Theorem 1.1). In the particular case of abelian varieties overQ
On the Rationality of the Zeta Function of an Algebraic Variety
Let p be a prime number, a2 the completion of the algebraic closure of the field of rational p-adic numbers and let A be the residue class field of Q. The field A is the algebraic closure of its
Padé approximations and diophantine geometry.
  • D. Chudnovsky, G. Chudnovsky
  • Mathematics, Medicine
    Proceedings of the National Academy of Sciences of the United States of America
  • 1985
Using methods of Padé approximations we prove a converse to Eisenstein's theorem on the boundedness of denominators of coefficients in the expansion of an algebraic function, for classes of
Ample subvarieties of algebraic varieties
Ample divisors.- Affine open subsets.- Generalization to higher codimensions.- The grothendieck-lefschetz theorems.- Formal-rational functions along a subvariety.- Algebraic geometry and analytic
Projective Geometry and Formal Geometry
I Special Chapters of Projective Geometry.- 1 Extensions of Projective Varieties.- 2 Proof of Theorem 1.3.- 3 Counterexamples and Further Consequences.- A Counterexample in Characteristic 2.- A
Cohomological Dimension of Algebraic Varieties
Let X be a scheme of finite type over a field k. The cohomological dimension of X is the smallest integer n > 0 such that H'(X, F) = 0 for all i > n, and for all quasi-coherent sheaves F on X. There
...
1
2
3
4
5
...